MISC - NOGAMI Yasuyuki

軽量暗号SIMONを用いたCAN通信におけるペイロード暗号化とMACの設計と実装

壷井智也, 小寺雄太, 野上保之, 日下卓也

電子情報通信学会技報   2022.7

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Consideration of an authentication using protocol CVMA parameters

三好茜音, 池坂和真, 小寺雄太, 日下卓也, 野上保之

電子情報通信学会技術研究報告(Web)   122 ( 128(IT2022 15-28) )   2022

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環境センサを用いた室内エアコンの精密制御システムの設計と実装

段 夢芳, 服部大地, 日下卓也, 野上保之

電子情報通信学会技報   2021

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これからの暗号化技術8+α—特集 組み込みコンピュータ技術512 ; 通信技術84+α

野上 保之

インターフェース = Interface   46 ( 2 )   83 - 85   2020.2

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Secure Utilization of IoT and AI Tools for Society 5.0

野上保之

電子情報通信学会技術研究報告(Web)   120 ( 268(IT2020 24-62) )   2020

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Proposal of Optimal Curve Parameters for Pairing Computation on BLS12 Curve and Its Evaluation of Implementation

服部大地, 高橋裕人, 多田羅友也, 南條由紀, 小寺雄太, 日下卓也, 野上保之

電子情報通信学会技術研究報告(Web)   120 ( 224(ISEC2020 32-41) )   2020

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逐次拡大体上のTwisted Montgomery曲線を用いた楕円ElGamal暗号のArduino実装と評価—Implementation and Evaluation of EC-ElGamal Encryption with a Twisted Montgomery Curve over Tower Field on Arduino—集積回路

橋本 優太, 金成 忠樹, 日下 卓也, 野上 保之

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 261 )   1 - 5   2019.11

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Other Link： http://id.ndl.go.jp/bib/030130100

埋め込み次数14のTate及びAteペアリングの実装—An Implementation of Tate and Ate Pairing of Embedding Degree 14

宋 子豪, 松村 陸矢, 南條 由紀, 野上 保之, 日下 卓也

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 260 )   7 - 12   2019.11

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Improvement of Miller's Algorithm of Pairing on Elliptic Curves with Embedding Degree 15 by Using Sparse Multiplication in Affine Coordinates

( 2019 )   1543 - 1550   2019.10

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A Performance Evaluation of Video Encryption by AES Cryptography on Raspberry Pi

2019   812 - 816   2019.10

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A Consideration of Side-Channel Attacks on Curve25519 using Order 4 Rational Points

2019 ( 2019 )   69 - 74   2019.8

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A Performance Analysis of Supersingular Isogeny Diffie-Hellman with Several Classes of the Quadratic Extension Fields (技術と社会・倫理)

NANJO Yuki, SHIRASE Masaaki, KUSAKA Takuya, NOGAMI Yasuyuki

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 141 )   207 - 214   2019.7

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A Performance Analysis of Supersingular Isogeny Diffie-Hellman with Several Classes of the Quadratic Extension Fields (バイオメトリクス)

NANJO Yuki, SHIRASE Masaaki, KUSAKA Takuya, NOGAMI Yasuyuki

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 142 )   207 - 214   2019.7

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A Performance Analysis of Supersingular Isogeny Diffie-Hellman with Several Classes of the Quadratic Extension Fields (ハードウェアセキュリティ)

NANJO Yuki, SHIRASE Masaaki, KUSAKA Takuya, NOGAMI Yasuyuki

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 143 )   207 - 214   2019.7

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A Performance Analysis of Supersingular Isogeny Diffie-Hellman with Several Classes of the Quadratic Extension Fields (情報セキュリティ)

NANJO Yuki, SHIRASE Masaaki, KUSAKA Takuya, NOGAMI Yasuyuki

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   119 ( 140 )   207 - 214   2019.7

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IoT・AI時代における情報セキュリティ研究と人材育成教育

野上 保之

電気学会誌   139 ( 5 )   313 - 313   2019.5

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IoT・AI時代のセキュリティ

現代ICT社会は，インターネットを仲介して様々な情報がクラウドサーバなどに蓄積され，それをAIが解析して自動的に処理を行うという時代に入ってきている。例えば自

DOI： 10.1541/ieejjournal.139.313

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2の冪乗位数をもつEdwards曲線の性質に関する考察—A Consideration of the Properties of Edwards Curve Whose Order is a Power of 2—情報理論

三浦 裕正, 小林 航也, 日下 卓也, 野上 保之

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 477 )   27 - 32   2019.3

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Other Link： http://id.ndl.go.jp/bib/029633293

BLS曲線上での高速化手法を用いたペアリング暗号ライブラリELiPSの実装と評価—An Implementation and Evaluation of Pairing Library ELiPS for BLS Curve with Several Technique

高橋 裕人, 金成 忠樹, 南條 由紀, カンダカル エムディ アルアミン, 日下 卓也, 野上 保之

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 479 )   21 - 26   2019.3

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CAN通信プロトコルに対するメッセージ認証機能の実装と計算遅延時間についての検討—A Consideration of Computational Delay Time by Using MAC Implemented on CAN

小椋 央都, 一色 竜之介, 日下 卓也, 野上 保之, 亀川 哲志, 前山 祥一, 荒木 俊輔

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 479 )   181 - 186   2019.3

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Other Link： http://id.ndl.go.jp/bib/029633004

マイコン実装したAES暗号に対するニューラルネットワークを用いた最終ラウンド鍵の解析—Analyzing Final Round Key of AES Implemented on Microcomputer using Neural Network—情報セキュリティ

小杉 聡志, 城市 翔, 生田 健, 日下 卓也, 野上 保之, 高橋 規一

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 478 )   71 - 76   2019.3

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エコー拡散法とオクターブ類似性を用いた会議録用音声データの改ざん検知の実装に関する研究—A Study on the System of Detecting Falsification for Conference Records Using Echo Spread Method and Octave Similarity—情報理論

大垣 翔矢, 荒木 俊輔, 宮崎 武, 上原 聡, 野上 保之

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 477 )   57 - 64   2019.3

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Analyzing Final Round Key of AES Implemented on Microcomputer using Neural Network

小杉聡志, 城市翔, 生田健, 日下卓也, 野上保之, 高橋規一

電子情報通信学会技術研究報告   118 ( 478(ISEC2018 81-134) )   2019

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Development of Collision-Based Attack Practicum for Security Evaluation by Using Next Generation Cryptosystem in IoT Era.

野上保之, 日下卓也, 小林航也, 橋本優太

コンピュータソフトウェア   36 ( 1 )   95 - 100   2019

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In the IoT era when a device is connected via the Internet, we can enjoy a great deal of convenience with the use of such devices, but a large number of users are not concerned about the attack from a third party. It is also meaningful for the user to have such a consciousness, but the person providing devices needs more specialized knowledge and awareness of the security. For such demands, a special education program called enPiT is held to provide an opportunity for undergraduate students to learn about cutting-edge information technologies and aims to cultivate human resources who have acquired the ability to solve specific problems in society. In this paper, we introduce the contents of \enPiT-Security advanced PBL exercise G" held in Okayama University and describe the helpfulness of the exercise and room for improvement based on the students' comments.

DOI： 10.11309/jssst.36.95

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Investigation of CAN transceiver and controller response to electromagnetic disturbance by using current injection probe: 電流注入プローブを用いた妨害波注入に対するCANトランシーバ・コントローラの挙動調査: 電磁環境 マグネティックス合同研究会 EMC一般(EMC Joint Workshop, 2018, Daejon)

イッシキ リュウノスケ, イオキベ ケンゴ, クサカ タクヤ, カメガワ テツシ, ノガミ ヤスユキ

2018 ( 120 )   23 - 28   2018.11

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Investigation of CAN transceiver and controller response to electromagnetic disturbance by using current injection probe: 電流注入プローブを用いた妨害波注入に対するCANトランシーバ・コントローラの挙動調査: 電磁環境 マグネティックス合同研究会 EMC一般(EMC Joint Workshop, 2018, Daejon)

イッシキ リュウノスケ, イオキベ ケンゴ, クサカ タクヤ, カメガワ テツシ, ノガミ ヤスユキ

2018 ( 48 )   23 - 28   2018.11

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計算効率の良い逐次拡大体の構成条件の下でのBLS曲線の係数決定法—Determining BLS Curves for Pairing over Efficient Tower of Extension Field—情報セキュリティ

南條 由紀, カンダカル エムディ アルアミン, 白勢 政明, 日下 卓也, 野上 保之

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   118 ( 30 )   9 - 16   2018.5

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Uniform Binary Sequence Generated over Odd Characteristic Field

Yuta Kodera, Takuya Kusaka, Ali Md. Arshad, Yasuyuki Nogami, Takeru Miyazak, Satoshi Uehara

International Journal of Information and Electronics Engineering   8 ( 1 )   5 - 9   2018.3

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DOI： 10.18178/ijiee.2018.8.1.685

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Uniform Binary Sequence Generated over Odd Characteristic Field

Yuta Kodera, Takuya Kusaka, Ali Md. Arshad, Yasuyuki Nogami, Takeru Miyazak, Satoshi Uehara

International Journal of Information and Electronics Engineering   8 ( 1 )   5 - 9   2018.3

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DOI： 10.18178/ijiee.2018.8.1.685

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Investigation of Reaction of Error Control of CAN Protocol in Case of Electromagnetic Disturbance Injection: 電磁妨害波注入を想定したCANプロトコルのエラー制御の挙動調査: 環境電磁工学

イッシキ リュウノスケ, クサカ タクヤ, イオキベ ケンゴ, ノガミ ヤスユキ

117 ( 384 )   25 - 30   2018.1

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An Implementation of ECC with Twisted Montgomery Curve over 32nd Degree Tower Field on Arduino Uno.

Yuta Hashimoto, Md. Al-Amin Khandaker, Yuta Kodera, Taehwan Park, Takuya Kusaka, Howon Kim, Yasuyuki Nogami

IJNC   8 ( 2 )   341 - 350   2018

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Secure Data Communication using File Hierarchy Attribute Based Encryption in Wireless Body Area Network

B. Chandrasekaran, R. Balakrishnan, Y. Nogami

Journal of Communications Software and Systems   14 ( 1 )   75 - 81   2018

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Wireless Body Area Networks (WBANs) play an important role in healthcare system by enabling medical experts to guide patients remotely. The unauthorized access of medical data from WBAN controller as well as the unreliable data communication may leads to risk for patients life. Currently, Chunqiang Hu et al., [1] proposed a data communication protocol by using Ciphertext-Policy Attribute-Based Encryption (CP-ABE) for a single file. The major limitation of Chunqiang Hu et al., [1] is that as the number of files increases, CP-ABE will suffer from parameters such as message size, energy consumption and computation cost. This paper proposes a more secure and efficient data communication scheme for WBANs by using an efficient File Hierarchy CP-ABE (FH-CP-ABE). The proposed scheme uses integrated access structure which is a combination of two or more access structures with hierarchical files encrypted. We evaluate the performance analysis of the proposed data communication protocol in terms of message size, energy consumption, computation cost and compared with Chunqiang Hu et al., [1].

DOI： 10.24138/jcomss.v14i1.446

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Highly Efficient GF(28) Inversion Circuit Based on Hybrid GF Arithmetic

Rei Ueno, Naofumi Homma, Yasuyuki Nogami, and Takafumi Aoki

Journal of Cryptographic Engineering   2018

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An Efficient Hierarchical Multi-Authority Attribute Based Encryption Scheme for Profile Matching using a Fast Ate Pairing in Cloud Environment

Balaji Chandrasekaran, Yasuyuki Nogami, and Ramadoss Balakrishnan

International Journal of Information and Electronics Engineering   14 ( 2 )   151 - 156   2018

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—In cloud environment, profile matching is a key technique in applications such as health care and social networks. Ciphertext-Policy Attribute-Based Encryption (CP-ABE) is a suitable technique for data sharing in such environments. In this paper, we propose an asymmetric pairing based Hierarchical Multi-Authority CP-ABE (HM-CP-ABE) construction for profile matching. We utilize the fast Ate pairing to make the proposed HM-CP-ABE scheme efficient. The performance analysis of the proposed scheme shows improved efficiency in terms of computational costs for initialization, key generation and encryption using ELiPS library when compared with existing works.

DOI： 10.24138/jcomss.v14i2.461

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Secure Data Communication using File Hierarchy Attribute Based Encryption in Wireless Body Area Network

B. Chandrasekaran, R. Balakrishnan, and Y. Nogami

Journal of Communications Software and Systems   14 ( 1 )   75 - 81   2018

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Wireless Body Area Networks (WBANs) play an important role in healthcare system by enabling medical experts to guide patients remotely. The unauthorized access of medical data from WBAN controller as well as the unreliable data communication may leads to risk for patients life. Currently, Chunqiang Hu et al., [1] proposed a data communication protocol by using Ciphertext-Policy Attribute-Based Encryption (CP-ABE) for a single file. The major limitation of Chunqiang Hu et al., [1] is that as the number of files increases, CP-ABE will suffer from parameters such as message size, energy consumption and computation cost. This paper proposes a more secure and efficient data communication scheme for WBANs by using an efficient File Hierarchy CP-ABE (FH-CP-ABE). The proposed scheme uses integrated access structure which is a combination of two or more access structures with hierarchical files encrypted. We evaluate the performance analysis of the proposed data communication protocol in terms of message size, energy consumption, computation cost and compared with Chunqiang Hu et al., [1].

DOI： 10.24138/jcomss.v14i1.446

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An Implementation of ECC with Twisted Montgomery Curve over 32nd Degree Tower Field on Arduino Uno.

Yuta Hashimoto, Md. Al-Amin Khandaker, Yuta Kodera, Taehwan Park, Takuya Kusaka, Howon Kim, Yasuyuki Nogami

IJNC   8 ( 2 )   341 - 350   2018

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Highly Efficient GF(28) Inversion Circuit Based on Hybrid GF Arithmetic

Rei Ueno, Naofumi Homma, Yasuyuki Nogami, Takafumi Aoki

Journal of Cryptographic Engineering   2018

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An Efficient Hierarchical Multi-Authority Attribute Based Encryption Scheme for Profile Matching using a Fast Ate Pairing in Cloud Environment

Balaji Chandrasekaran, Yasuyuki Nogami, Ramadoss Balakrishnan

International Journal of Information and Electronics Engineering   14 ( 2 )   151 - 156   2018

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Language：English   Publisher：CROATIAN COMMUNICATIONS & INFORMATION SOC  

In cloud environment, profile matching is a key technique in applications such as health care and social networks. Ciphertext-Policy Attribute-Based Encryption (CP-ABE) is a suitable technique for data sharing in such environments. In this paper, we propose an asymmetric pairing based Hierarchical Multi-Authority CP-ABE (HM-CP-ABE) construction for profile matching. We utilize the fast Ate pairing to make the proposed HM-CP-ABE scheme efficient. The performance analysis of the proposed scheme shows improved efficiency in terms of computational costs for initialization, key generation and encryption using ELiPS library when compared with existing works.

DOI： 10.24138/jcomss.v14i2.461

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CVMAを利用した部分体を用いるNTU系列の生成効率化

松本年史, 小寺雄太, ALI Md. Arshad, 日下卓也, 野上保之

情報理論とその応用シンポジウム予稿集(CD-ROM)   41st   2018

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KSS曲線を用いた効率的なペアリング暗号のための18次拡大体構成法の評価

南條由紀, KHANDAKER Md. Al-Amin, 日下卓也, 野上保之

情報処理学会シンポジウムシリーズ(CD-ROM)   2018 ( 2 )   2018

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線形複雑度が可変なある擬似乱数系列の部分的乱数性に対する評価

武田祐樹, 小寺雄太, 日下卓也, 野上保之

情報理論とその応用シンポジウム予稿集(CD-ROM)   41st   2018

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Interleaved sequences of geometric sequences binarized with legendre symbol of two types

Kazuyoshi Tsuchiya, Yasuyuki Nogami, Satoshi Uehara

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences   E100A ( 12 )   2720 - 2727   2017.12

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A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a binary sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol. They showed the geometric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric sequences of two types and show some properties of interleaved sequences of the geometric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the interleaved structure. Moreover, we show correlation properties and linear complexity of the interleaved sequences. A key of our observation is that the second type geometric sequence is the complement of the left shift of the first type geometric sequence by half-period positions.

DOI： 10.1587/transfun.E100.A.2720

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Improvement of Noise Source Amplitude Modulation Method to Identify Source Devices of Electromagnetic Interference

117 ( 319 )   35 - 40   2017.11

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Efficient Optimal-Ate Pairing on BLS-12 Curve Using Pseudo 8-Sparse Multiplication

2017 ( 2 )   2017.10

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Long period sequences generated by the logistic map over finite fields with control parameter four

Kazuyoshi Tsuchiya, Yasuyuki Nogami

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences   E100A ( 9 )   1816 - 1824   2017.9

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Pseudorandom number generators have been widely used in Monte Carlo methods, communication systems, cryptography and so on. For cryptographic applications, pseudorandom number generators are required to generate sequences which have good statistical properties, long period and unpredictability. ADickson generator is a nonlinear congruential generator whose recurrence function is the Dickson polynomial. Aly and Winterhof obtained a lower bound on the linear complexity profile of a Dickson generator. Moreover Vasiga and Shallit studied the state diagram given by the Dickson polynomial of degree two. However, they do not specify sets of initial values which generate a long period sequence. In this paper, we show conditions for parameters and initial values to generate long period sequences, and asymptotic properties for periods by numerical experiments. We specify sets of initial values which generate a long period sequence. For suitable parameters, every element of this set occurs exactly once as a component of generating sequence in one period. In order to obtain sets of initial values, we consider a logistic generator proposed by Miyazaki, Araki, Uehara and Nogami, which is obtained from a Dickson generator of degree two with a linear transformation. Moreover, we remark on the linear complexity profile of the logistic generator. The sets of initial values are described by values of the Legendre symbol. The main idea is to introduce a structure of a hyperbola to the sets of initial values. Our results ensure that generating sequences of Dickson generator of degree two have long period. As a consequence, the Dickson generator of degree two has some good properties for cryptographic applications.

DOI： 10.1587/transfun.E100.A.1816

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Multi-Valued Sequences Generated by Power Residue Symbols over Odd Characteristic Fields

Begum Nasima, Yasuyuki Nogami, Satoshi Uehara, Robert H. Moleros-Zaragoza

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E100A ( 4 )   922 - 929   2017.4

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This paper proposes a new approach for generating pseudo random multi-valued (including binary-valued) sequences. The approach uses a primitive polynomial over an odd characteristic prime field F-p, where p is an odd prime number. Then, for the maximum length sequence of vectors generated by the primitive polynomial, the trace function is used for mapping these vectors to scalars as elements in the prime field. Power residue symbol (Legendre symbol in binary case) is applied to translate the scalars to k-value scalars, where k is a prime factor of p-1. Finally, a pseudo random k-value sequence is obtained. Some important properties of the resulting multi-valued sequences are shown, such as their period, autocorrelation, and linear complexity together with their proofs and small examples.

DOI： 10.1587/transfun.E100.A.922

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An Improvement of Scalar Multiplication by Skew Frobenius Map with Multi-Scalar Multiplication for KSS Curve.

Md. Al-Amin Khandaker, Yasuyuki Nogami

IEICE Transactions   100-A ( 9 )   1838 - 1845   2017

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Scalar multiplication over higher degree rational point groups is often regarded as the bottleneck for faster pairing based cryp-tography. This paper has presented a skew Frobenius mapping technique in the sub-field isomorphic sextic twisted curve of Kachisa-Schaefer-Scott (KSS) pairing friendly curve of embedding degree 18 in the context of Ate based pairing. Utilizing the skew Frobenius map along with multi-scalar multiplication procedure, an efficient scalar multiplication method for KSS curve is proposed in the paper. In addition to the theoretic proposal, this paper has also presented a comparative simulation of the proposed approach with plain binary method, sliding window method and non-adjacent form (NAF) for scalar multiplication. The simulation shows that the proposed method is about 60 times faster than plain implementation of other compared methods.

DOI： 10.1587/transfun.E100.A.1838

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A Comparative Study of Twist Property in KSS Curves of Embedding Degree 16 and 18 from the Implementation Perspective.

Md. Al-Amin Khandaker, Taehwan Park, Yasuyuki Nogami, Howon Kim

J. Inform. and Commun. Convergence Engineering   15 ( 2 )   97 - 103   2017

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Implementation of faster pairing calculation is the basis of efficient pairing-based cryptographic protocol implementation. Generally, pairing is a costly operation carried out over the extension field of degree k ≥ 12. But the twist property of the pairing friendly curve allows us to calculate pairing over the sub-field twisted curve, where the extension degree becomes k/d and twist degree d = 2, 3, 4, 6. The calculation cost is reduced substantially by twisting but it makes the discrete logarithm problem easier if the curve parameters are not carefully chosen. Therefore, this paper considers the most recent parameters setting presented by Barbulescu and Duquesne [1] for pairing-based cryptography; that are secure enough for 128- bit security level; to explicitly show the quartic twist (d = 6) and sextic twist (d = 4) mapping between the isomorphic rational point groups for KSS (Kachisa-Schaefer-Scott) curve of embedding degree k = 16 and k = 18, receptively. This paper also evaluates the performance enhancement of the obtained twisted mapping by comparing the elliptic curve scalar multiplications.

DOI： 10.6109/jicce.2017.15.2.97

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移動ロボットにおけるCANの脆弱性をついたDOS攻撃となりすましの実証

藤井建人, 亀川哲志, 野上保之, 五福明夫

日本機械学会ロボティクス・メカトロニクス講演会講演論文集(CD-ROM)   2017   2017

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奇標数体上の原始多項式を用いた多値擬似乱数系列における係数分布

小川千晶, 小寺雄太, 宮崎武, 上原聡, 日下卓也, 野上保之

情報理論とその応用シンポジウム予稿集(CD-ROM)   40th   2017

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擬似乱数生成のための効率的なトレース計算法に関する考察

小寺雄太, 小川千晶, 村岡英之, 荒木俊輔, 宮崎武, 上原聡, 日下卓也, 野上保之

情報理論とその応用シンポジウム予稿集(CD-ROM)   40th   2017

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Improvement of Noise Source Amplitude Modulation Method to Identify Source Devices of Electromagnetic Interference

石田千晶, 吉野慎平, 小川千晶, 五百旗頭健吾, 豊田啓孝, 野上保之

電子情報通信学会技術研究報告   117 ( 319(EMCJ2017 64-75) )   2017

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Binary field multiplication on ARMv8

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

SECURITY AND COMMUNICATION NETWORKS   9 ( 13 )   2051 - 2058   2016.9

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In this paper, we show efficient implementations of binary field multiplication over ARMv8. We exploit an advanced 64-bit polynomial multiplication (PMULL) supported by ARMv8 and conduct multiple levels of asymptotically faster Karatsuba multiplication for polynomial multiplication. Finally, our method completed binary field multiplication within 57 and 153 clock cycles for B-251 and B-571 cases, respectively. Proposed method improves the speed-performance by a factor of 4.5 times than previous techniques on same target platform. Copyright (c) 2016 John Wiley & Sons, Ltd.

DOI： 10.1002/sec.1462

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Hybrid Montgomery Reduction

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

ACM TRANSACTIONS ON EMBEDDED COMPUTING SYSTEMS   15 ( 3 )   2016.7

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In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of n-limb Montgomery reduction from theta(n(2) + n) to asymptotic complexity theta(7n(2)/8 + n). Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11%.

DOI： 10.1145/2890502

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Hybrid Montgomery Reduction

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

ACM TRANSACTIONS ON EMBEDDED COMPUTING SYSTEMS   15 ( 3 )   58 - 13   2016.7

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In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of n-limb Montgomery reduction from theta(n(2) + n) to asymptotic complexity theta(7n(2)/8 + n). Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11%.

DOI： 10.1145/2890502

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Hybrid Montgomery Reduction

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

ACM TRANSACTIONS ON EMBEDDED COMPUTING SYSTEMS   15 ( 3 )   2016.7

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In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of n-limb Montgomery reduction from theta(n(2) + n) to asymptotic complexity theta(7n(2)/8 + n). Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11%.

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FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves

Yasuyuki Nogami, Hiroto Kagotani, Kengo Iokibe, Hiroyuki Miyatake, Takashi Narita

IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS   E99D ( 4 )   805 - 815   2016.4

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Pairing-based cryptography has realized a lot of innovative cryptographic applications such as attribute-based cryptography and semi homomorphic encryption. Pairing is a bilinear map constructed on a torsion group structure that is defined on a special class of elliptic curves, namely pairing-friendly curve. Pairing-friendly curves are roughly classified into supersingular and non supersingular curves. In these years, non supersingular pairing-friendly curves have been focused on from a security reason. Although non supersingular pairing-friendly curves have an ability to bridge various security levels with various parameter settings, most of software and hardware implementations tightly restrict them to achieve calculation efficiencies and avoid implementation difficulties. This paper shows an FPGA implementation that supports various parameter settings of pairings on non supersingular pairing-friendly curves for which Montgomery reduction, cyclic vector multiplication algorithm, projective coordinates, and Tate pairing have been combinatorially applied. Then, some experimental results with resource usages are shown.

DOI： 10.1587/transinf.2015ICP0018

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Successful Attack Detection of Elliptic Curve Cryptography with Dynamic DNS

三好俊介, 山井成良, 野上保之

情報処理学会研究報告(Web)   2016 ( IOT-32 )   2016

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素体上のロジスティック写像による系列の平均周期・リンク長期待値

宮崎武, 荒木俊輔, 上原聡, 野上保之

日本応用数理学会年会講演予稿集(CD-ROM)   2016   2016

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同じNTU系列を生成する異なる既約多項式についての考察

平郡聖士, 野上保之, 上原聡, 土屋和由

情報理論とその応用シンポジウム予稿集(CD-ROM)   39th   2016

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A Consideration of an Efficient Calculation over the Extension Field of Degree 4 for Elliptic Curve Pairing Cryptography

眞田晃宏, DUQUESNE Sylvain, 白勢政明, 野上保之

電子情報通信学会技術研究報告   116 ( 163(IT2016 21-33) )   2016

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A Consideration of an Efficient Calculation over the Extension Field of Degree 3 for Elliptic Curve Pairing Cryptography

小寺雄太, DUQUESNE Sylvain, 白勢政明, 野上保之

電子情報通信学会技術研究報告   116 ( 163(IT2016 21-33) )   2016

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Dynamic Job Scheduling Method Based on Expected Probability of Completion of Voting in Volunteer Computing

Yuto Miyakoshi, Shinya Yasuda, Kan Watanabe, Masaru Fukushi, Yasuyuki Nogami

IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS   E98D ( 12 )   2132 - 2140   2015.12

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This paper addresses the problem of job scheduling in volunteer computing (VC) systems where each computation job is replicated and allocated to multiple participants (workers) to remove incorrect results by a votingmechanism. In the job scheduling of VC, the number of workers to complete a job is an important factor for the system performance; however, it cannot be fixed because some of the workers may secede in real VC. This is the problem that existing methods have not considered in the job scheduling. We propose a dynamic job scheduling method which considers the expected probability of completion (EPC) for each job based on the probability of worker's secession. The key idea of the proposed method is to allocate jobs so that EPC is always greater than a specified value (SPC). By setting SPC as a reasonable value, the proposed method enables to complete jobs without excess allocation, which leads to the higher performance of VC systems. We assume in this paper that worker's secession probability follows Weibull-distribution which is known to reflect more practical situation. We derive parameters for the distribution using actual trace data and compare the performance of the proposed and the previous method under the Weibull-distribution model, as well as the previous constant probability model. Simulation results show that the performance of the proposed method is up to 5 times higher than that of the existing method especially when the time for completing jobs is restricted, while keeping the error rate lower than a required value.

DOI： 10.1587/transinf.2015PAP0027

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Dynamic Job Scheduling Method Based on Expected Probability of Completion of Voting in Volunteer Computing

Yuto Miyakoshi, Shinya Yasuda, Kan Watanabe, Masaru Fukushi, Yasuyuki Nogami

IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS   E98D ( 12 )   2132 - 2140   2015.12

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This paper addresses the problem of job scheduling in volunteer computing (VC) systems where each computation job is replicated and allocated to multiple participants (workers) to remove incorrect results by a votingmechanism. In the job scheduling of VC, the number of workers to complete a job is an important factor for the system performance; however, it cannot be fixed because some of the workers may secede in real VC. This is the problem that existing methods have not considered in the job scheduling. We propose a dynamic job scheduling method which considers the expected probability of completion (EPC) for each job based on the probability of worker's secession. The key idea of the proposed method is to allocate jobs so that EPC is always greater than a specified value (SPC). By setting SPC as a reasonable value, the proposed method enables to complete jobs without excess allocation, which leads to the higher performance of VC systems. We assume in this paper that worker's secession probability follows Weibull-distribution which is known to reflect more practical situation. We derive parameters for the distribution using actual trace data and compare the performance of the proposed and the previous method under the Weibull-distribution model, as well as the previous constant probability model. Simulation results show that the performance of the proposed method is up to 5 times higher than that of the existing method especially when the time for completing jobs is restricted, while keeping the error rate lower than a required value.

DOI： 10.1587/transinf.2015PAP0027

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A Consideration on Trace Sequence over Finite Field (情報理論)

野上 保之, 井納 弘人, 土屋 和由

電子情報通信学会技術研究報告 = IEICE technical report : 信学技報   115 ( 214 )   25 - 29   2015.9

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Montgomery multiplication and squaring for Optimal Prime Fields

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

COMPUTERS & SECURITY   52   276 - 291   2015.7

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Optimal Prime Fields (OPFs) are considered to be one of the best choices for lightweight elliptic curve cryptography implementations on resource-constraint embedded processors. In this paper, we revisit the efficient modular arithmetic over the special prime fields, and present improved implementations of modular multiplication and squaring for OPFs, called Optimal Prime Field Coarsely Integrated Operand Caching (OPF-CIOC) and Coarsely Integrated Sliding Block Doubling (OPF-CISBD) methods. The OPF-CIOC and OPF-CISBD methods follow the general ideas of (consecutive) operand caching and sliding block doubling techniques, respectively. The methods have been carefully optimized and redesigned for Montgomery multiplication and squaring in an integrated fashion. We then evaluate the practical performance of proposed methods on representative 8-bit AVR processor. Experimental results show that the proposed OPF-CIOC and OPF-CISBD methods outperform the previous best known results in ACNS'14 by a factor of 8% and 32%. Furthermore, our methods are implemented in a regular way which helps to reduce the leakage of side-channel information. (C) 2015 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.cose.2015.03.005

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Montgomery multiplication and squaring for Optimal Prime Fields

Hwajeong Seo, Zhe Liu, Yasuyuki Nogami, Jongseok Choi, Howon Kim

COMPUTERS & SECURITY   52   276 - 291   2015.7

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Optimal Prime Fields (OPFs) are considered to be one of the best choices for lightweight elliptic curve cryptography implementations on resource-constraint embedded processors. In this paper, we revisit the efficient modular arithmetic over the special prime fields, and present improved implementations of modular multiplication and squaring for OPFs, called Optimal Prime Field Coarsely Integrated Operand Caching (OPF-CIOC) and Coarsely Integrated Sliding Block Doubling (OPF-CISBD) methods. The OPF-CIOC and OPF-CISBD methods follow the general ideas of (consecutive) operand caching and sliding block doubling techniques, respectively. The methods have been carefully optimized and redesigned for Montgomery multiplication and squaring in an integrated fashion. We then evaluate the practical performance of proposed methods on representative 8-bit AVR processor. Experimental results show that the proposed OPF-CIOC and OPF-CISBD methods outperform the previous best known results in ACNS'14 by a factor of 8% and 32%. Furthermore, our methods are implemented in a regular way which helps to reduce the leakage of side-channel information. (C) 2015 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.cose.2015.03.005

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A Consideration of NTRU Whose Modular Polynomial Is All One Polynomial

MISUMI Koki, NOGAMI Yasuyuki

Technical report of IEICE. ISEC   114 ( 471 )   5 - 10   2015.3

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Quantum computer will break most of recent public key cryptosystems by Shor's algorithm. Post-quantum cryptosystem which is not broken by them has been studied. NTRU is one of the post-quantum cryptosystems proposed by Hoffstein et al. This work considers a variant of NTRU with All One Polynomial Ring based on All One Polynomial. This work applies CVMA:Cyclic Vector Multiplication Algorithm. For the security evaluation, the same approach of the original NTRU can be applied to the proposed variant.

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Efficient Calculation of Pairing with Supersingular Curve on 2-dimentional Extension Field

KUMANO Akito, NOGAMI Yasuyuki, SHIRASE Masaaki

Technical report of IEICE. ISEC   114 ( 471 )   11 - 17   2015.3

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Yoshida et al. have proposed the vector decomposition problem using supersingular curve whose embedding degree is 2. In order to apply the problem as a trapdoor of cryptosystem, it is necessary to speed up pairing calculations. This paper considers an efficient approach for Tate pairing with a supersingular curve on 2-dimentional extension field. This paper improves both Miller's algorithm and final exponentiation that is required for Tate pairing. In detail, one multiplication in the calculation of Miller's loop is eliminated by using twist map. This paper discusses how efficient the calculation of pairing becomes.

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Associative Rational Points for Improving Random Walkswith Collision-based Attack on Elliptic Curve Discrete Logarithm Problem

Yasuyuki Nogami, Thomas H. Austin

International Journal of Computer and Information Technology   2015

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Associative Rational Points for Improving Random Walkswith Collision-based Attack on Elliptic Curve Discrete Logarithm Problem

Yasuyuki Nogami, Thomas H. Austin

International Journal of Computer and Information Technology   2015

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A Relation between Autocorrelations of Sequences Generated by the Logistic Map over Prime Fields and Loops whose Periods are Almost Same as Precisions

宮崎武, 荒木俊輔, 上原聡, 野上保之

電子情報通信学会技術研究報告   114 ( 470(IT2014 62-101) )   2015

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素体上のロジスティック写像による生成系列の連続増加・減少回数について

宮崎武, 荒木俊輔, 上原聡, 野上保之

情報理論とその応用シンポジウム予稿集(CD-ROM)   38th   2015

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奇標数体上の多値擬似乱数系列の値の分布について

小池将太, 野上保之, 土屋和由, 上原聡

情報理論とその応用シンポジウム予稿集(CD-ROM)   38th   2015

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奇標数体上の非原始多項式を用いた擬似乱数系列の生成

小川千晶, 野上保之, 土屋和由, 上原聡

情報理論とその応用シンポジウム予稿集(CD-ROM)   38th   2015

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逐次拡大体F_x(x=(2⁴)²)における演算の効率化について

實井識人, 野上保之, 日下卓也

情報理論とその応用シンポジウム予稿集(CD-ROM)   38th   2015

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制御変数が4である有限体上のロジスティック写像による最大周期系列に対する線形複雑度プロファイル

土屋和由, 野上保之

日本応用数理学会年会講演予稿集(CD-ROM)   2015   2015

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Evaluation of parallelization for multiple-precision Cyclic Vector Multiplication Algorithm using CUDA

HARAMURA Satoshi, KAGOTANI Hiroto, NOGAMI Yasuyuki, SUGIYAMA Yuji

IEICE technical report   113 ( 418 )   109 - 112   2014.1

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Recently, studies on implementation of cryptographical algorithms on GPUs are widely conducted as GPGPU development environment provides easy access to GPUs for non-graphics programming. In this paper, we implement cyclic vector multiplication algorithm, or an efficient multiplication algorithm on extension fields, that will accelerate pairing cryptography. Since we adopt 256-bit characteristic for the base fields, we also implement Montgomery multiplication for 256-bit operands on GPU. As the result of implementation on NVIDIA GeForce GTX680, it is over 10 times slower than implementation on Core i7 3970X.

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Evaluation of parallelization for multiple-precision Cyclic Vector Multiplication Algorithm using CUDA

HARAMURA Satoshi, KAGOTANI Hiroto, NOGAMI Yasuyuki, SUGIYAMA Yuji

IEICE technical report. Computer systems   113 ( 417 )   109 - 112   2014.1

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Recently, studies on implementation of cryptographical algorithms on GPUs are widely conducted as GPGPU development environment provides easy access to GPUs for non-graphics programming. In this paper, we implement cyclic vector multiplication algorithm, or an efficient multiplication algorithm on extension fields, that will accelerate pairing cryptography. Since we adopt 256-bit characteristic for the base fields, we also implement Montgomery multiplication for 256-bit operands on GPU. As the result of implementation on NVIDIA GeForce GTX680, it is over 10 times slower than implementation on Core i7 3970X.

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Evaluation of parallelization for multiple-precision Cyclic Vector Multiplication Algorithm using CUDA

HARAMURA Satoshi, KAGOTANI Hiroto, NOGAMI Yasuyuki, SUGIYAMA Yuji

Technical report of IEICE. VLD   113 ( 416 )   109 - 112   2014.1

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Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

Recently, studies on implementation of cryptographical algorithms on GPUs are widely conducted as GPGPU development environment provides easy access to GPUs for non-graphics programming. In this paper, we implement cyclic vector multiplication algorithm, or an efficient multiplication algorithm on extension fields, that will accelerate pairing cryptography. Since we adopt 256-bit characteristic for the base fields, we also implement Montgomery multiplication for 256-bit operands on GPU. As the result of implementation on NVIDIA GeForce GTX680, it is over 10 times slower than implementation on Core i7 3970X.

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An Implementation of Elliptic Curve Scalar Multiplication with Improved Quadrupling of Rational Point

Gegerihu, NOGAMI Yasuyuki, SHIRASE Masaaki

IEICE technical report. Information theory   113 ( 411 )   45 - 49   2014.1

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This paper introduces a fast implementation of elliptic curve scalar multiplication with an improved quadrupling in Jacobian coordinates, since one quadrupling is more efficient than two doublings for which the scalar needs to be represented in double-base chains form. For point additions, this paper applies coordinates conversion technique which can save one field multiplication in each term of point addition in Mixed coordinates. After that, some experimental results on 256 bit elliptic curve cryptography are shown.

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Evaluation of parallelization for multiple-precision Cyclic Vector Multiplication Algorithm using CUDA

2014 ( 19 )   1 - 4   2014.1

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Pseudo 8-sparse multiplication for efficient ate-based pairing on Barreto-Naehrig curve

Yuki Mori, Shoichi Akagi, Yasuyuki Nogami, Masaaki Shirase

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)   8365   186 - 198   2014

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According to some recent implementation reports on Ate-based pairings such as optimal ate pairing with Barreto-Naehrig curve whose embedding degree is 12, sparse multiplication accelerates Miller's loop calculation in a pairing calculation. Especially, 7-sparse multiplication is available when the implementation uses affine coordinates, where 7-sparse means that the multiplicand or multiplier has 7 zeros among 12 coefficients. This paper extends it to pseudo 8-sparse multiplication. Then, some experimental results together with theoretic calculation costs are shown in order to evaluate its efficiency. © 2014 Springer International Publishing.

DOI： 10.1007/978-3-319-04873-4_11

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有限体上のロジスティック写像による生成系列に対する長周期を保証するための条件

土屋和由, 野上保之

日本応用数理学会年会講演予稿集(CD-ROM)   2014   2014

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A Binary Sequence Generated by Legendre Symbol and Primitive Polynomial over Odd Characteristic

NOGAMI Yasuyuki, UEHARA Satoshi, TADA Kazuki

IEICE technical report. Information theory   113 ( 58 )   5 - 8   2013.5

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Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in F^*_pm, then the sequence S={s_i}, s_i=Tr(w^i), i=0,1,2,… becomes a maximum length sequence, where Tr(・) is the trace function over F_p. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. Then, it is shown that the obtained binary sequence has the period L=2(p^m-1)/(p-1) and a certain periodic autocorrelation. This paper also shows a small example.

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A Smaller Final Exponentiation for Tate and Ate Pairings with Barreto-Naehrig Curve

Yuki Kono, Taichi Sumo, Yasuyuki Nogami

2013 16TH INTERNATIONAL CONFERENCE ON NETWORK-BASED INFORMATION SYSTEMS (NBIS 2013)   518 - 522   2013

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This paper shows an approach for reducing the size of the exponent of final exponentiation with multiplying some extra terms. In the case of Tate and Ate pairings with Barreto-Naehrig curve whose embedding degree is 12, the exponent is reduced to (p(4) - p(2) + 1)/r, where p is the characteristic of the base field and r is the order of pairing.

DOI： 10.1109/NBiS.2013.86

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A Binarization of Geometric Sequences with Legendre Symbol and Its Autocorrelation

Yasuyuki Nogami, Kazuki Tada, Satoshi Uehara

SIXTH INTERNATIONAL WORKSHOP ON SIGNAL DESIGN AND ITS APPLICATIONS IN COMMUNICATIONS   28 - 31   2013

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Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let omega be its zero, that is a primitive element in F-pm*, then the sequence S = {s(i)}, s(i) = Tr (omega(i)) for i = 0, 1, 2, ... becomes a maximum length sequence, where Tr (.) is the trace function over F-p. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2 (p(m) - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.

DOI： 10.1109/IWSDA.2013.6849054

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Safe Primeを法とした素体上のロジスティック写像による生成系列に関する一考察

宮崎武, 荒木俊輔, 上原聡, 野上保之

日本応用数理学会年会講演予稿集(CD-ROM)   2013   2013

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A Property of Norm Pairs of Rational Point on A Twisted Elliptic Curve and Adapting Pollard's Rho Method

2012 ( 3 )   493 - 500   2012.10

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Finding a Basis Conversion Matrix Using a Polynomial Basis Derived by a Small Multiplicative Cyclic Group

Yasuyuki Nogami, Hidehiro Kato, Kenta Nekado, Satoshi Uehara, Yoshitaka Morikawa

IEEE TRANSACTIONS ON INFORMATION THEORY   58 ( 7 )   4936 - 4947   2012.7

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Several methods for finding a basis conversion matrix between two different bases in an extension field F-p(m) have been proposed. Among them, the one based on Gauss period normal basis (GNB) is on average the most efficient. However, since it needs to construct a certain tower field F-(pm)n, some inefficient cases in which the towering degree n becomes large have been reported. This paper first determines that such inefficient cases are caused by the GNB condition. In order to overcome this inefficiency, we propose a method that does not use any GNB in the target extension field F-p(m), but instead uses a certain polynomial basis in F-p(m) derived by a certain small cyclic group in F-(pm)n. This causes relaxation of the condition for the towering degree n. In addition, our experimental results show that the proposed method substantially accelerates the computation time for finding a basis conversion matrix.

DOI： 10.1109/TIT.2012.2191477

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Finding a Basis Conversion Matrix Using a Polynomial Basis Derived by a Small Multiplicative Cyclic Group

Yasuyuki Nogami, Hidehiro Kato, Kenta Nekado, Satoshi Uehara, Yoshitaka Morikawa

IEEE TRANSACTIONS ON INFORMATION THEORY   58 ( 7 )   4936 - 4947   2012.7

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Several methods for finding a basis conversion matrix between two different bases in an extension field F-p(m) have been proposed. Among them, the one based on Gauss period normal basis (GNB) is on average the most efficient. However, since it needs to construct a certain tower field F-(pm)n, some inefficient cases in which the towering degree n becomes large have been reported. This paper first determines that such inefficient cases are caused by the GNB condition. In order to overcome this inefficiency, we propose a method that does not use any GNB in the target extension field F-p(m), but instead uses a certain polynomial basis in F-p(m) derived by a certain small cyclic group in F-(pm)n. This causes relaxation of the condition for the towering degree n. In addition, our experimental results show that the proposed method substantially accelerates the computation time for finding a basis conversion matrix.

DOI： 10.1109/TIT.2012.2191477

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An Approach for Constructing Primitive Polynomials of Degree 2^i over Odd Characteristic Prime Field

NOGAMI Yasuyuki, TAKAI Yusuke, KOBAYASHI Shigeki, SUGIMURA Tatsuo, UEHARA Satoshi

Technical report of IEICE. ISEC   112 ( 39 )   19 - 24   2012.5

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This paper considers an approach to obtain primitive polynomials over odd characteristic field F_p for which a variable transformation x&map;x+αx^<-1> called "self α reciprocal transform" is used, where α is a certain element in F^*_<p^<2^j>> of j<i. Self α reciprocal transforms give irreducible polynomials of degree 2^i under a certain condition to which this paper considers their primitivities. If they are not primitive polynomials over F_p, this paper also touches an efficient approach for translating them to primitive polynomials.

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SubBytes Transform for AES Adopting Basis Conversion with More Miscellaneously Mixed Basis in F_{(2^4)^2}

2011 ( 3 )   349 - 354   2011.10

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Freeman曲線を用いたXateおよびR–ateペアリングのための定義体における乗算アルゴリズム

根角 健太, 湯浅 達也, 野上 保之, 森川 良孝

コンピュータセキュリティシンポジウム2009 (CSS2009) 論文集   2009   1 - 6   2011.10

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SubBytes Transform for AES Adopting Inversion in F_{(2^4)^2} with Redundantly Represented Basis

2011 ( 3 )   343 - 348   2011.10

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

TAKAHASHI Ryosuke, NEKADO Kenta, TAKAI Yusuke, NOGAMI Yasuyuki, KAGOTANI Hiroto, NARITA Takashi

IEICE technical report   111 ( 125 )   145 - 150   2011.7

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The authors have proposed a cyclic vector multiplication algorithm abbreviated as CVMA that is flexible to the parameters of extension field, that is extension degree and charasteristic. For the hardware implementation of CVMA, the authors have improved the double loop structure of CVMA to a single loop one, and it has been presented at SITA2010. However, it has a problem that it needs a lot of memory. Thus, this paper proposes a method to save the memory use of the single loop version of CVMA.

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

TAKAHASHI Ryosuke, NEKADO Kenta, TAKAI Yusuke, NOGAMI Yasuyuki, KAGOTANI Hiroto, NARITA Takashi

IEICE technical report   111 ( 124 )   145 - 150   2011.7

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The authors have proposed a cyclic vector multiplication algorithm abbreviated as CVMA that is flexible to the parameters of extension field, that is extension degree and charasteristic. For the hardware implementation of CVMA, the authors have improved the double loop structure of CVMA to a single loop one, and it has been presented at SITA2010. However, it has a problem that it needs a lot of memory. Thus, this paper proposes a method to save the memory use of the single loop version of CVMA.

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

TAKAHASHI Ryosuke, NEKADO Kenta, TAKAI Yusuke, NOGAMI Yasuyuki, KAGOTANI Hiroto, NARITA Takashi

IEICE technical report   111 ( 126 )   145 - 150   2011.7

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The authors have proposed a cyclic vector multiplication algorithm abbreviated as CVMA that is flexible to the parameters of extension field, that is extension degree and charasteristic. For the hardware implementation of CVMA, the authors have improved the double loop structure of CVMA to a single loop one, and it has been presented at SITA2010. However, it has a problem that it needs a lot of memory. Thus, this paper proposes a method to save the memory use of the single loop version of CVMA.

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

TAKAHASHI Ryosuke, NEKADO Kenta, TAKAI Yusuke, NOGAMI Yasuyuki, KAGOTANI Hiroto, NARITA Takashi

IEICE technical report   111 ( 123 )   145 - 150   2011.7

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The authors have proposed a cyclic vector multiplication algorithm abbreviated as CVMA that is flexible to the parameters of extension field, that is extension degree and charasteristic. For the hardware implementation of CVMA, the authors have improved the double loop structure of CVMA to a single loop one, and it has been presented at SITA2010. However, it has a problem that it needs a lot of memory. Thus, this paper proposes a method to save the memory use of the single loop version of CVMA.

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

2011 ( 23 )   1 - 6   2011.7

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Memory Saving Implementation of Cyclic Vector Multiplication Algorithm

2011 ( 23 )   1 - 6   2011.7

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Classification of MRCPs and Type-I CVMA Modified for Its Prime Field Multiplication

TAKAI Yusuke, NEKADO Kenta, NOGAMI Yasuyuki, MORIKAWA Yoshitaka, KAGOTANI Hiroto

IEICE technical report   110 ( 137 )   19 - 24   2010.7

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Cyclic vector multiplication algorithm (CVMA) has been originally proposed as a vector multiplication algorithm for elements in a certain extension field. Recently, Granger et al. have proposed the primes called Minimal Redundancy Cyclotomic Primes (MRCPs) and they have applied CVMA for the modular multiplications with MRCPs. This paper shows that several detailed classes of MRCPs and their customized multiplication algorithms.

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Cost Estimates of CNTW Attack against EMV Signatures

SAKEMI Yumi, IZU Tetsuya, TAKENAKA Masahiko, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   110 ( 114 )   195 - 201   2010.6

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Specification of EMV signatures is an ISO/IEC 9796-2 compliant format with extra redundancy, and used for certifying credit cards. At CRYPTO 2009, Coron, Naccache, Tibouchi, and Weinmann proposed a new forgery attack against the signature ISO/IEC 9796-2. They also discussed the possibility when the attack is applied to the EMV signatures. As a results, they concluded that the attack can not forge them for operational reason. However, since they estimated the attack's cost under a specific condition, it is difficult to evaluate the attack's threat under other conditions. This paper gives the detailed cost of the attack under all conditions based on the fixed data in a message. Then, the conditions under which the attack can forge EMV signature with low costs are shown.

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Cost Estimates of CNTW Attack against EMV Signatures

SAKEMI Yumi, IZU Tetsuya, TAKENAKA Masahiko, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   110 ( 113 )   195 - 201   2010.6

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Specification of EMV signatures is an ISO/IEC 9796-2 compliant format with extra redundancy, and used for certifying credit cards. At CRYPTO 2009, Coron, Naccache, Tibouchi, and Weinmann proposed a new forgery attack against the signature ISO/IEC 9796-2. They also discussed the possibility when the attack is applied to the EMV signatures. As a results, they concluded that the attack can not forge them for operational reason. However, since they estimated the attack's cost under a specific condition, it is difficult to evaluate the attack's threat under other conditions. This paper gives the detailed cost of the attack under all conditions based on the fixed data in a message. Then, the conditions under which the attack can forge EMV signature with low costs are shown.

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Existing Probability of Gauss Period Normal Basis

NEKADO Kenta, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   109 ( 446 )   403 - 407   2010.2

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Recently, for several pairing-based cryptographic applications, convenient basis which can construct extension field for every pair of characteristic p and extension degree m has been required. Kato et al. have focused on Gauss period normal basis(GNB)as the basis, and proposed cyclic vector multiplication algorithm(CVMA)as an efficient multiplication algorithm in an extension field constructed by GNB. This paper considers how to elicite existing probability of GNB theoretically, and estimates the efficiency of CVMA by using the elicitation method.

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Anonymous IEEE802.1X Authentication System Using Group Signatures

A. Sudarsono, T. Nakanishi, Y. Nogami, and N. Funabiki

IPSJ Journal   51 ( 3 )   691 - 704   2010

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Mixed Bases for Efficient Inversion in F(((22)2)2) and Conversion Matrices of Sub Bytes of AES

Yasuyuki Nogami, Kenta Nekado, Tetsumi Toyota, Naoto Hongo, Yoshitaka Morikawa

CRYPTOGRAPHIC HARDWARE AND EMBEDDED SYSTEMS - CHES 2010   6225   234 - 247   2010

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A lot of improvements and optimizations for the hardware implementation of Sub Bytes of Rijndael, in detail inversion in F(28) have been reported. Instead of the Rijndael original F(28) it is known that its isomorphic tower field F(((22)2)2) has a more efficient inversion. For the towerings, several kinds of bases such as polynomial and normal bases can be used in mixture. Different from the meaning of this mixture of bases, this paper proposes another mixture that contributes to the reduction of the critical path delay of SubBytes. To the F((22)2)-inversion architecture, for example, the proposed mixture inputs and outputs elements represented with normal and polynomial bases, respectively.

DOI： 10.1007/978-3-642-15031-9_16

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Anonymous IEEE802.1X Authentication System Using Group Signatures

A. Sudarsono, T. Nakanishi, Y. Nogami, N. Funabiki

IPSJ Journal   51 ( 3 )   691 - 704   2010

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Mixed Bases for Efficient Inversion in F(((22)2)2) and Conversion Matrices of Sub Bytes of AES

Yasuyuki Nogami, Kenta Nekado, Tetsumi Toyota, Naoto Hongo, Yoshitaka Morikawa

CRYPTOGRAPHIC HARDWARE AND EMBEDDED SYSTEMS - CHES 2010   6225   234 - 247   2010

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A lot of improvements and optimizations for the hardware implementation of Sub Bytes of Rijndael, in detail inversion in F(28) have been reported. Instead of the Rijndael original F(28) it is known that its isomorphic tower field F(((22)2)2) has a more efficient inversion. For the towerings, several kinds of bases such as polynomial and normal bases can be used in mixture. Different from the meaning of this mixture of bases, this paper proposes another mixture that contributes to the reduction of the critical path delay of SubBytes. To the F((22)2)-inversion architecture, for example, the proposed mixture inputs and outputs elements represented with normal and polynomial bases, respectively.

DOI： 10.1007/978-3-642-15031-9_16

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GLV Method for Non-Supersingular Paring-Friendly Curves of Embedding Degree 1

TAKEUCHI Shoichi, IZUTA Tetsuya, SAKEMI Yumi, NISHII Kazushi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   109 ( 337 )   37 - 41   2009.12

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Recently, some pairing-based cryptographies that uses a non-supersingular pairing-friendly curve whose order has large prime factors as RSA does have been proposed. Since it generally takes a lot of calculation time for generating such a composite order pairing-friendly curve, several efficient methods have been also proposed. Then, using such a special pairing-friendly curve, pairing-based cryptographies will be implemented with not only pairing calculations but also scalar multiplications and exponentiations over a certain finite field. This paper first introduces a non-supersingular pairing-friendly curve of embedding degree 1 whose order has two large prime factors as RSA does. Then, it is shown that GLV method and multi scalar multiplication are efficiently applied for accelerating a scalar multiplication. After that, some experimental results are shown.

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A Construction Method of Basis Conversion Matrix by Using Small Multiplicative Group for Cryptographic Applications

KATO Hidehiro, NOGAMI Yasuyuki, UEHARA Satoshi, MORIKAWA Yoshitaka

IEICE technical report   109 ( 337 )   43 - 50   2009.12

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Several methods for finding a basis conversion matrix between two different bases in extension field F_<p^m> have been proposed. Among them, the one based on Gauss period normal basis (GNB-based method) is averagely the most efficient. However, since it needs to construct a certain tower field F_<(p^m)^n, some inefficient cases in which the towering degree becomes relatively large have been reported. It is caused from the condition of GNB. In order to overcome the inefficiency, this paper proposes a method that does not need any GNBs in the target extension field F_<p^m> but uses a certain polynomial basis in F_<p^m> derived by a certain small cyclic group in F_<(p^m)^n. Then, the condition for the towering degree is relaxed, some experimental results show that the proposed method substantially accelerates the average computation time for finding a basis conversion matrix.

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An Implementation of Tate Pairing with Non-Supersingular Curves of Embedding degree 1

IZUTA Tetsuya, SAKEMI Yumi, NISHII Kazushi, TAKEUCHI Shoichi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   109 ( 207 )   59 - 64   2009.9

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Recently, pairing-based cryptographic applications such as ID-based cryptography have received much attention. On the other hand, RSA cryptography has been widely used and it is defined over a certain composite order as the modulus. In detail, it generally needs to be a product of two large prime numbers. In order to apply RSA-based techniques to pairing-based cryptography, pairing friendly curve needs to have such a composite order. This paper focuses on some non-supersingular pairing friendly curves whose order r has two large prime factors and embedding degree k is equal to 1. This paper shows some experimental results on generating such a composite order curve and Tate pairing.

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Integer Variable chi-Based Cross Twisted Ate Pairing and Its Optimization for Barreto-Naehrig Curve

Yasuyuki Nogami, Yumi Sakemi, Hidehiro Kato, Masataka Akane, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 8 )   1859 - 1867   2009.8

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It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log(2) r/phi(k), where phi(.) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from left perpendicularlog(2) right perpendicular to left perpendicularlog(2)(t-1)right perpendicular, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "chi." For such a curve, this paper gives integer variable chi-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve. it reduces the number of loops to left perpendicularlog(2)chi right perpendicular. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

DOI： 10.1587/transfun.E92.A.1859

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Integer Variable chi-Based Cross Twisted Ate Pairing and Its Optimization for Barreto-Naehrig Curve

Yasuyuki Nogami, Yumi Sakemi, Hidehiro Kato, Masataka Akane, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 8 )   1859 - 1867   2009.8

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It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log(2) r/phi(k), where phi(.) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from left perpendicularlog(2) right perpendicular to left perpendicularlog(2)(t-1)right perpendicular, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "chi." For such a curve, this paper gives integer variable chi-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve. it reduces the number of loops to left perpendicularlog(2)chi right perpendicular. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

DOI： 10.1587/transfun.E92.A.1859

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Finding a Basis Conversion Matrix via Prime Gauss Period Normal Basis

Yasuyuki Nogami, Ryo Namba, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 6 )   1500 - 1507   2009.6

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This paper proposes a method to construct a basis conversion matrix between two given bases in F-pm. In the proposed method, Gauss period normal basis (GNB) works as a bridge between the two bases. The proposed method exploits this property and construct a basis conversion matrix mostly faster than EDF-based algorithm on average in polynomial time. Finally, simulation results are reported in which the proposed method compute a basis conversion matrix within 30 msec on average with Celeron (2.00 GHz) when m log p approximate to 160.

DOI： 10.1587/transfun.E92.A.1500

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A Consideration on Generating Ordinary Pairing Friendly Curve of Composite Order That Has Two Large Prime Factors

NISHII Kazushi, SAKEMI Yumi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   109 ( 42 )   51 - 54   2009.5

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Recently, pairing-based cryptographic applications such as ID-based cryptography have received much attention. On the other hand, RSA cryptography has been widely used and is defined over a certain composite order as the modulus. In detail, it generally needs to be a product of two large prime numbers. In order to apply RSA-based techniques to pairing-based cryptography, pairing-friendly elliptic curve also needs to have such a composite order. This paper proposes a method for generating ordinary pairing-friendly curves of such a composite order especially when the embedding degree k is equal to 3, the discriminant is equal to 3, in addition the elliptic curve is given in the form of y^2=x^3+a, a∈F_p.

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An Implementation of a Group Signature Scheme with Efficient Verifier-Local Revocation Check

SUDARSONO Amang, NAKANISHI Toru, SAKEMI Yumi, NOGAMI Yasuyuki, FUNABIKI Nobuo

IEICE technical report   109 ( 42 )   37 - 42   2009.5

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A group signature scheme with efficient verifier-local revocation check has been proposed. This scheme provides a more efficient revocation check, using the multi-pairing technique for a product of pairings. However, this scheme has not been implemented and evaluated. This paper presents an implementation of the group signature scheme and the experimental results.

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An Improvement of Cyclic Vector Multiplication Algorithm based on Gauss period Normal Basis

YANAGI Erika, NEKADO Kenta, NOGAMI Yasuyuki, MORIKAWA Yoshiktaka

IEICE technical report   109 ( 42 )   55 - 60   2009.5

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This paper first introduces cyclic vector multiplication algorithm (CVMA) that is a multiplication algorithm with Gauss period Normal Basis in extension field. Then, it is also introduced that CVMA is useful under the tight restrictions of pairing-based cryptographies. Then, this paper points out a problem about the calculation cost of CVMA. For this problem, this paper proposes an improvement. According to some simulation and experimental results, it is shown that the improvement makes CVMA much more efficient in some cases.

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Fast Ate Pairing Computation of Embedding Degree 12 Using Subfield-Twisted Elliptic Curve

Masataka Akane, Yasuyuki Nogami, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 2 )   508 - 516   2009.2

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This paper presents implementation techniques of fast Ate pairing of embedding degree 12. In this case, we have no trouble in finding a prime order pairing friendly curve E such as the Barreto-Naehrig curve y(2) = x(3) + a, a is an element of F-p. For the curve, an isomorphic substitution from G(2) is an element of E(F-p(12)) into G(2)'in subfield-twisted elliptic curve E'(F-p(2)) speeds up scalar multiplications over G(2) and wipes out denominator calculations in Miller's algorithm. This paper mainly provides about 30% improvement of the Miller's algorithm calculation using proper subfield arithmetic operations. Moreover, we also provide the efficient parameter settings of the BN curves. When p is a 254-bit prime, the embedding degree is 12, and the processor is Pentium4 (3.6 GHz), it is shown that the proposed algorithm computes Ate pairing in 13.3 milli-seconds including final exponentiation.

DOI： 10.1587/transfun.E92.A.508

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Fast Ate Pairing Computation of Embedding Degree 12 Using Subfield-Twisted Elliptic Curve

Masataka Akane, Yasuyuki Nogami, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 2 )   508 - 516   2009.2

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This paper presents implementation techniques of fast Ate pairing of embedding degree 12. In this case, we have no trouble in finding a prime order pairing friendly curve E such as the Barreto-Naehrig curve y(2) = x(3) + a, a is an element of F-p. For the curve, an isomorphic substitution from G(2) is an element of E(F-p(12)) into G(2)'in subfield-twisted elliptic curve E'(F-p(2)) speeds up scalar multiplications over G(2) and wipes out denominator calculations in Miller's algorithm. This paper mainly provides about 30% improvement of the Miller's algorithm calculation using proper subfield arithmetic operations. Moreover, we also provide the efficient parameter settings of the BN curves. When p is a 254-bit prime, the embedding degree is 12, and the processor is Pentium4 (3.6 GHz), it is shown that the proposed algorithm computes Ate pairing in 13.3 milli-seconds including final exponentiation.

DOI： 10.1587/transfun.E92.A.508

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A Multiplication Algorithm in F-pm Such That p &gt; m with a Special Class of Gauss Period Normal Bases

Hidehiro Kato, Yasuyuki Nogami, Tomoki Yoshida, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 1 )   173 - 181   2009.1

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In this paper, a multiplication algorithm in extension field F-pm is proposed. Different from the previous works, the proposed algorithm can be applied for an arbitrary pair of characteristic 1) and extension degree in only except for the case when 4p divides m(p-1) and m is an even number. As written in the title, when p &gt; m, 4p does not divice m(p - 1). The proposed algorithm is derived by modifying cyclic vector multiplication algorithm (CVMA). We adopt a special class of Gauss period normal bases. At first in this paper, it is formulated as an algorithm and the calculation cost of the modified algorithm is evaluated. Then, compared to those of the previous works, some experimental results are shown. Finally, it is shown that the proposed algorithm is sufficient practical when extension degree in is small.

DOI： 10.1587/transfun.E92.A.173

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Scalar Multiplication Using Frobenius Expansion over Twisted Elliptic Curve for Ate Pairing Based Cryptography

Yasuyuki Nogami, Yumi Sakemi, Takumi Okimoto, Kenta Nekado, Masataka Akane, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 1 )   182 - 189   2009.1

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For ID-based cryptography, not only pairing but also scalar multiplication must be efficiently computable. In this paper, we propose a scalar multiplication method on the circumstances that we work at Ate pairing with Barreto-Naehrig (BN) curve. Note that the parameters of BN curve are given by a certain integer, namely mother parameter. Adhering the authors' previous policy that we execute scalar multiplication on subfield-twisted curve (E) over tilde (F-p2) instead of doing on the original curve E(F-p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) in E(Fp2). On BN curves, note (phi) over tilde is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs (phi) over tilde. In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.

DOI： 10.1587/transfun.E92.A.182

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Scalar Multiplication Using Frobenius Expansion over Twisted Elliptic Curve for Ate Pairing Based Cryptography

Yasuyuki Nogami, Yumi Sakemi, Takumi Okimoto, Kenta Nekado, Masataka Akane, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E92A ( 1 )   182 - 189   2009.1

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For ID-based cryptography, not only pairing but also scalar multiplication must be efficiently computable. In this paper, we propose a scalar multiplication method on the circumstances that we work at Ate pairing with Barreto-Naehrig (BN) curve. Note that the parameters of BN curve are given by a certain integer, namely mother parameter. Adhering the authors' previous policy that we execute scalar multiplication on subfield-twisted curve (E) over tilde (F-p2) instead of doing on the original curve E(F-p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) in E(Fp2). On BN curves, note (phi) over tilde is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs (phi) over tilde. In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.

DOI： 10.1587/transfun.E92.A.182

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How to Generate a Secure Composite Order Ordinary Pairing-friendly Curve of Embedding Degree 3

Y.Nogami, K.Nishii, Y.Sakemi, H.Kato, Y.Morikawa

ITC-CSCC2009   1474 - 1447   2009

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Thread Computing for Miller's algorithm of Pairing

Shoichi Takeuchi, Yumi Sakemi, Yasuyuki Nogami, Yoshitaka Morikawa

ISCE: 2009 IEEE 13TH INTERNATIONAL SYMPOSIUM ON CONSUMER ELECTRONICS, VOLS 1 AND 2   587 - 591   2009

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This paper shows an improvement of Miller's algorithm so as to be efficient for thread programming by using divisor theorem. Then, this paper implements a thread program of the improved Miller's algorithm by which it is shown that the proposed technique is more efficient than ordinary Miller's algorithm.

DOI： 10.1109/ISCE.2009.5156956

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Cost Evaluation of The Improvement of Twisted Ate Pairing That Uses Integer Variable X of Small Hamming Weight

Sakemi Yumi, Kato hidehiro, Nogami Yasuyuki, Morikawa Yoshikawa

Memoirs of the Faculty of Engineering, Okayama University   43 ( 15 )   113 - 116   2009

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A High-Speed Square Root Algorithm for Extension fields -Especially for Fast Extension Fields-

Kato Hidehiro, Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   43   99 - 107   2009

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Inversion with Normal Bases in Tower Field F_{((2^{2})^{2})^2} for S-Box of AES

Y. Nogami, M. Hagio(Oki Network LSI, E. Yanagi, Y. Morikawa

ITC-CSCC2009   1337 - 1338   2009

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Cross Twisted Xate Pairing with Barreto-Naehrig Curve for Multi-pairing Technique

Yumi Sakemi, Yasuyuki Nogami, Hidehiro Kato, Yoshitaka Morikawa

2009 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, VOLS 1- 4   2386 - 2390   2009

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Recently, efficient pairings such as optimal-ate, R-ate, and Xate pairings have been proposed. On the other hand, some pairing-based cryptographic protocols need several pairing calculations for one procedure. Thus, multi-pairing technique will be efficiently applied. This paper optimizes Xate pairing for multi-pairing and then evaluates its efficiency with comparing to the other efficient pairings. Then, it is shown that the multi-pairing with the optimized Xate pairing is the most efficient.

DOI： 10.1109/ISIT.2009.5205994

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Determining Basis Conversion Matrix without Gauss Period Normal Basis

Y.Nogami, E.Yanagi, M.Hagio, Oki Network LSI, Y.Morikawa

ITC-CSCC2009   1331 - 1332   2009

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Zero Correlation Distribution of ZCZ Sequences Obtained from a Perfect Sequence and a Unitary Matrix

Satoshi Uehara, Shuichi Jono, Yasuyuki Nogami

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E91A ( 12 )   3745 - 3748   2008.12

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A class of zero-correlation zone (ZCZ) sequences constructed by the recursive procedure from a perfect sequence and a unitary matrix was proposed by T, Nakamura, and Suehiro [1]. In the reference [1], three parameters, s.t., the Sequence length, the family size and the length of the ZCZ, were evaluated for a general estimate of the performance of the ZCZ sequences. In this letter, we give more detailed distributions of that correlation values are zero on their ZCZ sequence sets.

DOI： 10.1093/ietfec/e91-a.12.3745

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Efficient Exponentiation in Extensions of Finite Fields without Fast Frobenius Mappings

Yasuyuki Nogami, Hidehiro Kato, Kenta Nekado, Yoshitaka Morikawa

ETRI JOURNAL   30 ( 6 )   818 - 825   2008.12

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This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

DOI： 10.4218/etrij.08.0108.0178

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A method for constructing a self-dual normal basis in odd characteristic extension fields

Yasuyuki Nogami, Hiroaki Nasu, Yoshitaka Morikawa, Satoshi Uehara

FINITE FIELDS AND THEIR APPLICATIONS   14 ( 4 )   867 - 876   2008.11

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This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F-p(m) such that 4p does not divide m(p - 1) and m is odd. In detail, when the characteristic p and extension degree in satisfies the following conditions (1) and either (2a) or (2b); (1) 2km + 1 is a prime number, (2a) the order of p in F2km+ 1 is 2km, (2b) 2 dagger km and the order of p in F2km + 1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F-p(m). (C) 2008 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.ffa.2008.04.001

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A method for constructing a self-dual normal basis in odd characteristic extension fields

Yasuyuki Nogami, Hiroaki Nasu, Yoshitaka Morikawa, Satoshi Uehara

FINITE FIELDS AND THEIR APPLICATIONS   14 ( 4 )   867 - 876   2008.11

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This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F-p(m) such that 4p does not divide m(p - 1) and m is odd. In detail, when the characteristic p and extension degree in satisfies the following conditions (1) and either (2a) or (2b); (1) 2km + 1 is a prime number, (2a) the order of p in F2km+ 1 is 2km, (2b) 2 dagger km and the order of p in F2km + 1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F-p(m). (C) 2008 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.ffa.2008.04.001

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A Consideration on Efficient Exponentiation in Extension Field for Pairing-based Cryptography

YOSHIDA Tomoki, KATO Hidehiro, NEKADO Kenta, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   108 ( 162 )   101 - 107   2008.7

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In recent years, pairing-based cryptographies such as ID-base cryptography and group signature have been studied. These cryptographies require exponentiations in multiplicative group G_T. In this paper, the author proposes an efficient exponentiation method for the case of Barreto-Naehrig (BN) curve. Additionally, the author shows implementation results of the proposed method with conventional techniques such as binary method, window method, NAF method and Avanzi method. Then, the author shows the proposed method carries out exponentiation fast.

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A Consideration on Efficient Exponentiation in Extension Field for Pairing-based Cryptography

YOSHIDA Tomoki, KATO Hidehiro, NEKADO Kenta, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2008 ( 71 )   235 - 241   2008.7

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In recent years, pairing-based cryptographies [2] such as ID-base cryptography [3] and group signature [1]have been studied. These cryptographies require exponentiations in multiplicative group G_T In this paper, the author proposes an efficient exponentiation method for the case of Barreto-Naehrig (BN) curve [9] . Additionally, the author shows implementation results of the proposed method with conventional techniques such as binary method, window method, NAF method and Avanzi method [12] [15]. Then, the author shows the proposed method carries out exponentiation fast.

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Basis translation matrix between two isomorphic extension fields via optimal normal basis

Yasuyuki Nogami, Ryo Namba, Yoshitaka Morikawa

ETRI JOURNAL   30 ( 2 )   326 - 334   2008.4

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This paper proposes a method for generating a basis translation matrix between isomorphic extension fields. To generate a basis translation matrix, we need the equality correspondence of a basis between the isomorphic extension fields. Consider an extension field F-p(m) where p is characteristic. As a brute force method, when p(m) is small, we can check the equality correspondence by using the minimal polynomial of a basis element; however, when P, is large, it becomes too difficult. The proposed methods are based on the fact that Type I and Type H optimal normal bases (ONBs) can be easily identified in each isomorphic extension field. The proposed methods efficiently use Type I and Type II ONBs and can generate a pair of basis translation matrices within 15 ms on Pentium 4 (3.6 GHz) when mlog(2)p = 160.

DOI： 10.4218/etrij.08.0107.0182

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A necessary condition for Gauss period normal bases to be the same normal basis

Yasuyuki Nogami, Ryo Namba, Yoshitaka Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E91A ( 4 )   1229 - 1231   2008.4

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This paper shows a necessary condition for type-&lt; k, m &gt; and &lt; k', m &gt; Gauss period normal bases in F-p(m) to be the same normal basis by using their traces.

DOI： 10.1093/ietfec/e91-a.4.1229

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Extension Field for Ate Pairing with Freeman Curve

K.Nekado, H.Kato, M.Akane, Y.Nogami, Y.Morikawa

ITC-CSCC2008   653 - 656   2008

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Fast Exponentiation in Extension Field with Frobenius Mappings

Kato Hidehiro, Nekado Kenta, Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   42 ( 4 )   36 - 43   2008

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A Method for Checking the Parity of (#Jc - 1)=2 of Genus 2 and 3 Hyperelliptic Curves

Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   42 ( 14 )   110 - 114   2008

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Systematic Generation of An Irreducible Polynomial of An Arbitrary Degree m over F-p Such That p &gt; m

Hiroaki Nasu, Yasuyuki Nogami, Yoshitaka Morikawa, Shigeki Kobayashi, Tatsuo Sugimura

THIRD 2008 INTERNATIONAL CONFERENCE ON CONVERGENCE AND HYBRID INFORMATION TECHNOLOGY, VOL 2, PROCEEDINGS   2   478 - +   2008

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This paper proposes a method for generating an irreducible polynomial of an arbitrary degree m over an arbitrary prime field F, such that p &gt; m. The proposed method is closely related to the minimal polynomial determination and therefore it has the following features: its complexity has little dependency on the size of characteristic p, its calculation cost is explicitly given with degree m, and it can generate primitive polynomials when p(m) - 1 is factorized as the product of prime numbers. The restriction p &gt; m comes from using Newton's formula.

DOI： 10.1109/ICCIT.2008.171

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An Implementation of Anonymous IEEE802.1X Authentication System for Wireless Networks

A. Sudarsono, T. Nakanishi, Y. Nogami, N. Funabiki

Proc. the 10th Industrial Electronics Seminar 2008 (IES2008)   2008

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Skew Frobenius Map and Efficient Scalar Multiplication for Pairing-Based Cryptography

Yumi Sakemi, Yasuyuki Nogami, Katsuyuki Okeya, Hidehiro Kato, Yoshitaka Morikawa

CRYPTOLOGY AND NETWORK SECURITY   5339   226 - +   2008

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This paper considers a new skew Frobenius endomorphism with pairing-friendly elliptic curve E(F-p) defined over prime field F-p. Then, using the new skew Frobenius map, an efficient scalar multiplication method for pairing-friendly elliptic curve E(F-p) is shown. According to the simulation result, a scalar multiplication by the proposed method with multi-exponentiation technique is about 40% faster than that by plain binary method.

DOI： 10.1007/978-3-540-89641-8_16

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An Improvement of Twisted Ate Pairing with Barreto-Naehrig Curve by using Frobenius Mapping

Yumi Sakemi, Hidehiro Kato, Yasuyuki Nogami, Yoshitaka Morikawa

Third 2008 International Conference on Convergence and Hybrid Information Technology, Vol 2, Proceedings   2   406 - 410   2008

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This paper proposes an improvement of twisted-Ate pairing with Barreto-Naehrig curve so as to efficiently use Frobenius mapping with respect to prime field. Then, this paper shows some simulation results by which it is shown that the improvement accelerates twisted-Ate pairing.

DOI： 10.1109/ICCIT.2008.193

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Efficient Pairings on Twisted Elliptic Curve

Yasuyuki Nogami, Masataka Akane, Yumi Sakemi, Yoshitaka Morikawa

THIRD 2008 INTERNATIONAL CONFERENCE ON CONVERGENCE AND HYBRID INFORMATION TECHNOLOGY, VOL 2, PROCEEDINGS   2   430 - +   2008

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This paper proposes an efficient implementation of Ate pairing on twisted elliptic curve. Suppose that a pairing-friendly elliptic curve E has a twisted elliptic curve E' of degree d, and let psi(d) be an isomorphic map from E'(F-p(e)) to the corresponding subgroup of E(F-p(k)). Then, consider G' = psi(-1)(d) (G1) and G(2)' = psi(-1)(d) (G(2)) for G(1), G(2) at Ate pairing alpha. Let P is an element of G(1), Q is an element of G(2), P' is an element of G'(1) and Q' is an element of G'(2), the authors have shown alpha(Q,P) = F-t-q,F-Q (P)((pk-1)/r) = f(t-1,Q')(P')((pk-1)/r). This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.

DOI： 10.1109/ICCIT.2008.172

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An Improvement of Cyclic Vector Multiplication Algorithm

Hidehiro Kato, Yasuyuki Nogami, Tomoki Yoshida, Kenta Nekado, Shoichi Takeuchi, Yoshitaka Morikawa

Third 2008 International Conference on Convergence and Hybrid Information Technology, Vol 2, Proceedings   2   401 - 405   2008

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This paper first introduces cyclic vector multiplication algorithm (CVMA) that is a multiplication algorithm in extension field. Then, it is also introduced that CVMA is useful under the tight restrictions of pairing-based cryptographies. Then, this paper points out a problem about the calculation cost of CVMA. For this problem, this paper proposes an improvement. According to some simulation results, it is shown that the improvement makes CVMA much more efficient.

DOI： 10.1109/ICCIT.2008.166

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A method for constructing a pseudo self-dual normal basis

Hiroaki Nasu, Yasuyuki Nogami, Satoshi Uehara, Ryo Namba, Yoshitaka Morikawa

CYBERNETICS AND SYSTEMS   39 ( 6 )   563 - 582   2008

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Self-dual normal basis is efficient for the arithmetic operations in extension field and especially trace calculation. However, self-dual normal bases do not exist in [image omitted] when characteristic p is odd and degree m is even. This paper proposes a method to construct an efficient normal basis for trace calculation when extension degree is even. In this paper, we call it pseudo self-dual normal basis.

DOI： 10.1080/01969720802188201

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Generating Irreducible Self-reciprocal Polynomials by Using Even Polynomial over Fq

Shigeki Kobayashi, Yasuyuki Nogami, Tatsuo Sugimura

The 23rd International Technical Conference on Circuits/Systems, Computers and Communications   121 - 124   2008

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An improvement of twisted ate pairing using integer variable with small hamming weight

SAKEMI Y.

2008 Symposium on Cryptography and Information Security (SCIS2008), Jan.   269 - 272   2008

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Fast Squaring in TypeI All One Polynomial Field

Hidehiro Kato, Yasuyuki Nogami, Yoshitaka Morikawa

The 23rd International Technical Conference on Circuits/Systems, Computers and Communications   273 - 276   2008

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Integer Variable chi-Based Ate Pairing

Yasuyuki Nogami, Masataka Akane, Yumi Sakemi, Hidehiro Kato, Yoshitaka Morikawa

PAIRING-BASED CRYPTOGRAPHY - PAIRING 2008   5209   178 - +   2008

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In implementing an efficient pairing calculation, it is said that the lower bound of the number of iterations of Miller's algorithm is log(2)r/phi(k), where phi(.) is the Euler's function. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from left perpendicularlog(2)rright perpendicular to left perpendicularlog(2)(t - 1)right perpendicular. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "chi". For the curve, this paper gives integer variable chi-based Ate pairing that achieves the lower bound by reducing it to left perpendicularlog(2) chi right perpendicular.

DOI： 10.1007/978-3-540-85538-5_13

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Basis translation matrix between two isomorphic extension fields via optimal normal basis

Yasuyuki Nogami, Ryo Namba, Yoshitaka Morikawa

ETRI Journal   30 ( 2 )   326 - 334   2008

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This paper proposes a method for generating a basis translation matrix between isomorphic extension fields. To generate a basis translation matrix, we need the equality correspondence of a basis between the isomorphic extension fields. Consider an extension field Fpm where p is characteristic. As a brute force method, when pm is small, we can check the equality correspondence by using the minimal polynomial of a basis element
however, when pm is large, it becomes too difficult The proposed methods are based on the fact that Type I and Type II optimal normal bases (ONBs) can be easily identified in each isomorphic extension field. The proposed methods efficiently use Type I and Type II ONBs and can generate a pair of basis translation matrices within 15 ms on Pentium 4 (3.6 GHz) when mlog2p = 160.

DOI： 10.4218/etrij.08.0107.0182

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Cyclic vector multiplication algorithm based on a special class of Gauss period normal basis

Hidehiro Kato, Yasuyuki Nogami, Tomoki Yoshida, Yoshitaka Morikawa

ETRI JOURNAL   29 ( 6 )   769 - 778   2007.12

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This paper proposes a multiplication algorithm for F-pm, which can be efficiently applied to many pairs of characteristic p and extension degree m except for the case that 8p divides m(p-1). It uses a special class of type-&lt; k, m &gt; Gauss period normal bases. This algorithm has several advantages: it is easily parallelized; Frobenius mapping is easily carried out since its basis is a normal basis; its calculation cost is clearly given; and it is sufficiently practical and useful when parameters k and m are small.

DOI： 10.4218/etrij.07.0107.0040

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An Implementation of a Pairing-Based Group Signature Scheme with Verifier-Local Revocation

HIRA Yuta, KATO Hidehiro, NAKANISHI Toru, NOGAMI Yasuyuki, FUNABIKI Nobuo, MORIKAWA Yoshitaka

IEICE technical report   107 ( 209 )   69 - 76   2007.8

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Group signature scheme is one of signature schemes which keep signer's identity information secret. A pairing-based group signature scheme which can make a signer sign without any revocation information and has the short length of the signature was proposed. However this scheme has a problem that the anonymity cannot be proved in case of using the twisted pairing which is known for an efficient implementation of pairing. Then, in this paper, we extend the previous scheme to have the provable anonymity in case of using twisted pairing. In addition, we efficiently implemented the extended scheme by transforming some equations and introducing parallel processing, and then we show that the extended scheme can be executed for a practical time on a server with multi-core CPU even if about 500 members are revoked.

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Constructing Self-Dual Normal Bases in Odd-Degree Extension Field

SAKEMI Yumi, NASU Hiroaki, NANBA Ryo, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   107 ( 143 )   23 - 27   2007.7

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A self-dual normal basis only exists in odd-degree extension field when characteristic p is an odd prime. However, a method to construct it systematically has not been given. In this paper, we show a method for constructing a self-dual normal basis in an arbitrary extension field F_<p^m> such that 4p does not divide m(p-1) and m is odd by using a special class of Gauss period normal bases. Then, it is also shown that the vector representations with the self-dual and Gauss period normal bases are easily translated to each other.

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Normal Basis Efficient for Trace Calculation in Odd Characteristic Extension Field of Even Degree

NEKADO Kenta, NASU Hiroaki, NANBA Ryo, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   107 ( 143 )   19 - 22   2007.7

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Language：Japanese   Publisher：The Institute of Electronics, Information and Communication Engineers  

Some public-key cryptographies require an extension field as the definition field. When we implement arithmetic operations in the extension field, we need to prepare an efficient basis. In this paper, we deal with normal bases efficient for trace calculations. First, we go over that there are no self-dual normal bases [2]-[4] in odd characteristic extension fields of even degree. Then, we introduce a normal basis in odd characteristic extension field of degree 2 as efficient as self-dual normal basis.

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自己相反逆変換を用いたF2上の高次既約多項式の生成法

小林茂樹, 野上保之, 杉村立夫, 難波諒

電子情報通信学会論文誌A   J90-A ( 5 )   460 - 469   2007

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A Multiplication Algorithm in F_{p^m} for An Arbitrary Pair of The Characteristic p and Degree m Such That p>m,

Hidehiro Kato, Yasuyuki Nogami, Yoshitaka Morikawa, Tomoki Yoshida

ETRI journal   採録済み   2007

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An Algorithm for Generating Irreducible Cubic Trinomials over Prime Field

Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   41 ( 1 )   11 - 19   2007

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The Number of the Irreducible Cubic Polynomials in the Form of x(3) + ax + b with a Certain Fixed Element a

Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   41 ( 1 )   1 - 10   2007

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A method for constructing an efficient basis for trace calculation

Hiroaki Nasu, Yasuyuki Nogami, Ryo Namba, Yoshitaka Morikawa

2007 International Conference on Convergence Information Technology, ICCIT 2007   229 - 234   2007

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Self-dual normal basis is efficient for the arithmetic operations in extension field and especially trace calculation. However, self-dual normal bases do not exist in Fpm when characteristic p is odd and degree m is even. This paper proposes a method to construct a normal basis of even degree that is efficient for trace calculation. © 2007 IEEE.

DOI： 10.1109/ICCIT.2007.4420265

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A multiplication algorithm in F_<p^m> for arbitrary pairs of characteristic p and degree m such that p>m

YOSHIDA T.

Joint Workshop on Information Security, 2007   469 - 483   2007

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A method for distinguishing the two candidate elliptic curves in the complex multiplication method

Yasuyuki Nogami, Mayumi Obara, Yoshitaka Morikawa

ETRI JOURNAL   28 ( 6 )   745 - 760   2006.12

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In this paper, we particularly deal with no F-p-rational two-torsion elliptic curves, where F-p is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E' - 1)/2 are reciprocal to each other, where #E and #E' are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

DOI： 10.4218/etrij.06.0106.0059

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Extension Fields Efficient for Pairing Operation

OBARA Mayumi, AKANE Masataka, OKIMOTO Takumi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

29 ( 1 )   391 - 394   2006.11

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Representation and multiplication of finite field that used TypeII ONB like normal basis

KOBAYASHI Shigeki, NOGAMI Yasuyuki, SUGIMURA Tatsuo

29 ( 1 )   169 - 172   2006.11

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An extension field for variable extension degree and characteristic

NAMBA Ryo, KATOU Hidehiro, YOSHIDA Tomoki, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

29 ( 1 )   395 - 398   2006.11

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Square Root Calculation Algorithm over Type-II All One Polynomial Field

KATOU Hidehiro, WANG FENG, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2006 ( 81 )   7 - 12   2006.7

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The authors proposes a square root (SQRT) algorithm in F_p^m (m = r_0r_1…r_<n-1>2^d,r_2: odd prime, d>0: integer). We compute the inverse SQRT in F_p^2^d using MW-ST algorithm. Then the Frobenius mappings with an addition chain are adopted for this SQRT algorithm, in which a lot of computations in a given extension field F_p^m are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increases efficiency in the SQRT implementation. More specifically the Smart algorithm and proposed algorithm in F_p^6, for example, were implemented on a Pentium4 (3.8FHz) computer using the C++ programming language and NTL Library. The computer simulations showed that, on average, the proposed algoithm accelerates the SQRT computation by 3 times in F_p^6, compared to the Smart algorithm.

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Other Link： http://id.nii.ac.jp/1001/00044566/

Pairing Computation with MNT Curve over All One Polynomial Field

AKANE Masataka, OKIMOTO Takumi, NOGAMI Ysuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2006 ( 81 )   13 - 18   2006.7

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In recent years, many cryptographic applications with bilinear-pairing over elliptic curves have been proposed. The well-known MNT curves, that are non-supersingular elliptic curves, provide bilinear-pairings over extension fields of degree 3, 4, and 6. When the embedding degree is equal to 3, MNT curves cannot be defined over optimal extension field (OEF). Even when the embedding degree is equal to 4 or 6, MNT curves cannot be always defined over OEF. For some of such cases, it can be defined over all one polynomial field (AOPF). Since Frobenius mapping can be fast carried out in the AOPFs, this paper gives considered some improvements for Tate pairing calculation. Then, some examples and simulation results are shown.

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Other Link： http://id.nii.ac.jp/1001/00044567/

A Translation Matrix between Two Isomorphic Extension Fields via Optimal Normal Basis Representation

NAMBA Ryo, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2006 ( 81 )   1 - 6   2006.7

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Some extension fields efficient for fast implementation have been proposed. Such extension fields adopt unique modular polynomial and basis. Therefore, an element can have some different vector representations in the isomorphic extension fields. This paper proposes a method for generating a basis translation matrix between two isomorphic extension fields. First, this paper shows that the translation matrix can be obtained via TypeI Optimal Normal Basis (ONB). TypeI ONB plays key role since it has the following properties; TypeI ONB is a set of conjugate elements and of course a normal basis, these conjugates have the same order, they are zeros of a certain irreducible all one polynomial. Then, some examples of translation matrix are shown. From the experimental result, it is shown that the proposed method is enough practical.

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Other Link： http://id.nii.ac.jp/1001/00044565/

Square Root Calculation Algorithm over Type-II All One Polynomial Field

KATOU Hidehiro, WANG FENG, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report   106 ( 174 )   7 - 12   2006.7

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The authors proposes a square root (SQRT) algorithm in F_<p^m> (m=r_0r_1…r_<n-1>2^d, r_i: odd prime, d>0: integer). We compute the inverse SQRT in F_<p^2^d> using MW-ST algorithm. Then the Frobenius mappings with an addition chain are adopted for this SQRT algorithm, in which a lot of computations in a given extension field F_<p^m> are also reduce to those in a proper sub field by the norm computations. Those reductions of the field degree increases efficiency in the SQRT implementation. More specifically the Smart algorithm and proposed algorithm in F_<p^6>, for example, were implemented on a Pentium4 (3.8FHz) computer using the C++ programming language and NTL Library. The computer simulations showed that, on average, the proposed algoithm accelerates the SQRT computation by 3 times in F_<p^6>, compared to the Smart algorithm.

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A high-speed square root algorithm in extension fields

Hidehiro Katou, Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

INFORMATION SECURITY AND CRYPTOLOGY - ICISC 2006, PROCEEDINGS   4296   94 - +   2006

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A square root (SQRT) algorithm in GF(p(m)) (m = r(0)r(1)center dot center dot center dot r(n-1)2(d), r(i): odd prime, d &gt; 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF (p(2d)), where most of the computations are performed in the corresponding subfields GF(p(2d)) for 0 &lt;= i &lt;= d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p(m)) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p(22)), GF(P-44) and GF(P-88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF (P-22), by 45 times in GF (P-44), and by 70 times in GF(p(88)), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

DOI： 10.1007/11927587_10

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A high-speed square root algorithm in extension fields

Hidehiro Katou, Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

INFORMATION SECURITY AND CRYPTOLOGY - ICISC 2006, PROCEEDINGS   4296   94 - +   2006

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A square root (SQRT) algorithm in GF(p(m)) (m = r(0)r(1)center dot center dot center dot r(n-1)2(d), r(i): odd prime, d &gt; 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF (p(2d)), where most of the computations are performed in the corresponding subfields GF(p(2d)) for 0 &lt;= i &lt;= d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p(m)) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p(22)), GF(P-44) and GF(P-88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF (P-22), by 45 times in GF (P-44), and by 70 times in GF(p(88)), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

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A Method for Checking the Parity of (#Jc-1)/2

M.Akane, Y.Nogami, Y.Morikawa

The 2006 International Symposium on Information Theory and its Applications   2006

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The Orders of Elliptic Curves y(2) = x(3) + b, b ∈ F(* q)

Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   40 ( 1 )   83 - 94   2006

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Cyclic vector multilication algorithm makes an inversion in F_{p^3} fastest

Yasuyuki Nogami, Hidehiro Katou, Yoshitaka Morik

JWIS2006(Joint workshop on information security)   2006

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Cyclic Vector Multiplication is Efficient for Small Extension Degrees

Hidehiro Katou, Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

Second International Congress on Mathematical Software ICMS2006   2006

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A Consideration on Cyclic Vector Multiplication Algorithm

KATOU Hidehiro, NOGAMI Yasuyuki, NANBA Ryou, MORIKAWA Yoshitaka

28 ( 1 )   295 - 298   2005.11

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A Consideration on the Order of Genus 2 Hyperelliptic Curve

NOGAMI Yasuyuki, MORIKAWA Yoshitaka

28 ( 2 )   889 - 892   2005.11

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An efficient square root computation in finite fields GF(p(2d))

F Wang, Y Nogami, Y Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E88A ( 10 )   2792 - 2799   2005.10

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This paper focuses on developing a square root (SQRT) algorithm in finite fields GF(p(2d)) (d &gt;= 0). Examining the Smart algorithm, a well-known SQRT algorithm, we can see that there is some computation overlap between the Smart algorithm and the quadratic residue (QR) test, which must be implemented before a SQRT computation. It makes the Smart algorithm inefficient. In this paper, we propose a new QR test and a new SQRT algorithm in GF(p(2d)), in which not only there is no computation overlap, but also most of computations required for the proposed SQRT algorithm in GF(p2d) can be implemented in the corresponding subfields GF(p(2d-i)) for 1 &lt;= i &lt;= d, which yields many reductions in the computational time and complexity. The computer simulation also shows that the proposed SQRT algorithm is much faster than the Smart algorithm.

DOI： 10.1093/ietfec/e88-a.10.2792

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The Representation of typeII ONB by self-reciprocal transformation

KOBAYASHI Shigeki, NOGAMI Yasuyuki, SUGIMURA Tatsuo

IEICE technical report. Information theory   105 ( 311 )   25 - 30   2005.9

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This paper make relation of All One Polynomials and Irreducible Polynomials with TypeIIONB clear, by using polynomial conversion technique from polynomial of degree m to polynomial of degree 2m called self-reciprocal transformation and polynomial conversion technique from polynomial of degree 2m to polynomial of degree m called self-reciprocal reverse transformation. If irreducible polynomial of degree m become All One Polynomial by self-reciprocal traneformation, we show that original irreducible polynomial of degree m is irreducible polynomial with TypeIIONB. When there is TypeIIONB, we show that we can find irreducible polynomial with TypeIIONB by doing self-reciprocal reverse tranceformation for All One Polynomial. As application, we give technique to make an irreducible polynomial of prime degree.

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Performance of Prime Order Elliptic Curve Generation based on y-twist

OBARA Mayumi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

Technical report of IEICE. ISEC   105 ( 193 )   59 - 66   2005.7

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This paper proposes a new twist technique and then shows some necessary conditions for prime order curves in the form y^2=x^3+a. Then, by combining x-twist and y-twist, we consider six elliptic curves. For these six elliptic curves, when the characteristic p of the definition field F_q satisfies that p>3 and 3|(p-1), we show that it is possible for only two elliptic curves among the six curves defined over F_q, q=p^<2^i3^j> to have prime orders, where i, j are non-negative integers. Then, we show an example of prime order curve. After that, compared to the complex multiplication method, we evaluate the performance of the proposed method.

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Fast implementation of extension fields with TypeII ONB and cyclic vector multiplication algorithm

Y Nogami, S Shinonaga, Y Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E88A ( 5 )   1200 - 1208   2005.5

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This paper proposes an extension field named TypeII AOPF. This extension field adopts TypeII optimal normal basis, cyclic vector multiplication algorithm, and Itoh-Tsujii inversion algorithm. The calculation costs for a multiplication and inversion in this field is clearly given with the extension degree. For example, the arithmetic operations in TypeII AOPF F-p5 is about 20% faster than those in OEF F-p5. Then, since CVMA is suitable for parallel processing, we show that TypeII AOPF is superior to AOPF as to parallel processing and then show that a multiplication in TypeII AOPF becomes about twice faster by parallelizing the CVMA computation in TypeII AOPF.

DOI： 10.1093/ietfec/e88-a.5.1200

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XTRを用いた暗号とその高速実装 (符号と暗号の代数的数理)

野上 保之

数理解析研究所講究録   1420   183 - 192   2005.4

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Other Link： http://hdl.handle.net/2433/47175

An algorithm for systematically generating irreducible cubic trinomials over prime field

Yasuyuki Nogami, Yoshitaka Morikawa

Proceeding of The 2005 International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC2005)   2005

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Generating prime degree irreducible polynomials by using irreducible all-one polynomial over F-2

K Makita, Y Nogami, T Sugimura

ELECTRONICS AND COMMUNICATIONS IN JAPAN PART III-FUNDAMENTAL ELECTRONIC SCIENCE   88 ( 7 )   23 - 32   2005

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In most of the methods of public key cryptography devised in recent years, a finite field of a large order is used as the field of definition. In contrast, there are many studies in which a higher-degree extension field of characteristic 2 is fast implemented for easier hardware realization. There are also many reports of the generation of the required higher-degree irreducible polynomial, and of the construction of a basis suited to fast implementation, such as an optimal normal basis (ONB). For generating higher-degree irreducible polynomials, there is a method in which it 2m-th degree self-reciprocal irreducible polynomial is generated from an m-th degree irreducible polynomial by a simple polynomial transformation (called the self-reciprocal transformation). This paper considers this transformation and shows that When the set of zeros of the m-th degree irreducible polynomial forms a normal basis, the set of zeros of the generated 2m-th order self-reciprocal irreducible polynomial also forms a normal base. Then it is clearly shown that there is a one-to-one correspondence between the transformed irreducible polynomial and the generated self-reciprocal irreducible polynomial. Consequently, the inverse transformation of the self-reciprocal transformation (self-reciprocal inverse transformation) can be applied to a self-reciprocal irreducible polynomial. It is shown that an m-th degree irreducible polynomial can always be generated from a 2m-th degree self-reciprocal irreducible polynomial by the self-reciprocal inverse transformation. We can use this fact for generating 1/2-degree irreducible polynomials. As an application of 1/2-degree irreducible polynomial generation, this paper proposes a method which generates a prime degree irreducible polynomial with a Type II ONB as its zeros. (c) 2005 Wiley Periodicals, Inc.

DOI： 10.1002/ecjc.20151

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Generating prime degree irreducible polynomials by using irreducible all-one polynomial over F-2

K Makita, Y Nogami, T Sugimura

ELECTRONICS AND COMMUNICATIONS IN JAPAN PART III-FUNDAMENTAL ELECTRONIC SCIENCE   88 ( 7 )   23 - 32   2005

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Language：English   Publisher：SCRIPTA TECHNICA-JOHN WILEY & SONS  

In most of the methods of public key cryptography devised in recent years, a finite field of a large order is used as the field of definition. In contrast, there are many studies in which a higher-degree extension field of characteristic 2 is fast implemented for easier hardware realization. There are also many reports of the generation of the required higher-degree irreducible polynomial, and of the construction of a basis suited to fast implementation, such as an optimal normal basis (ONB). For generating higher-degree irreducible polynomials, there is a method in which it 2m-th degree self-reciprocal irreducible polynomial is generated from an m-th degree irreducible polynomial by a simple polynomial transformation (called the self-reciprocal transformation). This paper considers this transformation and shows that When the set of zeros of the m-th degree irreducible polynomial forms a normal basis, the set of zeros of the generated 2m-th order self-reciprocal irreducible polynomial also forms a normal base. Then it is clearly shown that there is a one-to-one correspondence between the transformed irreducible polynomial and the generated self-reciprocal irreducible polynomial. Consequently, the inverse transformation of the self-reciprocal transformation (self-reciprocal inverse transformation) can be applied to a self-reciprocal irreducible polynomial. It is shown that an m-th degree irreducible polynomial can always be generated from a 2m-th degree self-reciprocal irreducible polynomial by the self-reciprocal inverse transformation. We can use this fact for generating 1/2-degree irreducible polynomials. As an application of 1/2-degree irreducible polynomial generation, this paper proposes a method which generates a prime degree irreducible polynomial with a Type II ONB as its zeros. (c) 2005 Wiley Periodicals, Inc.

DOI： 10.1002/ecjc.20151

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A Method for Generating Prime Order Elliptic Curves over F(q(2c))

Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   39 ( 1 )   71 - 81   2005

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A High-Speed Square Root Computation in Finite Fields with Application to Elliptic Curve Cryptosystem

Wang Feng, Nogami Yasuyuki, Morikawa Yoshitaka

Memoirs of the Faculty of Engineering, Okayama University   39 ( 1 )   82 - 92   2005

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A Twist Technique Based on Third Power Residue and Non Residue

HIDAKA Yoshito, OBARA Mayumi, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Office Information Systems   104 ( 423 )   1 - 6   2004.11

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In this paper, we propose a twist technique based on third power residue and non residue. In addition, with the conventional twist technique, we consider the six curves. We show a relation of these curves and the conditions such that they are not super-singular curves.

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A Fast Square Root Computation in Some Finite Fields

WANG Feng, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2004 ( 75 )   7 - 13   2004.7

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It is well known that quadratic residue (QR) test should be implemented in advance of a square root (SQRT) computation. Smart algorithm, the previously known fastest algorithm for SQRT computation, only got the idea how to compute SQRT through QR test. However there is a lot of computation overlap in QR test and Smart algorithm. The essence of our proposition is thus to present a new QR test and SQRT algorithm to avoid all the overlapping computations. In this paper the authors devised a SQRT algorithm for which most of the data required have been computed in QR test. This yields many reductions in the computational time and amount. In GF(p) and GF(p^2), we implemented Smart algorithm and the proposed algorithm in C++ language on Pentium4 (2.6GHz), where p=2^<16>+1 (4|p-1) and p=2^<16>+3 (4〓p-1). The computer simulations showed that for p=2^16+1 the proposed algorithm on average accelerates the SQRT computation 2 times and 10 times faster than Smart algorithm over GF(p) and GF(p^2), respectively and for p=2^<16>+3 the proposed algorithm on average accelerates the SQRT computation 20 times and 6 times faster than Smart algorithm over GF(p) and GF(p^2), respectively.

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Other Link： http://id.nii.ac.jp/1001/00044873/

A Fast Square Root Computation in Some Finite Fields

WANG Feng, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

Technical report of IEICE. ISEC   104 ( 199 )   7 - 13   2004.7

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Language：English   Publisher：The Institute of Electronics, Information and Communication Engineers  

It is well known that quadratic residue (QR) test should be implemented in advance of a square root (SQRT) computation. Smart algorithm, the previously known fastest algorithm for SQRT computation, only got the idea how to compute SQRT through QR test. However there is a lot of computation overlap in QR test and Smart algorithm. The essence of our proposition is thus to present a new QR test and SQRT algorithm to avoid all the overlapping computations. In this paper the authors devised a SQRT algorithm for which most of the data required have been computed in QR test. This yields many reductions in the computational time and amount. In GF(p) and GF(p^2), we implemented Smart algorithm and the proposed algorithm in C++ language on Pentium4 (2.6GHz), where p=2^<16>+1(4|p-1) and p = 2^<16>+3(4&nmid;p-1). The computer simulations showed that for p=2^<16>+1 the proposed algorithm on average accelerates the SQRT computation 2 times and 10 times faster than Smart algorithm over GF(p) and GF(p^2), respectively and for p=2^<16>+3 the proposed algorithm on average accelerates the SQRT computation 20 times and 6 times faster than Smart algorithm over GF(p) and GF(p^2), respectively.

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A Classification of Irreducible Cubic Polynomials over Prime Field

Y.Nogami, Y.Morikawa

Proc. of The 2005 International Technical Conference on Circuits/Systems, Computers and Communications 2004 (CD-ROM)   2004

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The number of xs such that x^2+u u\in F_p^* becomes a quadratic power residue in F_p

W.Feng, Y.Nogami, Y.Morikawa

Proc. of Proceeding of The 2005 International Technical Conference on Circuits/Systems, Computers and Communications 2004 (CD-ROM)   2004

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F_2上の既約 All One Polynomial を用いた素数次の既約多項式の組織的な生成法

牧田 慶, 野上 保之, 杉村 立夫

電子情報通信学会論文誌（A)   2004

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A method for distinguishing the two candidate elliptic curves in CM method

Y Nogami, Y Morikawa

INFORMATION SECURITY AND CRYPTOLOGY - ICISC 2004   3506   249 - 260   2004

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In this paper, we first introduce a shift product-based polynomial transformation. Then, we show that the parities of (#E - 1)/2 and (#E' - 1)/2 are reciprocal to each other, where #E and #E' are the orders of the two candidate curves obtained at the last step of CM method algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transformation. For a 160-bits prime number as the characteristic, the proposed method carries out the parity check about 20 times faster than the conventional method when 4 divides the characteristic minus 1.

DOI： 10.1007/11496618_19

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The parity of (#E-1)/2

Y.Nogami, Y.Morikawa

Proc. of The 2004 International Symposium on Information Theory and Its Application (ISITA2004) CD-ROM   2004

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A Relation between Irreducible Cubic Polynomials and the Number of Solutions on y^2=x^2+a, a∈F_p

NAKASHIMA Jun, OKAMURA Masato, NOGAMI Yasuyuki, MORIKAWA Yoshiktaka

Technical report of IEICE. ISEC   103 ( 315 )   89 - 94   2003.9

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We have researched on the systematic generation of irreducible cubic polynomials for use in elliptic curve cryptosystem. But we couldn't make it clear that the number of irreducible cubic polynomials whose coefficient of degree 1 is a quadratic ersidue. In this paper, we show that this number is given with the number of quadratic residues in the set {X^2+3|X&isins;F_p}, then we give the number of quadratic residues in the set.

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A Fast Implementation of Fp^<6m> for XTR

SHINONAGA Shigeru, FUJII Yoshihiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

Technical report of IEICE. ISEC   103 ( 315 )   81 - 88   2003.9

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XTR (Efficient and Compact Subgroup Trace Representation) is a method to efficiently represent elements ina certain multiplicative subgroup of F_<p^6> as elements inF_<p^2> with using the trace function, where p is the characteristic. By applying the XTR onto a certain cryptosystem that uses some multiplicative group in a finite field, we can fast implement the calculations needed for its encryption and decryption processes, moreover the cryptosystem requires a shorter key length for security. And, the generalized XTR which is defined over F_<p^<6m>> has been also proposed. To achieve a fast implementation of the definition field of the cryptosystem using the XTR technique, this paper proposes a method to fast implement an extension field using the successive extension technique.

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Finite extension field with modulus of all-one polynomial and representation of its elements for fast arithmetic operations

Y Nogami, A Saito, Y Morikawa

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES   E86A ( 9 )   2376 - 2387   2003.9

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In many cryptographic applications, a large-order finite field is used as a definition field, and accordingly, many researches on a fast implementation of such a large-order extension field are reported. This paper proposes a definition field F(p)m with its characteristic p a pseudo Mersenne number, the modular polynomial f(x) an irreducible all-one polynomial (AOP), and using a suitable basis. In this paper, we refer to this extension field as an all-one polynomial field (AOPF) and to its basis as pseudo polynomial basis (PPB). Among basic arithmetic operations in AOPF, a multiplication between non-zero elements and an inversion of a non-zero element are especially time-consuming. As a fast realization of the former, we propose cyclic vector multiplication algorithm (CVMA), which can be used for possible extension degree m and exploit a symmetric structure of multiplicands in order to reduce the number of operations. Accordingly, CVMA attains a 50% reduction of the number of scalar multiplications as compared to the usually adopted vector multiplication procedure. For fast realization of inversion, we use the Itoh-Tsujii algorithm (ITA) accompanied with Frobenius mapping (FM). Since this paper adopts the PPB, FM can be performed without any calculations. In addition to this feature, ITA over AOPF can be composed with self reciprocal vectors, and by using CVMA this fact can also save computation cost for inversion.

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Other Link： https://dblp.uni-trier.de/db/journals/ieiceta/ieiceta86.html#NogamiSM03

Fast generation of elliptic curves with prime order over extension field of even extension degree

Y Nogami, Y Morikawa

2003 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY - PROCEEDINGS   18 - 18   2003

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DOI： 10.1109/isit.2003.1228032

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大学における研究活動と特許

野上保之

日本弁理士会論文誌パテント   2003

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A Fast Implementation of Elliptic Curve Cryptosystem with Prime Order Defined over F(p8)

NOGAMI Y.

Memoirs of the Faculty of Engineering, Okayama University   37 ( 2 )   73 - 87   2003

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Fast Generation of Elliptic Curves with Prime Order over F_{p^{2^c } }

Y.Nogami, Y.Morikawa

Proceeding of The International Workshop on Coding and Cryptography (WCC2003)   2003

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A Fast Square Root Calculation for Elliptic Curve Cryptosystem

Y.Nogami, Y.Morikawa

Proceeding of The 2003 International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC2003)   2003

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A fast square root computation using the Frobenius mapping

W Feng, Y Nogami, Y Morikawa

INFORMATION AND COMMUNICATIONS SECURITY, PROCEEDINGS   2836   1 - 10   2003

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The objective of this paper is to give a fast square root computation method. First the Frobenius mapping is adopted. Then a lot of calculations over an extension field are reduced to that over a proper subfield by the norm computation. In addition a inverse square root algorithm and an addition chain are adopted to save the computation cost. All of the above-mentioned steps have been proven to make the proposed algorithm much faster than the conventional algorithm. From the table which compares the computation between the conventional and the proposed algorithm, it is clearly shown that the proposed algorithm accelerates the square root computation 10 times and 20 times faster than the conventional algorithm in F(p)11 and F(p)22 respectively. At the same time, the proposed algorithm reduces the computation cost 10 times and 20 times less than the conventional algorithm.

DOI： 10.1007/978-3-540-39927-8_1

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Other Link： https://dblp.uni-trier.de/db/conf/icics/icics2003.html#FengNM03

A Successive Extension of Degree 2 for Fast Frobenius Map

FUJII Yoshihiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Information theory   102 ( 331 )   37 - 42   2002.9

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Elliptic Curve Cryptosystem has attracted much attentions as a public key cryptosystem in the next generation and an extension field has been uesd as the definition field. When the degree of an extension field is the composition number, fundamental arithmetics in the field can be fast implemented by using successive extensions. On the other hand, a fast implementation of Frobenius Map in such an extension field has not been explicitly presented. In this paper, two successive extension methods, in which a binomial or a trinomial is used as the modular polynomial, are introduced, and then, we evaluate the calculation costs of the Frobenius Maps implemented by these two methods. Finally, it is clearly shown that the method using a binomial as the modular polynomial is suitable for a fast Frobenius Map implementation.

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A Systematic Generation of Irreducible Polynomials of Degree 3 for Use in Elliptic Curve Cryptosystem

OKAMURA Masato, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Information theory   102 ( 331 )   49 - 54   2002.9

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The order of elliptic curve which is adopted in the elliptic curve cryptosystem is prefered to be a prime except for the characteristic of the definition field. Accordingly, some algorithms for generating such a prime-order elliptic curve has been proposed. Among these algorithms, there is an algorithm using irreducible polynomials of degree 3 effectively in order to generate no-two torsion elliptic curves in which all prime-order elliptic curves are contained. In this paper, let us consider that an irreducible polynomial of degree 3 is given, an algorithm to generate a lot of irreducible polynomials of degree 3 is proposed by use of a relation between the zeros and the coefficients of the given irreducible polynomial. And then, it is experimentally shown that the preceding algorithm can quickly generate prime order elliptic curves by using this algorithm and some properties between an elliptic curve and its twisted version. In the case of the characteristic p = 2^<24> - 3, for example, the number of possible prime-order elliptic curves for this algorithm is 164, and all kinds of these prime-order elliptic curves are generated within 3 minutes by using 10 processors, such as Pentium4 (1.7GHz).

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A Fast Implementation of Elliptic Curve Addition over F_<p^8> on 16-bits Microcontroller

IKEDA Takaaki, FUJII Yoshihiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Information theory   102 ( 331 )   43 - 48   2002.9

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Elliptic Curve Cryptosystem(ECC) has been received much attention, and implemented on a terminal even with scarce computation resources. In this paper, by using a successive extension field as the definition field of ECC, it is shown that a scalar multiplication, which is needed in the encryption/decryption process, can be fast implemented. And, we generate an elliptic curve with a prime order defined over a successive extension field of degree 8, and then a result of an implementation of Elgamal Elliptic Curve Cryptosystem on micro-controller is reported.

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Watermarking Method Using PN Diffusion and 1D-Fourier Transform of Multi-Resolution Approximation for Image

TSUGE Koichiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IPSJ SIG Notes   2001 ( 121 )   49 - 54   2001.12

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In this paper, we propose image digital watermark method which has robustness against image compression and clipping. In this method, we first at first take discrete wavelet transform of original image, and get out multi-resolution approximation (MRA). Next, we spread spectrum of the MRA by Pseudo Noise (PN) sequence, and then take 1D-discrete Fourier transform (DFT) of each column. We vary the DFT coefficients according to each bit of watermark information. This method has the features which the third eavesdropper, who don't know the PN sequence, can not decipher the information data and alter them. IN detection process, we take the DFT difference between PN-spread MRA of original image and that of watermark-embedded image and calculate shift of the DFT coefficients. Therefore, this method has the advantage that we can detect the watermark information even from the clipping image.

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Other Link： http://id.nii.ac.jp/1001/00041426/

Watermarking Method Using PN Diffusion and 1D-Fourier Transform of Multi-Resolution Approximation for Image

TSUGE Koichiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Communication systems   101 ( 494 )   49 - 54   2001.12

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In this paper, we propose image digital watermark method which has robustness against image compression and clipping. In this method, we first at first take discrete wavelet transform of original image, and get out multi-resolution approximation(MRA). Next, we spread spectrum of the MRA by Pseudo Noise(PN)sequence, and then take 1D-discrete Fourier transform(DFT)of each column. We vary the DFT coefficients according to each bit of watermark information. This method has the features which the third eavesdropper, who don't know the PN sequence, can not decipher the information data and alter them. In detection process, we take the DFT difference between PN-spread MRA of original image and that of watermark-embedded image and calculate shift of the DFT coefficients. Therefore, this method has the advantage that we can detect the watermark information even from the clipping image.

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An Image Watermarking Method Using 1-D DFT Phase of Multi-Resolution Approximation

TSUGE Koichiro, NOGAMI Yasuyuki, MORIKAWA Yoshitaka

24 ( 1 )   215 - 218   2001.12

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An Implementation of Elliptic Curve Cryptosystem Using An Extended AOPE

HIRAMOTO T., SAITO A., NOGAMI Y., MORIKAWA Y.

24 ( 1 )   351 - 354   2001.12

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Conditions of Characteristic and Trace for Rank One Elliptic Curve Twisted over F_<p^2^m>

DANNO T., NOGAMI Y., MORIKAWA Y.

SITA2001   24 ( 1 )   355 - 358   2001.12

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A Consideration on Elliptic Curve Cryptosystem Using Irreducible Polynomial of Degree 3

NOGAMI Yasuyuki, MORIKAWA Yoshitaka

IEICE technical report. Information theory   101 ( 407 )   7 - 12   2001.11

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For realization of the elliptic curve cryptosystem(ECC), enough considerations on its cryptological strength and implementation must be done. On cryptological strength, the elliptic curve order must have a prime factor of about 160-bits to be enough safe. In general, the ECC is estimated to be enough safe in the case that the order is a prime number which is not equal to the characteristic. Even if prime, however, it is not always safe, for example super singular elliptic curve, MOV-reduction and FR-reduction work on the elliptic curve of prime order. In this paper, we consider elliptic curve with coefficient over the prime field and defined over its extension field. Then, the parameter conditions for the elliptic curve to be enough safe and with fast implementation are given. Finally, construction examples, which satisfy all of the given conditions, are introduced.

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High-Speed Algorithm for Taking Square Root in an Extension Field by Using (x^<m+1>-1)/(x-1) as the Modulus

DANNO T., SAITO A., NOGAMI Y., MORIKAWA Y.

IEICE technical report. Information theory   101 ( 234 )   31 - 36   2001.7

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ECC(Elliptic Curve Cryptosystem)is expected as a next generation public-key cryptosystem. Four arithmetic operations on finite field including the inverse operation, quadratic residue test, and square root derivation are necessary to implement ECC. These operations must be enough speedy to make ECC practical. In this paper, we propose a high speed algorithm for taking square root in an extension field AOPF(All One Polynomial Field)by using(x^<m+1>-1)/(x-1)as the moduls.

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Steganography in Wavelet Domain by Use of Visual Complexity

TSUGE K., NOGAMI Y., MORIKAWA Y.

Technical report of IEICE. SST   100 ( 694 )   35 - 40   2001.3

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Extension Fields by Using (x^<m+1>-1)/(x-1) as the Modulus for High-Speed Arithmetic

SAITO A., HIRAMOTO T., DANNO T., NOGAMI Y., MORIKAWA Y.

Technical report of IEICE. SST   100 ( 693 )   129 - 134   2001.3

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Elliptic Curve Cryptosystem has attracted much attentions as a public-key cryptosystem in next generation, and an extension field has been used as the definition field. The arithmetic speed in extension feild is drastically different by the selection of characteristic, extension degree or basis, and therefore, it is necessary to pay the attention to this selection. In such meaning, OEF proposed recently is said that it is optimal. In this paper, we propose a class of finite fields which uses {ω,ω^2,…,ω^m) as a basis in order to execute multiplication and inversion faster than OEF, where ω is a zero of (x^<m+1>-1)/(x+1).

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GF(P)における3次多項式の高速既約判定アルゴリズム

平本琢士, 野上保之, 森川良孝

電子情報通信学会 論文誌A   2001

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Determining Minimal Polynomial of Proper Element by Using Higher Degree Traces

Y.Nogami, Y.Morikawa

MEMOIRS OF THE FACULTY OF ENGINEERING OKAYAMA UNIVERSITY   2001

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Testing Irreducibility of Polynomial of Degree 3 over GF(P)

HIRAMOTO T., YANO T., NOGAMI Y., MORIKAWA Y.

IEICE technical report. Information theory   99 ( 562 )   19 - 24   2000.1

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Properties of elliptic curve y^2=f(x)=^^△x^3+ax+b over prime field;GF(P)have been studied from many aspects, since the curve may yield a diverse of public key cryptsystems with short length key. Whereas for secure cryptststem the cublic polynomial f(x) has to be irreducible over GF(P), only the theorem of Stickelberger is known concerning decision of the reducibility. By means of the theorem, we can decide reducibility if the discriminant of f(x)is quadratic power residue, but not otherwise. In this paper, we apply Cardano's formula for solution of third order equation over complex field to the one over GF(P)and derive the condition for all solutions to be proper elements of cubic extension field GF(P^3). We also show the proposed algorithm for test of reducibility are very fast.

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On Derivation of Primitive Polynomials in the Case of (P^<P^<i+1>>-1)/(P^<P^i>-1) Prime

NOGAMI Y., MORIKAWA Y.

IEICE technical report. Information theory   99 ( 562 )   31 - 35   2000.1

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In the area of information communication, Reed Solomon code(RS code)is often used to encode and decode per a byte. Therefore, RS code is defined over GF(2^8), a primitive polynomial of degree 8 over GF(2) is needed, so that a primitive polynomial of degree 2^i over GF(2). In this paper, Fermat prime is generalized into(P^P^<i+1>-1) / (P^P^i-1), where P is an arbitrary prime which includes P=2: Fermat prime. Then a method to derive primitive polynomials of degree P^<i+1>, P^<i+1> in the case of (P^P^<i+1>-1) / (P^P^i-1)prime is proposed. This method uses variable transformation;x:=x^P-x+s, s∈GF(P).

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Deriving infinite number of irreducible polynomials by repetition of variable transformations x:=x^P-x+s and x:=x^k

SAITO A., OKA H., NOGAMI Y., MORIKAWA Y.

IEICE technical report. Information theory   99 ( 562 )   25 - 30   2000.1

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Two variable transformation methods are known to generate a number of irreducible polynomials; x:=x^k on condition of k-th power residue and x:=x^P-x on condition of trace. Those two transformations might not be consistent with each other owing to each conditions. In this paper, we consider the case of k|(P-1). Then we present a method to derive infinite number of irreducible polynomials of degree mk^jP^i from a irreducible polynomial of degree m by repetition of above variable transformations.

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A Consideration on Specifying Minimal Polynomials with nth Trace

NOGAMI Y., YAMANE N., MORIKAWA Y.

IEICE technical report. Information theory   99 ( 296 )   7 - 11   1999.9

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Deriving Infinite Number of Irreducible Polynomials by Variable Transformation x^p-x+s

THE TRANSACTIONS OF THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS A   J82-A ( 4 )   587 - 590   1999

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Deriving Infinite Number of Irreducible Polynomials by Variable Transformation x^P-x+s

NOGAMI Yasuyuki, TANAKA Kiyoshi, SUGIMURA Tatsuo, OSHITA Shinjiro

The Transactions of the Institute of Electronics,Information and Communication Engineers. A   J82-A ( 4 )   587 - 590   1999

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Deriving Infinite Number of Irreducible Polynomials by Variable Transformation x^p-x+s

NOGAMI Yasuyuki, TANAKA Kiyoshi, SUGIMURA Tatsuo, OSHITA Shinjiro

THE TRANSACTIONS OF THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS A   J82-A ( 4 )   587 - 590   1999

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Deriving Infinite Number of Irreducible Polynomials by Variable Transformation x^P-x+s

NOGAMI Yasuyuki, TANAKA Kiyoshi, SUGIMURA Tatsuo, OSHITA Shinjiro

The Transactions of the Institute of Electronics,Information and Communication Engineers. A   J82-A ( 4 )   587-590 - 590   1999

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A Consideration on Normal Bases over the Extension Fields of Prime Degree by P-Polynomials

NOGAMI Y., TANAKA K., SUGIMURA T., OSHITA S.

IEICE technical report. Information theory   97 ( 80 )   25 - 30   1997.5

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In recent years, in the area of information security many considerations on the base of the finite field theory are proceeded. So considering finite fields into details is very interesting and important. In this paper, we consider on finite fields from the viewpoints of self recursive polynomials and P-polynomials. Consequently, in the extension field whose extension degree is a prime not its characteristic, normal bases and their minimal polynomials are determined uniquely. And the number of the normal bases over such extension fields is obtained. Furthermore, a new method to test for an irreducible polynomial whether a pair of its zeros is a normal base or not.

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原始多項式の判定および導出

電子情報通信学会論文誌   J79-A ( 3 )   761 - 767   1996

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Testing and Deriving Primitive Polynomial

THE TRANSACTIONS OF THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS A   J79-A ( 3 )   761 - 767   1996

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Testing and Deriving Primitive Polynomial

THE TRANSACTIONS OF THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS A   J79-A ( 3 )   761 - 767   1996

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原始多項式の判定および導出

野上保之, 杉村立夫

電子情報通信学会論文誌(A)   1996

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