Faculty Profiles - NOGAMI Yasuyuki

MISC - NOGAMI Yasuyuki

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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

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

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

Language：English Publisher：IEICE-INST ELECTRONICS INFORMATION COMMUNICATIONS ENG

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

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

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

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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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

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

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

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

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

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

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

Language：Japanese Publisher：Information Processing Society of Japan (IPSJ)

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

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

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

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

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

Language：English Publisher：SPRINGER-VERLAG BERLIN

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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Other Link： https://dblp.uni-trier.de/db/conf/icisc/icisc2004.html#NogamiM04

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

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

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

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

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

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

Language：English Publisher：IEICE-INST ELECTRONICS INFORMATION COMMUNICATIONS ENG

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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