JaLCDOI | 10.18926/14126 |
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FullText URL | Mem_Fac_Eng_OU_40_1_84.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper particularly deals with elliptic curves in the form of E(x, y) = y(2) − x(3) −b = 0, b ∈ F(* q) , where 3 divides q−1. In this paper, we refer to the well-known twist technique as x-twist and propose y-twist. By combining x-twist and y-twist, we can consider six elliptic curves and this paper proposes a method to obtain the orders of these six curves by counting only one order among the six curves. |

Keywords | elliptic curve twist third power residue/non-residue |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2006-01 |

Volume | volume40 |

Issue | issue1 |

Start Page | 83 |

End Page | 94 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308548 |

JaLCDOI | 10.18926/14079 |
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FullText URL | Mem_Fac_Eng_OU_41_1_1.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | In this paper, we first show the number of x's such that x(2) +u, u ∈ F(*)(p) , becomes a quadratic residue in F(p), and then this number is proven to be equal to (p+1)/2 if −u is a quadratic residue in Fp, which is a necessary fact for the following. With respect to the irreducible cubic polynomials over Fp in the form of x(3)+ax+b, we give a classification based on the trace of an element in F(p3) and based on whether or not the coefficient of x, i.e. the parameter a, is a quadratic residue in Fp. According to this classification, we can know the minimal set of the irreducible cubic polynomials, from which all the irreducible cubic polynomials can be generated by using the following two variable transformations: x=x + i, x=j−1x, i, j ∈ Fp, j ≠ 0. Based on the classification and that necessary fact, we show the number of the irreducible cubic polynomials in the form of x(3)+ax+b, b ∈ F(p), where a is a certain fixed element in F(p). |

Keywords | Irreducible cubic polynomial trace quadratic residue |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2007-01 |

Volume | volume41 |

Issue | issue1 |

Start Page | 1 |

End Page | 10 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308562 |

Squaring Algorithm Efficient for Cubic Extension Field Derived with Pseudo Gauss Period Normal Basis

JaLCDOI | 10.18926/44500 |
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FullText URL | mfe_045_054_059.pdf |

Author | Nekado, Kenta| Takai, Yusuke| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | Recently, pairing–based cryptographies have attracted much attention. For fast pairing calculation, not only pairing algorithms but also arithmetic operations in extension field should be efficient. Especially for final exponentiation included in pairing calculation, squaring is more important than multiplication. This paper considers squaring algorithms efficient for cubic extension field which is often used for pairing implementaions. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2011-01 |

Volume | volume45 |

Start Page | 54 |

End Page | 59 |

ISSN | 1349-6115 |

language | 英語 |

Copyright Holders | Copyright © by the authors |

File Version | publisher |

NAID | 80021759252 |

JaLCDOI | 10.18926/49321 |
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FullText URL | mfe_047_019_024.pdf |

Author | Nogami, Yasuyuki| Sumo, Taichi| |

Abstract | Recent efficient pairings such as Ate pairing use two efficient rational point subgroups such that π(P) = P and π(Q) = [p]Q, where π, p, P, and Q are the Frobenius map for rational point, the characteristic of definition field, and torsion points for pairing, respectively. This relation accelerates not only pairing but also pairing–related operations such as scalar multiplications. It holds in the case that the embedding degree k divides r − 1, where r is the order of torsion rational points. Thus, such a case has been well studied. Alternatively, this paper focuses on the case that the degree divides r + 1 but does not divide r − 1. Then, this paper shows a multiplicative representation for r–torsion points based on the fact that the characteristic polynomial f(π) becomes irreducible over Fr for which π also plays a role of variable. |

Keywords | pairing–friendly curve torsion point group structure rank |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2013-01 |

Volume | volume47 |

Start Page | 19 |

End Page | 24 |

ISSN | 1349-6115 |

language | 英語 |

Copyright Holders | Copyright © by the authors |

File Version | publisher |

NAID | 120005232373 |

JaLCDOI | 10.18926/19960 |
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FullText URL | Mem_Fac_Eng_OU_44_60.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper proposes a method for generating a certain composite order ordinary pairing–friendly elliptic curve of embedding degree 3. In detail, the order has two large prime factors such as the modulus of RSA cryptography. The method is based on the property that the order of the target pairing–friendly curve is given by a polynomial as r(X) of degree 2 with respect to the integer variable X. When the bit size of the prime factors is about 500 bits, the proposed method averagely takes about 15 minutes on Core 2 Quad (2.66Hz) for generating one. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2010-01 |

Volume | volume44 |

Start Page | 60 |

End Page | 68 |

ISSN | 1349-6115 |

language | 英語 |

File Version | publisher |

NAID | 120002309063 |

JaLCDOI | 10.18926/44499 |
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FullText URL | mfe_045_046_053.pdf |

Author | Nogami, Yasuyuki| Yanagi, Erika| Izuta, Tetsuya| Morikawa, Yoshitaka| |

Abstract | Recently, composite order pairing–based cryptographies have received much attention. The composite order needs to be as large as the RSA modulus. Thus, they require a certain pairing–friendly elliptic curve that has such a large composite order. This paper proposes an efficient algorithm for generating an ordinary pairing–friendly elliptic curve of the embedding degree 1 whose order has two large prime factors as the RSA modulus. In addition, the generated pairing–friendly curve has an efficient structure for the Gallant–Lambert–Vanstone (GLV) method. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2011-01 |

Volume | volume45 |

Start Page | 46 |

End Page | 53 |

ISSN | 1349-6115 |

language | 英語 |

Copyright Holders | Copyright © by the authors |

File Version | publisher |

NAID | 120002905955 |

JaLCDOI | 10.18926/49322 |
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FullText URL | mfe_047_025_032.pdf |

Author | Nekado, Kenta| Takai, Yusuke| Nogami, Yasuyuki| |

Abstract | Pairing–based cryptosystems are well implemented with Ate–type pairing over Barreto–Naehrig (BN) curve. Then, for instance, their securities depend on the difficulty of Discrete Logarithm Problem (DLP) on the so–denoted G3 over BN curve. This paper, in order to faster solve the DLP, first proposes to utilize Gauss period Normal Basis (GNB) for Pollard’s rho method, and then considers to accelerate the solving by an adoption of lazy random walk, namely tag tracing technique proposed by Cheon et al. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2013-01 |

Volume | volume47 |

Start Page | 25 |

End Page | 32 |

ISSN | 1349-6115 |

language | 英語 |

Copyright Holders | Copyright © by the authors |

File Version | publisher |

NAID | 120005232374 |

JaLCDOI | 10.18926/14057 |
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FullText URL | Mem_Fac_Eng_OU_42_36.pdf |

Author | Kato, Hidehiro| Nekado, Kenta| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper proposes an exponentiation method with Frobenius mappings. Our method is closely related to so-called interleaving exponentiation. Different from the interleaving exponentiation methods, our method can carry out several exponentiations using same base at the same time. The efficiency to use Frobenius mappings for an exponentiation in extension field is well introduced by Avanzi and Mihailescu. This exponentiation method is based on so-called simultaneous exponentiation and uses many Frobenius mappings. Their method more decreased the number of multiplications; however, the number of Frobenius mappings inversely increased. Compared to their method , the number of multiplications needed for the proposed method becomes about 20% larger; however, that of Frobenius mappings becomes small enough. |

Keywords | exponentiation Frobenius mapping extension field |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2008-01 |

Volume | volume42 |

Issue | issue1 |

Start Page | 36 |

End Page | 43 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308105 |

JaLCDOI | 10.18926/17851 |
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FullText URL | Mem_Fac_Eng_OU_43_108.pdf |

Author | Nekado, Kenta| Kato, Hidehiro| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | Recently, pairing-based cryptographies such as ID-based cryptography and group signature have been studied. For fast pairing calculation, not only pairing algorithms but also arithmetic operations in extension field must be efficiently carried out. The authors show efficient arithmetic operations of extension field for Xate pairing especially with Freeman curve. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2009-01 |

Volume | volume43 |

Start Page | 108 |

End Page | 112 |

ISSN | 1349-6115 |

language | 英語 |

File Version | publisher |

NAID | 120002308904 |

JaLCDOI | 10.18926/19961 |
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FullText URL | Mem_Fac_Eng_OU_44_69.pdf |

Author | Nekado, Kenta| Kato, Hidehiro| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | Recently, pairing–based cryptographies have attracted much attention. For fast pairing calculation, not only pairing algorithms but also arithmetic operations in extension field should be efficient. Especially for final exponentiation included in pairing calculation, squaring is more important than multiplication. This paper proposes an efficient squaring algorithm in extension field for Freeman curve. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2010-01 |

Volume | volume44 |

Start Page | 69 |

End Page | 72 |

ISSN | 1349-6115 |

language | 英語 |

File Version | publisher |

NAID | 120002309070 |

JaLCDOI | 10.18926/15380 |
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FullText URL | Mem_Fac_Eng_OU_35_197.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | Modern communication engineerings, such as elliptic curve cryptographies, often requires algebra on finite extension field defined by modulus arithmetic with an irreducible polynomial. This paper provides a new method to detemine the minimal (irreducible) polynomial of a given proper element in finite extension field. In the conventional determination method, as we have to solve the simultaneous equations, the computation is very involved. In this paper, the well known "trace" is extended to higher degree traces. Using the new traces, we yield the coefficient formula of the desired minimal polynomial. The new method becomes very simple without solving the simultaneous equations, and about twice faster than the conventional method in computation speed. |

Keywords | finite field minimal polynomial irreducible polynomial higher degree trace trace cryptography |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2001-03-27 |

Volume | volume35 |

Issue | issue1-2 |

Start Page | 197 |

End Page | 205 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002307992 |

JaLCDOI | 10.18926/17853 |
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FullText URL | Mem_Fac_Eng_OU_43_113.pdf |

Author | Sakemi, Yumi| Kato, hidehiro| Nogami, Yasuyuki| Morikawa, Yoshikawa| |

Abstract | Barreto–Naehrig (BN) curve has been introduced as an efficient pairing-friendly elliptic curve over prime field F(p) whose embedding degree is 12. The characteristic and Frobenius trace are given as polynomials of integer variable X. The authors proposed an improvement of Miller's algorithm of twisted Ate pairing with BN curve by applying X of small hamming weight in ITC–CSCC2008; however, its cost evaluation has not been explicitly shown. This paper shows the detail of the cost evaluation. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2009-01 |

Volume | volume43 |

Start Page | 113 |

End Page | 116 |

ISSN | 1349-6115 |

language | 英語 |

File Version | publisher |

NAID | 120002308945 |

JaLCDOI | 10.18926/14080 |
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FullText URL | Mem_Fac_Eng_OU_41_1_11.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper proposes an algorithm for generating irreducible cubic trinomials in the form x(3) + ax + b, b ∈ F(p), where a is a certain fixed non-zero element in the prime field F(p). The proposed algorithm needs a certain irreducible cubic trinomial over F(p) to be previously given as a generator; however, the proposed algorithm can generate irreducible cubic polynomials one after another by changing a certain parameter in F(p). In this paper, we compare the calculation cost and the average computation time for generating an irreducible cubic polynomial, especially trinomial, among Hiramoto et al. irreducibility testing algorithm, Berlekamp-Massey minimal polynomial determining algorithm, and the proposed algorithm. From the experimental results, it is shown that the proposed algorithm is the fastest among the three algorithms for generating irreducible cubic trinomials. |

Keywords | irreducible cubic polynomial minimal polynomial |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2007-01 |

Volume | volume41 |

Issue | issue1 |

Start Page | 11 |

End Page | 19 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308381 |

JaLCDOI | 10.18926/14156 |
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FullText URL | Mem_Fac_Eng_39_1_71.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper proposes an algorithm for generating prime order elliptic curves over extension field whose extension degree is a power of 2. The proposed algorithm is based on the fact that the order of the twisted elliptic curve is able to be a prime number when the extension degree for the twist operation is a power of 2. When the definition field is F(2(40)−87)(4) , the proposed algorithm can generate a prime order elliptic curve within 5 seconds on PentiumIII (800MHz) with C language. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2005-01 |

Volume | volume39 |

Issue | issue1 |

Start Page | 71 |

End Page | 81 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308036 |

JaLCDOI | 10.18926/14071 |
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FullText URL | Mem_Fac_Eng_OU_42_110.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | This paper shows a method for checking the parity of (#Jc − 1)/2 without calculating the order #Jc, where #Jc is the order of genus 2 or 3 hyperelliptic curve. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2008-01 |

Volume | volume42 |

Issue | issue1 |

Start Page | 110 |

End Page | 114 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308489 |

JaLCDOI | 10.18926/14157 |
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FullText URL | Mem_Fac_Eng_39_1_82.pdf |

Author | Wang, Feng| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | In this paper, we focus on developing a high-speed square root (SQRT) algorithm required for an elliptic curve cryptosystem. Examining Smart algorithm, the previously well-known SQRT algorithm, we can see that there is a lot of computation overlap in Smart algorithm and the quadratic residue (QR) test, which must be implemented prior to a SQRT computation. It makes Smart algorithm inefficient. The essence of our proposition is thus to present a new QR test and an efficient SQRT algorithm to avoid all the overlapping computations. The authors devised a SQRT algorithm for which most of the data required have been computed in the proposed QR test. Not only there is no computation overlap in the proposed algorithm and the proposed QR test, but also in the proposed algorithm over GF(p(2)) (4 | p − 1) some computations can be executed in GF(p); whereas in Smart algorithm over GF(p(2)) all the computations must be executed in GF(p(2)). These yield many reductions in the computational time and complexity. We implemented the two QR tests and the two SQRT algorithms over GF(pm) (m=1, 2) in C++ language with NTL (Number Theory Library) on Pentium4 (2.6GHz), where the size of p is around 160 bits. The computer simulations showed that the proposed QR test and the proposed algorithm over GF(p(m)) were about 2 times faster than the conventional QR test and Smart algorithm over GF(p(m)). |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2005-01 |

Volume | volume39 |

Issue | issue1 |

Start Page | 82 |

End Page | 92 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 120002308422 |

JaLCDOI | 10.18926/17849 |
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FullText URL | Mem_Fac_Eng_OU_43_99.pdf |

Author | Kato, Hidehiro| Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | A square root (SQRT) algorithm in extension field F(p(m))(m = r(0)r(1)･･･r(n−1)･2(d), r(i) : odd prime, d : positive integer) is proposed in this paper. First, a conventional SQRT algorithm, the Tonelli-Shanks algorithm, is modified to compute the inverse SQRT in F(p(2d)), where most of the computations are performed in the corresponding subfields F(p(2i)) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field F(p(m)) are also reduced to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. The Tonelli-Shanks algorithm and the proposed algorithm in F(p(6)) and F(p(10)) were implemented on a Core2 (2.66 GHz) using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerated the SQRT computation by 6 times in F(p(6)), and by 10 times in F(p(10)), compared to the Tonelli-Shanks algorithm. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2009-01 |

Volume | volume43 |

Start Page | 99 |

End Page | 107 |

ISSN | 1349-6115 |

language | 英語 |

File Version | publisher |

NAID | 120002308980 |

JaLCDOI | 10.18926/46982 |
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FullText URL | mfe_37_2_073_087.pdf |

Author | Nogami, Yasuyuki| Morikawa, Yoshitaka| |

Abstract | Public key cryptosystem has many uses, such as to sign digitally, to realize electronic commerce. Especially, RSA public key cryptosystem has been the most widely used, but its key for ensuring sufficient security reaches about 2000 bits long. On the other hand, elliptic curve cryptosystem(ECC) has the same security level with about 7-fold smaller length key. Accordingly, ECC has been received much attention and implemented on various processors even with scarce computation resources. In this paper, we deal with an elliptic curve which is defined over extension field F(p2c) and has a prime order, where p is the characteristic and c is a non negative integer. In order to realize a fast software implementation of ECC adopting such an elliptic curve, a fast implementation method of definition field F(p2c) especially F(p8) is proposed by using a technique called successive extension. First, five fast implementation methods of base field F(p2) are introduced. In each base field implementation, calculation costs of F(p2)-arithmetic operations are evaluated by counting the numbers of F(p)-arithmetic operations. Next, a successive extension method which adopts a polynomial basis and a binomial as the modular polynomial is proposed with comparing to a conventional method. Finally, we choose two prime numbers as the characteristic, and consider several implementations for definition field F(p8) by using five base fields and two successive extension methods. Then, one of these implementations is especially selected and implemented on Toshiba 32-bit micro controller TMP94C251(20MHz) by using C language. By evaluating calculation times with comparing to previous works, we conclude that proposed method can achieve a fast implementation of ECC with a prime order. |

Publication Title | Memoirs of the Faculty of Engineering, Okayama University |

Published Date | 2003-03 |

Volume | volume37 |

Issue | issue2 |

Start Page | 73 |

End Page | 87 |

ISSN | 0475-0071 |

language | 英語 |

File Version | publisher |

NAID | 80015999992 |