The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
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The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
Récemment, les schémas d’application cryptographiques basés sur le couplage ont attiré beaucoup d’attention. Afin de rendre les schémas plus efficaces, non seulement l’algorithme d’appariement mais aussi les opérations arithmétiques dans le champ d’extension doivent être efficaces. À cette fin, les auteurs ont proposé une série d’algorithmes de multiplication vectorielle cyclique (CVMA) correspondant aux bases adoptées telles que la base normale optimale (ONB) de type I. Notez ici que chaque base adaptée aux CVMA conventionnels ne sont que des classes spéciales de bases normales de la période de Gauss (GNB). En général, GNB est caractérisé par un certain nombre entier positif h en plus de la caractéristique p et diplôme d'extension m, à savoir tapez-⟨h.m⟩ GNB dans le champ d'extension Fpm. Le paramètre h doit satisfaire certaines conditions et un tel entier positif h existe infiniment. Du point de vue du coût de calcul du CVMA, il est préférable qu'il soit faible. Ainsi, le minimum noté hm. sera adapté. Cet article se concentre sur deux problèmes restants : 1) le CVMA n'a pas encore été étendu aux GNB généraux et 2) le minimum hm. devient parfois important et donne lieu à un dossier inefficace. Premièrement, cet article étend la CVMA aux GNB généraux. Cela améliorera certains cas critiques avec de grandes hm. rapportée dans les ouvrages conventionnels. Après cela, cet article montre un théorème qui, pour un nombre premier fixe r, d'autres nombres premiers modulo r répartir uniformément entre 1 et r-1. Ensuite, sur la base de ce théorème, la probabilité d'existence du type-⟨hm.,m⟩ GNB dans Fpm et aussi la valeur attendue de hm. sont explicitement données.
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Kenta NEKADO, Yasuyuki NOGAMI, Hidehiro KATO, Yoshitaka MORIKAWA, "Cyclic Vector Multiplication Algorithm and Existence Probability of Gauss Period Normal Basis" in IEICE TRANSACTIONS on Fundamentals,
vol. E94-A, no. 1, pp. 172-179, January 2011, doi: 10.1587/transfun.E94.A.172.
Abstract: Recently, pairing-based cryptographic application sch-emes have attracted much attentions. In order to make the schemes more efficient, not only pairing algorithm but also arithmetic operations in extension field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) corresponding to the adopted bases such as type-I optimal normal basis (ONB). Note here that every basis adapted for the conventional CVMAs are just special classes of Gauss period normal bases (GNBs). In general, GNB is characterized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-⟨h.m⟩ GNB in extension field Fpm. The parameter h needs to satisfy some conditions and such a positive integer h infinitely exists. From the viewpoint of the calculation cost of CVMA, it is preferred to be small. Thus, the minimal one denoted by hmin will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmin sometimes becomes large and it causes an inefficient case. First, this paper expands CVMA for general GNBs. It will improve some critical cases with large hmin reported in the conventional works. After that, this paper shows a theorem that, for a fixed prime number r, other prime numbers modulo r uniformly distribute between 1 to r-1. Then, based on this theorem, the existence probability of type-⟨hmin,m⟩ GNB in Fpm and also the expected value of hmin are explicitly given.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E94.A.172/_p
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@ARTICLE{e94-a_1_172,
author={Kenta NEKADO, Yasuyuki NOGAMI, Hidehiro KATO, Yoshitaka MORIKAWA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Cyclic Vector Multiplication Algorithm and Existence Probability of Gauss Period Normal Basis},
year={2011},
volume={E94-A},
number={1},
pages={172-179},
abstract={Recently, pairing-based cryptographic application sch-emes have attracted much attentions. In order to make the schemes more efficient, not only pairing algorithm but also arithmetic operations in extension field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) corresponding to the adopted bases such as type-I optimal normal basis (ONB). Note here that every basis adapted for the conventional CVMAs are just special classes of Gauss period normal bases (GNBs). In general, GNB is characterized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-⟨h.m⟩ GNB in extension field Fpm. The parameter h needs to satisfy some conditions and such a positive integer h infinitely exists. From the viewpoint of the calculation cost of CVMA, it is preferred to be small. Thus, the minimal one denoted by hmin will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmin sometimes becomes large and it causes an inefficient case. First, this paper expands CVMA for general GNBs. It will improve some critical cases with large hmin reported in the conventional works. After that, this paper shows a theorem that, for a fixed prime number r, other prime numbers modulo r uniformly distribute between 1 to r-1. Then, based on this theorem, the existence probability of type-⟨hmin,m⟩ GNB in Fpm and also the expected value of hmin are explicitly given.},
keywords={},
doi={10.1587/transfun.E94.A.172},
ISSN={1745-1337},
month={January},}
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TY - JOUR
TI - Cyclic Vector Multiplication Algorithm and Existence Probability of Gauss Period Normal Basis
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 172
EP - 179
AU - Kenta NEKADO
AU - Yasuyuki NOGAMI
AU - Hidehiro KATO
AU - Yoshitaka MORIKAWA
PY - 2011
DO - 10.1587/transfun.E94.A.172
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E94-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2011
AB - Recently, pairing-based cryptographic application sch-emes have attracted much attentions. In order to make the schemes more efficient, not only pairing algorithm but also arithmetic operations in extension field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) corresponding to the adopted bases such as type-I optimal normal basis (ONB). Note here that every basis adapted for the conventional CVMAs are just special classes of Gauss period normal bases (GNBs). In general, GNB is characterized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-⟨h.m⟩ GNB in extension field Fpm. The parameter h needs to satisfy some conditions and such a positive integer h infinitely exists. From the viewpoint of the calculation cost of CVMA, it is preferred to be small. Thus, the minimal one denoted by hmin will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmin sometimes becomes large and it causes an inefficient case. First, this paper expands CVMA for general GNBs. It will improve some critical cases with large hmin reported in the conventional works. After that, this paper shows a theorem that, for a fixed prime number r, other prime numbers modulo r uniformly distribute between 1 to r-1. Then, based on this theorem, the existence probability of type-⟨hmin,m⟩ GNB in Fpm and also the expected value of hmin are explicitly given.
ER -