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
Recentemente, esquemas de aplicativos criptográficos baseados em emparelhamento têm atraído muita atenção. Para tornar os esquemas mais eficientes, não apenas o algoritmo de emparelhamento, mas também as operações aritméticas no campo de extensão precisam ser eficientes. Para tanto, os autores propuseram uma série de algoritmos de multiplicação de vetores cíclicos (CVMAs) correspondentes às bases adotadas, como a base normal ótima tipo I (ONB). Observe aqui que todas as bases adaptadas para os CVMAs convencionais são apenas classes especiais de bases normais do período de Gauss (GNBs). Em geral, o GNB é caracterizado por um certo número inteiro positivo h além de característico p e extensão m, ou seja, digite-⟨h.m⟩ GNB no campo de extensão Fpm. O parâmetro h precisa satisfazer algumas condições e um número inteiro positivo h existe infinitamente. Do ponto de vista do custo de cálculo do CVMA, prefere-se que seja pequeno. Assim, o mínimo denotado por hminutos será adaptado. Este artigo concentra-se em dois problemas restantes: 1) o CVMA ainda não foi expandido para BGNs gerais e 2) o mínimo hminutos às vezes torna-se grande e causa um caso ineficiente. Primeiro, este artigo expande o CVMA para BGNs gerais. Irá melhorar alguns casos críticos com grandes hminutos relatado nos trabalhos convencionais. Depois disso, este artigo mostra um teorema que, para um número primo fixo r, outros números primos módulo r distribuir uniformemente entre 1 a r-1. Então, com base neste teorema, a probabilidade de existência do tipo-⟨hminutos,m⟩ GNB em Fpm e também o valor esperado de hminutos são dados explicitamente.
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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 -