Cyclic Vector Multiplication Algorithm and Existence Probability of Gauss Period Normal Basis

Kenta NEKADO; Yasuyuki NOGAMI; Hidehiro KATO; Yoshitaka MORIKAWA

doi:10.1587/transfun.E94.A.172

Cyclic Vector Multiplication Algorithm and Existence Probability of Gauss Period Normal Basis

Kenta NEKADO, Yasuyuki NOGAMI, Hidehiro KATO, Yoshitaka MORIKAWA

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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 F_p^m. 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 h_min will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal h_min 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 h_min 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-⟨h_min,m⟩ GNB in F_p^m and also the expected value of h_min are explicitly given.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E94-A No.1 pp.172-179

Publication Date: 2011/01/01

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Online ISSN: 1745-1337

DOI: 10.1587/transfun.E94.A.172

Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)

Category: Mathematics

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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 F_p^m. 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 h_min will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal h_min 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 h_min 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-⟨h_min,m⟩ GNB in F_p^m and also the expected value of h_min 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 F_p^m. 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 h_min will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal h_min 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 h_min 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-⟨h_min,m⟩ GNB in F_p^m and also the expected value of h_min 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 F_p^m. 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 h_min will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal h_min 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 h_min 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-⟨h_min,m⟩ GNB in F_p^m and also the expected value of h_min are explicitly given.
ER -