### 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 exten sion field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) correspond ing 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 charac terized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-(h.m) GNB in extension field F 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 hmjn will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmjn 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 ito r - 1. Then, based on this theorem, the existence probability of type-(hmjn,m) GNB in F_{p}^{m} and also the expected value of h_{min} are explicitly given.

Original language | English |
---|---|

Pages (from-to) | 172-179 |

Number of pages | 8 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E94-A |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2011 |

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### Keywords

- All one polynomial field
- Cyclic vector multiplication algorithm
- Extension field
- Gauss period normal basis

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Graphics and Computer-Aided Design
- Applied Mathematics
- Signal Processing

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E94-A*(1), 172-179. https://doi.org/10.1587/transfun.E94.A.172

**Cyclic vector multiplication algorithm and existence probability of gauss period normal basis.** / Nekado, Kenta; Nogami, Yasuyuki; Kato, Hidehiro; Morikawa, Yoshitaka.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E94-A, no. 1, pp. 172-179. https://doi.org/10.1587/transfun.E94.A.172

}

TY - JOUR

T1 - Cyclic vector multiplication algorithm and existence probability of gauss period normal basis

AU - Nekado, Kenta

AU - Nogami, Yasuyuki

AU - Kato, Hidehiro

AU - Morikawa, Yoshitaka

PY - 2011/1

Y1 - 2011/1

N2 - 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 exten sion field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) correspond ing 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 charac terized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-(h.m) GNB in extension field F 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 hmjn will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmjn 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 ito r - 1. Then, based on this theorem, the existence probability of type-(hmjn,m) GNB in Fpm and also the expected value of hmin are explicitly given.

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 exten sion field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) correspond ing 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 charac terized with a certain positive integer h in addition to characteristic p and extension degree m, namely type-(h.m) GNB in extension field F 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 hmjn will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmjn 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 ito r - 1. Then, based on this theorem, the existence probability of type-(hmjn,m) GNB in Fpm and also the expected value of hmin are explicitly given.

KW - All one polynomial field

KW - Cyclic vector multiplication algorithm

KW - Extension field

KW - Gauss period normal basis

UR - http://www.scopus.com/inward/record.url?scp=78650941856&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650941856&partnerID=8YFLogxK

U2 - 10.1587/transfun.E94.A.172

DO - 10.1587/transfun.E94.A.172

M3 - Article

AN - SCOPUS:78650941856

VL - E94-A

SP - 172

EP - 179

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 1

ER -