Finite Extension Field with Modulus of All-One Polynomial and Representation of Its Elements for Fast Arithmetic Operations

Yasuyuki NOGAMI  Akinori SAITO  Yoshitaka MORIKAWA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E86-A   No.9   pp.2376-2387
Publication Date: 2003/09/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Information Theory
optimal extension field,  Frobenius mapping,  normal basis,  inversion,  

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In many cryptographic applications, a large-order finite field is used as a definition field, and accordingly, many researches on a fast implementation of such a large-order extension field are reported. This paper proposes a definition field Fpm with its characteristic p a pseudo Mersenne number, the modular polynomial f(x) an irreducible all-one polynomial (AOP), and using a suitable basis. In this paper, we refer to this extension field as an all-one polynomial field (AOPF) and to its basis as pseudo polynomial basis (PPB). Among basic arithmetic operations in AOPF, a multiplication between non-zero elements and an inversion of a non-zero element are especially time-consuming. As a fast realization of the former, we propose cyclic vector multiplication algorithm (CVMA), which can be used for possible extension degree m and exploit a symmetric structure of multiplicands in order to reduce the number of operations. Accordingly, CVMA attains a 50% reduction of the number of scalar multiplications as compared to the usually adopted vector multiplication procedure. For fast realization of inversion, we use the Itoh-Tsujii algorithm (ITA) accompanied with Frobenius mapping (FM). Since this paper adopts the PPB, FM can be performed without any calculations. In addition to this feature, ITA over AOPF can be composed with self reciprocal vectors, and by using CVMA this fact can also save computation cost for inversion.