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Shigeo Tsujii

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I&C Journal 1989 Journal Article

Structure of parallel multipliers for a class of fields GF(2m)

Toshiya Itoh
Shigeo Tsujii

This paper presents a configuration of parallel multipliers for GF(2 m ) based on canonical bases. The possible parallel multipliers by the proposed configuration are limited to a class of fields GF(2 m ). However they can be constructed by O(m2) AND-gates and O(m2) EOR-gates with the structural modularity (this is a desirable feature for the hardware implementation), and their operation time is about (log m) T, where m is the dimension of GF(2 m ) and T is the delay time of an EOR-gate. In order to construct such parallel multipliers, we define two types of polynomials of special form over GF(2), one is called all one polynomial (denoted by AOP) and the other is called equally spaced polynomial (denoted by ESP). Furthermore, we show a necessary and sufficient condition for ESPs to be irreducible over GF(2) and the uniqueness of the irreducible ESPs over GF(2). Finally, we propose the configuration of parallel multipliers for a class of fields GF(2 m ) based on irreducible AOPs and ESPs over GF(2).

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I&C Journal 1988 Journal Article

A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases

Toshiya Itoh
Shigeo Tsujii

This paper proposes a fast algorithm for computing multiplicative inverses in GF(2 m ) using normal bases. Normal bases have the following useful property: In the case that an element x in GF(2 m ) is represented by normal bases, 2 k power operation of an element x in GF(2 m ) can be carried out by k times cyclic shift of its vector representation. C. C. Wang et al. proposed an algorithm for computing multiplicative inverses using normal bases, which requires (m − 2) multiplications in GF(2 m ) and (m − 1) cyclic shifts. The fast algorithm proposed in this paper also uses normal bases, and computes multiplicative inverses iterating multiplications in GF(2 m ). It requires at most 2[log2(m − 1)] multiplications in GF(2 m ) and (m − 1) cyclic shifts, which are much less than those required in the Wang's method. The same idea of the proposed fast algorithm is applicable to the general power operation in GF(2 m ) and the computation of multiplicative inverses in GF(q m ) (q = 2 n ).

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