Topic

# GF(2)

About: GF(2) is a research topic. Over the lifetime, 1041 publications have been published within this topic receiving 16659 citations. The topic is also known as: Z/2Z & Z2.

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TL;DR: The fast algorithm proposed in this paper also uses normal bases, and computes multiplicative inverses iterating multiplications in GF(2 m ).

Abstract: 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[log 2 ( 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 ).

663 citations

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Bell Labs

^{1}TL;DR: This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n), finding that in order to be safe from attacks using these algorithms, the value of n for which GF( 2n) is used in a cryptosystem has to be very large and carefully chosen.

Abstract: Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ? GF(q) is that integer k, 1 ? k ? q-1, for which u = gk. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.

384 citations

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TL;DR: In this article, a pipeline structure is developed to realize the Massey-Omura multiplier in the finite field GF(2m) with the simple squaring property of the normal basis representation used together with this multiplier.

Abstract: Finite field arithmetic logic is central in the implementation of Reed-Solomon coders and in some cryptographic algorithms. There is a need for good multiplication and inversion algorithms that can be easily realized on VLSI chips. Massey and Omura [1] recently developed a new multiplication algorithm for Galois fields based on a normal basis representation. In this paper, a pipeline structure is developed to realize the Massey-Omura multiplier in the finite field GF(2m). With the simple squaring property of the normal basis representation used together with this multiplier, a pipeline architecture is also developed for computing inverse elements in GF(2m). The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.

373 citations

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TL;DR: This work has applications in crytography and coding theory since a reduction in the complexity of multiplying and exponentiating elements of GF(2n) is achieved for many values of n, some prime.

334 citations

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TL;DR: It is shown that the multiplication operation c=a · b · r-1 in the field GF(2k can be implemented significantly faster in software than the standard multiplication, where r is a special fixed element of the field.

Abstract: We show that the multiplication operation c=a · b · r^-1 in the field GF(2^k can be implemented significantly faster in software than the standard multiplication, where r is a special fixed element of the field. This operation is the finite field analogue of the Montgomery multiplication for modular multiplication of integers. We give the bit-level and word-level algorithms for computing the product, perform a thorough performance analysis, and compare the algorithm to the standard multiplication algorithm in GF(2^k. The Montgomery multiplication can be used to obtain fast software implementations of the discrete exponentiation operation, and is particularly suitable for cryptographic applications where k is large.

312 citations