Showing papers on "GF(2) published in 1985"
1 Dec 1985
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.
422 citations
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.
378 citations
18 Aug 1985
TL;DR: This paper presents some results for obtaining a sub exponential time algorithms for the remaining cases GF(pm) for p ?
Abstract: The problem of computing logarithms over finite fields has proved to be of interest in different fields [4]. Subexponential time algorithms for computing logarithms over the special cases GF(p), GF(p2) and GF(pm) for a fixed p and m ? ? have been obtained. In this paper, we present some results for obtaining a sub exponential time algorithms for the remaining cases GF(pm) for p ? ? and fixed m ? 1, 2. The algorithm depends on mapping the fieLd GF(pm) into a suitable cyclotomic extension of the integers (or rationals). Once an isomorphism between GF(pm) and a subset of the cyclotomic field Q(?q) is obtained, the algorithms becomes similar to the previous algorithms for m = 1, 2.A rigorous proof for subexponential time is not yet available, but using some heuristic arguments we can show how it could be proved. If a proof would be obtained, it would use results on the distribution of certain classes of integers and results on the distribution of some ideal classes in cyclotomic fields.
39 citations
23 Aug 1985
TL;DR: The discrete logarithm problem is stated as follows: for any element b≠0 there exists an integer x, 0≤x≤q−2, such that b=αx, and x is the discrete logrithm of b to the base α and is called x=log α b and more simply by log b when the base is fixed.
Abstract: Consider the finite field having q elements and denote it by GF(q). Let α be a generator for the nonzero elements of GF(q). Hence, for any element b≠0 there exists an integer x, 0≤x≤q−2, such that b=αx. We call x the discrete logarithm of b to the base α and we denote it by x=log α b and more simply by log b when the base is fixed for the discussion. The discrete logarithm problem is stated as follows:
19 citations