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Discrete Logarithms in Finite Fields and Their Cryptographic Significance
, 1984
"... 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 = g k . The wellknown problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its appl ..."
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Cited by 87 (6 self)
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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 = g k . The wellknown 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(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) 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(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...
Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme
, 1984
"... The basic MerkleHellman additive trapdoor knapsack publickey cryptosystem was recently shown to be insecure, and attacks have also been developed on stronger variants of it, such as the GrahamShamir system and the iterated knapsack cryptosystem. This paper shows that some simple variants of anoth ..."
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Cited by 16 (3 self)
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The basic MerkleHellman additive trapdoor knapsack publickey cryptosystem was recently shown to be insecure, and attacks have also been developed on stronger variants of it, such as the GrahamShamir system and the iterated knapsack cryptosystem. This paper shows that some simple variants of another MerkleHellman system, the multiplicative knapsack cryptosystem, are insecure. It is also shown that the Shamir fast signature scheme can be broken quickly. Similar attacks can also be used to break the Scho .. biMassey authentication scheme. These attacks have not been rigorously proved to succeed, but heuristic arguments and empirical evidence indicate that they work on systems of practical size. Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme A. M. Odlyzko AT&T Bell Laboratories Murray Hill, New Jersey 07974 1. Introduction One of the bestknown publickey cryptosystems, the basic MerkleHellman additive trapdoor knapsack sys...
Finite fields and cryptology
 COMPUTER SCIENCE JOURNAL OF MOLDOVA, VOL.11, NO.2(32)
, 2003
"... The problem of a computationally feasible method of finding the discrete logarithm in a (large) finite field is discussed, presenting the main algorithms in this direction. Some cryptographic schemes based on the discrete logarithm are presented. Finally, the theory of linear recurring sequences is ..."
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The problem of a computationally feasible method of finding the discrete logarithm in a (large) finite field is discussed, presenting the main algorithms in this direction. Some cryptographic schemes based on the discrete logarithm are presented. Finally, the theory of linear recurring sequences is outlined, in relation to its applications in cryptology.