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 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(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, ...

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For the first time a number of more than 100 decimal digits has been factorized on a single computer by means of the Multiple Polynomial Quadratic Sieve method of Kraïtchik and Pomerance (with improvements by Montgomery and Silverman). This method (MPQS) is the best one known to handle numbers which are the product of two large, approximately equal prime factors. These numbers are being used in cryptography as keys in public-key cryptosystems. The safety of such cryptosystems depends on our ability to factorize these keys. The computer used is the four-processor Cray Y-MP4/464 which was installed

1.4 Simple XOR

this paper we present a concise survey of the research in Computational