C. Pomerance, `Fast, rigorous factorization and discrete logarithm algorithms,' Discrete Algorithms and Complexity , Academic Press, 1987, 119--143  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that is, the unique non negative integer u with g u # x (mod p) 0 # u # p 2. The importance of the discrete logarithm for modern cryptography is well known, see [13, 23] Surveys of many e#cient algorithms (including heuristic ones) for computing the discrete logarithm can be found in [1, 10, 11, 13, 17, 18, 20, 21, 23]. Some algebraic and number theoretic characteristics of the discrete logarithm, including the degree of its polynomial representation and linear complexity have been studied in [2, 4, 5, 6, 7, 8, 9, 14, 16, 22, 24] Nevertheless, despite the recent theoretical and practical progress in studying ....

C. Pomerance, `Fast, rigorous factorization and discrete logarithm algorithms', Discrete algorithms and complexity , Academic Press, San Diego, CA, 1987, 119--143.

....as the IC PR algorithm, the letters standing for Index Calculus Polynomial Ring. The running time of the IC PR algorithm depends on the probability that a polynomial of degree n is B smooth. In 1985, Odlyzko ( 52] determined this probability in the case that p = 2. As a result, Pomerance ([56]) was able to prove in 1987 that the algorithm, in the case that q is a power of 2, has an expected running time of L q [1=2; p 2 o(1) where the limit implicit in the o(1) is for q 1. Since then Odlyzko s arguments have been extended and modified. In 1992 Lovorn Bender ( 39] used the same ....

....approach is necessary. For n = 1, it is best to let R = ZZ, in which case one obtains the algorithm already described in the introduction to this section. We do not give the details here of the analysis of this version of the index calculus method as there have been no recent developments. See [56] for Pomerance s proof that this algorithm has an expected running time of L q [1=2; p 2 o(1) In the case that n = 2, we encounter in the work of El Gamal ( 26] and subsequently Lovorn Bender ( 39] 40] a new choice for R, namely the ring of integers of a quadratic extension of Q. We ....

C. Pomerance, Fast rigorous factorization and discrete logarithms algorithms, Discrete algorithms and complexity (D.S. Johnson, T. Nishizeki, A. Nozaki and H. Wilf, eds.), Academic Press, pp. 119--143, 1987

....were invented especially for this purpose, although Simon s problem does not appear contrived and could conceivably be useful. Discrete logarithms and integer factoring are two number theory problems which have been studied extensively but for which no polynomial time algorithms are known [16, 20, 21, 26]. In fact, these problems are so widely believed to be hard that cryptosystems based on their hardness have been proposed, and the RSA public key cryptosystem [27] based on the hardness of fac2 toring, is in use. We show that these problems can be solved in BQP. Currently, nobody knows how to ....

C. Pomerance, "Fast, rigorous factorization and discrete logarithm algorithms," in Discrete Algorithms and Complexity (Proc. Japan-US Joint Seminar) , pp. 119-143, Academic Press (1986).

....we may conclude that the hardness of the discrete logarithm problem modulo p in base g, for an integer g, is majorised 1. by q where q is the largest prime divisor of the multiplicative order of g modulo p, see [14, 24] 2. by L p (1 2, 2 ) for a rigorous unconditional algorithm, see [19]; 3. by L p 1 3, 64 9) 1 3 for the heuristic number field sieve algorithm, see [21, 22] where as usual we denote by Lm (#, #) any quantity of the form Lm (#, #) exp (# o(1) log m) log log m) 1 # with the o(1) expression tending to 0 as the variable m tends to #. The ....

C. Pomerance, `Fast, rigorous factorization and discrete logarithm algorithms,' Discrete Algorithms and Complexity , Academic Press, 1987, 119--143

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