D.M. Gordon. Discrete Logarithms in GF #p# Using the Number Field Sieve. In SIAM J. of Disc. Math., 6, 124#138, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Let s relate these cryptosystems to a mathematical problem. It is strongly believed that a discrete logarithm problem is difficult. However, it has not been proven how difficult such a problem is yet. The best method for solving this problem is to use the index calculus method [1] 11] 18] 52] 12][19][20] This method runs in sub exponential time. The first Elliptic Curve Cryptosystem was proposed [26] 41] by substituting an operation over elliptic curves for an operation over a finite field in the ElGamal cryptosystem. The security of this cryptosystem is based on the difficulty of an ....

D. M. Gordon, "Discrete Logarithm in GF (p) Using the Number Field Sieve," to appear in SIAM Journal on Discrete Math.

.... against ECDLP undP theindS calculus methodtho2G which runs over anyfield F q in time L q [1 ,c] exp( c O(1) log q) 1 2 (log log q) 1 2 ) On the other hand the extensiondtens k (log q) 2 means thatFR redE82gL gives a subexponential attack against ECDLP undP the numberfield sieve([14]) which runs over some field F q in time L q [1 3,c] exp( c O(1) log q) 1 3 (log log q) 2 3 ) Therefore inordV to construct enough secure elliptic curve cryptosystems it would bedSDAS02g that k # (log q) 2 . However thecond888G of k # log q in Corollary 4 is not highly optimistic ....

D. M. Gordon, "Discrete logarithms in GF (p) using the number fieldsiev e", SIAM J. on Discrete Math., 6(1993), 124-138.

....if v = r. Security Analysis. Since r and s are each integers less than q, DSA signatures are 320 bits in size. The security of the DSA relies on two distinct but related discrete logarithm problems. One is the discrete logarithm problem in Z p where the number field sieve algorithm (see Gordon [27] and Schirokauer [78] applies; this algorithm has a subexponential running time. More precisely, the expected running time of the algorithm is O i exp i (c o(1) lnp) 1=3 (ln ln p) 2=3 jj ; 1) The Elliptic Curve Digital Signature Algorithm (ECDSA) 6 where c 1:923, and ln n denotes ....

D. Gordon, "Discrete logarithms in GF (p) using the number field sieve", SIAM Journal on Discrete Mathematics, 6 (1993), 124-138.

....problem looks quite natural; Simon s algorithm inspired the work presented in this paper. Two number theory problems which have been studied extensively but for which no polynomial time algorithms have yet been discovered are finding discrete logarithms and factoring integers [Pomerance 1987, Gordon 1993, Lenstra and Lenstra 1993, Adleman and McCurley 1995] These problems are so widely believed to be hard that several cryptosystems based on their difficulty have been proposed, including the widely used RSA public key cryptosystem developed by Rivest, Shamir and Adleman [1978] We show that these ....

....111, Knuth 1981] Suppose we are given a prime p and such a generator g. The discrete logarithm of a number x with respect to p and g is the integer r with 0 r p Gamma 1 such that g r j x (mod p) The fastest algorithm known for finding discrete logarithms modulo arbitrary primes p is Gordon s [1993] adaptation of the number field sieve, which runs in time exp(O(log p) 1=3 (log log p) 2=3 ) We show how to find discrete logarithms on a quantum computer with two modular exponentiations and two quantum Fourier transforms. This algorithm will use three quantum registers. We first find q a ....

D. M. Gordon (1993) "Discrete logarithms in GF(p) using the number field sieve," SIAM J. Discrete Math. 6, 124--139.

....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 ....

D. M. Gordon, "Discrete logarithms in GF(p) using the number field sieve," SIAM J. Discrete Math. Vol. 6, pp. 124--139 (1993).

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D.M. Gordon. Discrete Logarithms in GF #p# Using the Number Field Sieve. In SIAM J. of Disc. Math., 6, 124#138, 1993.

No context found.

D. M. Gordon, "Discrete Logarithms in GF(P) Using the Number Field Sieve," SIAM J. Discret. Math., vol. 6, no. 1, pp. 124--138, 1993.

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Gordon, D., "Discrete logarithms in GF(p) using the number field sieve", SIAM Journal on Discrete Mathematics, 6 (1993), 124-138.

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