W.R. Alford and C. Pomerance, Implementing the self-initializing quadratic sieve on a distributed network, Proceedings of International Conference \Number Theoretic and Algebraic Methods in Computer Science" (Moscow,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....generating system of the relation lattice is produced. The solution employed by earlier algorithms is to attempt to factor randomly produced ideals over the factor base. We replace this step by a sieve based strategy similar to that used in the self initializing quadratic sieve factoring algorithm [1]. We refer the interested reader to [16] for more details. Once the structure of Cl( is computed, we have to compute representations of a and b over a system of generators of Cl( As shown in [17] the main work involved is essentially computing a single relation corresponding to a and b: ....

W.R. Alford and C. Pomerance, Implementing the self-initializing quadratic sieve on a distributed network, Proceedings of International Conference \Number Theoretic and Algebraic Methods in Computer Science" (Moscow,

....by sieving. Pomerance s quadratic sieve is a simpli cation of Schroeppel s linear sieve. Each method seems to always succeed in time y 2 o(1) with y as above. See [140] 79] 168] 62] 141] 65] 63] 165] 46] 150] 64] 155] 106] 13] 144] 156] 166] 133] 66] [9], 11] 27] and [52] The algorithm in this paper can be used to indirectly speed up sieving, as described above. Furthermore, a reduction in the sieve array size allows a reduction in the size of n; see, e.g. 52] Pollard s number eld sieve, as generalized by Buhler, Lenstra, and Pomerance, ....

W. R. Alford, Carl Pomerance, Implementing the self-initializing quadratic sieve on a distributed network, in [153] (1995), 163-174. MR 96k:11152.

....a Gamma2 mod q = g Gamma2 i 1 g Gamma2 i 2 : g Gamma2 i k mod q: Therefore, with the generation of the g primes we also compute and store the numbers g Gamma2 i mod q, i = 1; 2; r) for all the prime powers q in the factor base. For a fixed a, Alford and Pomerance [AP] developed a method to compute iteratively all the other values of b (and thus c) from a given initial value of b (see also the work of Peralta [Per] They also pointed out how the two solutions in the interval [0; q) of the congruence equation W (x) j 0 mod q can be calculated from the zeros mod ....

W.R. Alford and C. Pomerance. Implementing the self initializing quadratic sieve on a distributed network. To appear.

....N = a 2 c, a 2 p 2N=M , and jbj 1 2 a 2 . Then we have U 2 (x) j W (x) mod N and there are many possible choices for a and b (c follows from a and b) each choice yielding a new polynomial. For details about efficient polynomial generation in the quadratic sieve method, we refer to [100, 127, 104, 3]. In Table 6 we give some figures about record factorizations found at CWI on vector computers. All the results were obtained on one processor of the vector computer listed. On the Cray Y MP we could have used four CPUs, thus reducing the sieving time by a factor of about four, since Steps 2 5 ....

W.R. Alford and Carl Pomerance. Implementing the self initializing quadratic sieve on a distributed network. Manuscript, received Nov. 11, 1993.

....value at 0; GammaM; M . This value is approximately M p N= p 2. A slightly better value than 2 can be calculated for this constant, but this is only of theoretical interest and beyond the scope of this paper. 3 HMPQS was invented independently by this author [7] and by Alford and Pomerance [1]. The latter authors called this algorithm The Self Initializing Quadratic Sieve . 4 1. Define b j to be the unique positive integer which is less than t 2 , congruent to 1 mod q 2 j , and congruent to 0 mod q 2 i for 1 j 6= i n. That is b j = t 2 q j 2 Delta t 2 q j ....

Alford, W., Pomerance, C.: Implementing the self initializing quadratic sieve on a distributed network. In Number theoretic and algebraic methods in computer science (1995) Z. Poorten, Shparlinski, Ed. World Scientific publishing Co. Pte. Ltd. pp. 163--174.

....about 1:7 and 2:0 respectively. In 1988, several methods for speeding up the initializing stage of mpqs were published in [23] At the end of section 5 of this paper, the authors comment about one particularly fast way of initializing, but suppress most of the details. This was later published in [1], and is known as the self initializing quadratic sieve (siqs) 2.1 The Self Initializing Quadratic Sieve The self initializing quadratic sieve provides us with a fast way to change polynomials. This makes it beneficial to use a smaller M than in mpqs. Since M is smaller, the residues are ....

....not store all the values of 2B j a Gamma1 mod p if memory is limited. The higher indices are used much less frequently. A speedup over mpqs can still be obtained if one can only store the values of a Gamma1 mod p and 2B j a Gamma1 mod p for j = 1 and 2, for example. This is explained in [1]. Our goal is to show the practical side of siqs. We have programmed both mpqs and siqs, and have repeatedly experimented with these programs on several numbers of different sizes. We give several tables showing the results of our experiments. These tables suggest that factoring a large number ....

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W. R. Alford, Carl Pomerance, "Implementing the self initializing quadratic sieve on a distributed network," pp. 163-174; in: A.J. van der Poorten, I. Shparlinski, and H.G. Zimmer (eds), Number theoretic and algebraic methods in computer science, World Scientific, 1995.

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