A compact algorithm for Gaussian elimination over GF(2) implemented on highly parallel computers

Factoring with the quadratic sieve on large vector computers

“…Here a(-) denotes the sum of the divisors function. In Table 4.5 below we use the following notation: e.g., C48 2017;\16 means a cofactor of a(2017 16 ) of 48 decimal digits. …”

Implementing the Continued Fraction Factoring Algorithm on Parallel Machines

“…An algorithm which performs Gaussian elimination on a 0, 1 matrix in GF(2) using minimal storage is discussed in a separate paper by Parkinson and Wunderlich [5]. This step uses little computer time compared to the factoring in Step 3 and is not included in the subsequent analysis.…”

“…We will assume that sufficient Q must be factored so that NF, the number of factored Q, will equal F, the number of elements in the factor base. To do this, we must attempt to factor (5) NQ = F/r values of Q, where r is the fraction of Q 's which factor over the factor base and this will require ND division operations where The ratio r can be approximated by Dickman's function r(a) which is the limiting fraction of integers n for which all prime factors of n are less than na. To this end, we let a = (logx)/logv/M and using the prime number theorem and the fact that roughly half the prime numbers p have the property that the Legendre symbol (M/p) = 1, we obtain M«/2 F = which gives a running time estimate The effect of using the large prime variation can be analyzed in a similar fashion.…”

“…This is because a bit matrix of 4096 X 4096 will conveniently fit in the memory of the DAP and the elimination can be carried out in core without any disk swapping. Parkinson and Wunderlich [5] have shown how this can be done without the need for allocating additional memory for a history matrix. If one really has an unbounded number of processors, then one could compute the optimum number of processors and hence the number of primes in the factor base by minimizing the value of NQ = F/r given in (5).…”

“…Parkinson and Wunderlich [5] have shown how this can be done without the need for allocating additional memory for a history matrix. If one really has an unbounded number of processors, then one could compute the optimum number of processors and hence the number of primes in the factor base by minimizing the value of NQ = F/r given in (5). One then has to take into consideration the time taken to perform the elimination step and this makes the result hardware dependent.…”