GCD of Random Linear Combinations

Abstract

We show that for arbitrary positive integers a 1 , . . . , a m , with probability 6/π 2 + o(1), the gcd of two linear combinations of these integers with rather small random integer coefficients coincides with gcd(a 1 , . . . , a m ). This naturally leads to a probabilistic algorithm for computing the gcd of several integers, with probability 6/π 2 + o(1), via just one gcd of two numbers with about the same size as the initial data (namely the above linear combinations). This algorithm can be repeated to achieve any desired confidence level.

Probabilistic analyses of the plain multiple gcd algorithm

“…We now wish to compare more precisely the random strategy described in (von zur Gathen and Shparlinski, 2006) with the present analysis. The discussion is not completely clear as the size of the entries is not the same, and we know that changing the size may strongly change the results.…”

“…The discussion is not completely clear as the size of the entries is not the same, and we know that changing the size may strongly change the results. We deal here with the sum-size while von zur Gathen and Shparlinski (2006) deal with the supsize, and more precisely with the height, defined as 9 h(x) := sup(|x i |). The following is proved in (von zur Gathen and Shparlinski, 2006): the gcd of two linear combinations of asymptotically the same size as the inputs coincides with their gcd with probability 6/π 2 .…”

“…There exists also a probabilistic algorithm proposed in von zur Gathen and Shparlinski (2006) for computing gcd's. It replaces a gcd computation on entries by a unique gcd on two random linear combinations of the initial input.…”

“…It replaces a gcd computation on entries by a unique gcd on two random linear combinations of the initial input. The approach is different: we perform here a probabilistic analysis of a deterministic algorithm where the distribution of the inputs is chosen a priori, whereas the algorithm developed in von zur Gathen and Shparlinski (2006) is probabilistic, designed as requiring few steps, and handles the worstcase. In Section 10, we return to the comparison between the two strategies, and make more precise the comparison done in (von zur Gathen and Shparlinski, 2006, Section 3).…”