GCD of Random Linear Forms

Abstract

We show that for arbitrary positive integers a ₁, ..., a _m , with probability at least 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 at least 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). Naturally, this algorithm can be repeated to achieve any desired confidence level.