A randomized sublinear time parallel GCD algorithm for the EREW PRAM

New Result EREW PRAM: Compute gcd(x, y) with probability 1 − o(1) in O(n log log n/ log n) time using n6+ processors. [16] Reduction Our inputs are integers x, y with x ≥ y > 0. • Choose a prime bound B > 0, and assume p | x or p | y implies p > B. • Choose a at random, 1 ≤ a ≤ y − 1. • Compute r := ax mod y. • Remove all prime divisors ≤ B from r producing s. Thus P (r/s) ≤ B. We use (x, y) → (y, s) for our reduction. We claim: • gcd(x, y) = gcd(y, s) with probability 1− o(1). (This fails only if gcd(a, y) > 1, or gcd(a, y) > B, which is unlikely.) •with probability at least 1/B, we have