Chinese remaindering with errors

Abstract

The Chinese Remainder Theorem states that a positive integer m is uniquely specified by its remainder modulo k relatively prime integers p1,...,pk, provided m n-(n-k)log p1/log p1+log pn. In such a case there is a unique integer which has such agreement with the sequence of residues. One consequence of our result is a strengthening of the relationship between average-case complexity of computing the permanent and its worst-case complexity. Specifically we show that if a polynomial time algorithm is able to guess the permanent of a random n×n matrix on 2n-bit integers modulo a random n-bit prime with inverse polynomial success rate, then then P#P = BPP. Previous results of this nature typically worked over a fixed prime moduli or assumed success probability very close to one (as opposed to bounded away from zero).