Chinese remaindering with errors

The Chinese Remainder Theorem states that a positive integer m is uniquely specified by its remainder modulo k relatively prime integers p₁,...,p_k, provided m<∏_{i = 1}^k p_i. Thus the residues of m modulo relatively prime integers p₁<p₂<...<p_n form a redundant representation of m if m <∏_{i = 1}^k p_i and k<n. This suggests a number-theoretic construction of an `error-correcting code' that has been implicitly considered often in the past. In this paper we provide a new algorithmic tool to go with this error-correcting code: namely, a polynomial-time algorithm for error-correction. Specifically, given n residues r₁,...,r_n and an agreement parameter t, we find a list of all integers m<∏_{i = 1}^k p_i such that (m mod p₁) = r_i for at least t values of iε{1,...,n}, provided t = Ω(√kn log p_n/log p₁). We also give a simpler algorithm, with a nearly linear time implementation, to decode from a smaller number of errors, i.e., when t>n-(n-k)log p₁/log p₁+log p_n. 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).