Finding smooth integers in short intervals using CRT decoding

Abstract

We present a new algorithm for CRT list decoding. Given B, 〈p 1,...,pn〉 and 〈r1,...,r n〉, where the pi's are relatively prime, the CRT list decoding problem asks for all positive integers x < B such that x = ri mod pi for all but e values of i ∈ {1,...,n}. Suppose B = Πi=1k pi for some integer k. Goldreich, Ron, and Sudan recently gave several applications for this problem and presented an efficient algorithm whenever e (approximately) satisfies e n/3. The bounds we obtain are identical to the bounds obtained by Guruswami and Sudan for Reed-Solomon list decoding. Hence, our algorithm closes the gap between CRT list decoding and list decoding of Reed-Solomon codes. In addition, we give a new application for CRT list decoding: finding smooth integers in short intervals. This problem is relevant to factoring large integers. We define and solve a generalized CRT list decoding problem and show how it can be used within the quadratic sieve factoring method. © 2000 ACM.