Finding smooth integers in short intervals using CRT decoding

Abstract

We present a new algorithm for CRT list decoding. An instance of the, CRT list decoding problem consists of integers B, 〈P1,...,P n〉 and 〈r1,..., rn〉, where p1 〈p2 < pn is a sequence of relatively prime integers. The CRT list decoding problem is to find all positive integers x < B such that x = ri, mod Pi for all but e values of ie {1,..., n}. Suppose B = ∏i-ir Pi for some integer k. Goldreich, Ron, and Sudan (in "Proc. of STOC'99", pp. 225-234, 1999) recently gave several applications for this problem and presented the first efficient algorithm that works whenever e (approximately) satisfies e n/3. The bounds we obtain are similar to the bounds obtained by Guruswami and Sudan for Reed-Solomon list decoding. Hence, our algorithm reduces 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. Problems of this type come up in several algorithms for factoring large integers. We define and solve a generalized CRT list decoding problem and discuss how it might be used within the quadratic sieve factoring method. © 2002 Elsevier Science (USA).