Finding smooth integers in short intervals using CRT decoding

TLDR: This work defines and solves a generalized CRT list decoding problem and discusses how it might be used within the quadratic sieve factoring method, and gives a new application for CRt list decoding: finding smooth integers in short intervals.

Abstract: We present a new algorithm for CRT list decoding. An instance of the, CRT list decoding problem consists of integers B, 〈p1, ..., pn〉 and 〈r1, ..., rn〉, where p1 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.