## Cryptology ePrint Archive: Report 2004/339

Divisors in Residue Classes, Constructively

Don Coppersmith and Nick Howgrave-Graham and S. V. Nagaraj

Abstract: Let $r,s,n$ be integers satisfying $0 \leq r < s < n$, $s \geq n^{\alpha}$, $\alpha > 1/4$, and $\gcd(r,s)=1$. Lenstra showed that the number of integer divisors of $n$ equivalent to $r \pmod s$ is upper bounded by $O((\alpha-1/4)^{-2})$. We re-examine this problem; showing how to explicitly construct all such divisors and incidentally improve this bound to $O((\alpha-1/4)^{-3/2})$.

Category / Keywords: foundations / lattice divisors LLL