Mathematics of Computation

Abstract:

According to a theorem of Coppersmith, Howgrave-Graham, and Nagaraj, relying on lattice basis reduction, the divisors of an integer $n$ which lie in some fixed residue class modulo a given integer $A$ can be computed efficiently if $A$ is large enough. We extend their algorithm to the setting when the modulus is a product $A\cdot B$, where $A$ is given and the unknown $B$ divides an integer whose prime factors are known. The resulting tool is applied in the context of reducing integer factorization to computing Euler’s totient function $\varphi$. Our reduction is deterministic, runs in at most $\exp \left (\left (72^{-\frac {1}{3}}+o(1)\right ) (\ln n)^{\frac {1}{3}}(\ln \ln n)^{\frac {2}{3}}\right )$ time, and requires no more than $\ln _8 n$ chosen values of $\varphi$. This improves upon a previous recent result both in terms of the factor $72^{-\frac {1}{3}}$ and the number of values of $\varphi$ needed.

In a more concrete setting, another algorithmic extension of the theorem of Coppersmith et al. may be worth noting. We can make use of the (unknown) smooth part of a shifted divisor $d$ of $n$ (or even several shifts of $d$) to compute a suitably large modulus $A$ and the corresponding residue class $d\bmod A$ via Chinese remaindering.