AN EXTENSION OF A RESULT ABOUT DIVISORS IN A RESIDUE CLASS AND ITS APPLICATION TO REDUCING INTEGER FACTORIZATION TO COMPUTING EULER'S TOTIENT

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 . 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 phi. Our reduction is deterministic, runs in at most exp ((72(-1/3) + o(1)) (ln n)(1/3)(ln ln n)(2/3)) time, and requires no more than In(8 )n chosen values of phi. This improves upon a previous recent result both in terms of the factor 72(-1/3) and the number of values of phi 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 mod A via Chinese remaindering.