Paper 2007/424

When e-th Roots Become Easier Than Factoring

Antoine Joux, David Naccache, and Emmanuel Thomé

Abstract

We show that computing $e$-th roots modulo $n$ is easier than factoring $n$ with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form $x_i + c$. Here $c$ is fixed and $x_i$ denotes small integers of the attacker's choosing. Several variants of the attack are presented, with varying assumptions on the oracle, and goals ranging from selective to universal forgeries. The computational complexity of the attack is $L_n(\frac{1}{3}, \sqrt[3]{\frac{32}{9}})$ in most significant situations, which matches the {\sl special} number field sieve's ({\sc snfs}) complexity. This sheds additional light on {\sc rsa}'s malleability in general and on {\sc rsa}'s resistance to affine forgeries in particular -- a problem known to be polynomial for $x_i > \sqrt[3]{n}$, but for which no algorithm faster than factoring was known before this work.