The RSA group is pseudo-free

Abstract

We prove, under the strong RSA assumption, that the group of invertible integers modulo the product of two safe primes is pseudo-free. More specifically, no polynomial-time algorithm can output (with non negligible probability) an unsatisfiable system of equations over the free Abelian group generated by the symbols g 1,⋯,g n, together with a solution modulo the product of two randomly chosen safe primes when g 1,⋯,g n are instantiated to randomly chosen quadratic residues. Ours is the first provably secure construction of pseudo-free Abelian groups under a standard cryptographic assumption and resolves a conjecture of Rivest (Theory of Cryptography Conference-Proceedings of TCC 2004, LNCS, vol. 2951, pp. 505-521, 2004).