Journal of Cryptology

, Volume 23, Issue 2, pp 169–186 | Cite as

The RSA Group is Pseudo-Free

Open Access


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).


Cryptographic assumptions Pseudo-free Abelian group Strong RSA problem Safe primes 


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of California at San DiegoLa JollaUSA

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