Cryptanalysis of RSA Using Algebraic and Lattice Methods

Abstract

This paper applies tools from the geometry of numbers to solve several problems in cryptanalysis. We use algebraic techniques to cryptanalyze several public key cryptosystems. This paper focuses on RSA and RSA-like schemes, and use tools from the theory of integer lattices to get our results. We believe that this field is still underexplored, and that much more work can be done utilizing connections between lattices and cryptography. This paper studies the security of the RSA public key cryptosystem under partial key exposure. We show that for short public exponent RSA, given a quarter of the bits of the private key an adversary can recover the entire private key. Similar results (though not as strong) are obtained for larger values of the public exponent e. Our results point out the danger of partial key exposure in the RSA public key cryptosystem. This paper shows that if the secret exponent d used in the RSA public key cryptosystem is less than N0.292, then the system is insecure. This is the first improvement over an old result of Wiener showing that when d is less than N0.25 the RSA system is insecure.