Knapsack Public Key Cryptosystems and Diophantine Approximation

Abstract

This paper presents and analyzes cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation. Shamir’s attack on the basic Merkle-Hellman knapsack cryptosystem is shown to depend on the existence of “unusually good” simultaneous Diophantine approximations to a vector constructed from the public key. This aspect of Shamir’s attack carries over to multiply iterated knapsack cryptosystems: there are “unusually good” simultaneous Diophantine approximations to an analogous vector constructed from the public key. These “unusually good” simultaneous Diophantine approximations can be used to break multiply iterated knapsaçk cryptosystems provided one can solve a certain nonlinear Diophantine approximation problem. This nonlinear problem is solved in the simplest case and then used to give a new cryptanalytic attack on doubly iterated knapsack cryptosystems.