Quantum cryptanalysis of hidden linear functions

Abstract

Recently there has been a great deal of interest in the power of “Quantum Computers” [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor’s to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a ‘hidden linear form’ can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois fields and elliptic curves. Finally, we introduce the notion of ‘junk bits’ which are helpful when performing classical computations that are not injective.