Attacks on public key cryptosystems based on free partially commutative monoids and groups

At INDOCRYPT 2003, Abisha, Thomas and Subramanian have proposed a public key encryption scheme and a zero-knowledge authentication protocol based on the word problem on monoids, as well as a group variant of these systems. We here present a total break attack on each of the two encryption schemes. The complexity bounds of our algorithms show that these schemes are insecure for practical parameter sizes. In the monoid setting, we go one step further by proposing an algorithm that breaks the NP-hard problem underlying both the encryption scheme and the zero-knowledge protocol, as well as an upper bound on its complexity.