Symbolic protocol analysis with products and Diffie-Hellman exponentiation

We demonstrate that for any well-defined cryptographic protocol, the symbolic trace reachability problem in the presence of an Abelian operator (e.g., multiplication) can be reduced to solvability of a particular system of quadratic Diophantine equations. This result enables formal analysis of protocols that employ primitives such as Diffie-Hellman exponentiation, products, and xor, with a bounded number of role instances, but without imposing any bounds on the size of terms created by the attacker. In the case of xor, the resulting system of Diophantine equations is decidable. In the case of a general Abelian group, decidability remains an open equation, but our reduction demonstrates that standard mathematical techniques for solving systems of Diophantine equations are sufficient for the discovery of protocol insecurities.