Modular multiplication and exponentiation are common operations in modern cryptography. Unification problems with respect to some equational theories that these operations satisfy are investigated. Two different but related equational theories are analyzed. A unification algorithm is given for one of the theories which relies on solving syzygies over multivariate integral polynomials with noncommuting indeterminates. For the other theory, in which the distributivity property of exponentiation over multiplication is assumed, the unifiability problem is shown to be undecidable by adapting a construction developed by one of the authors to reduce Hilbert's 10th problem to the solvability problem for linear equations over semi-rings. A new algorithm for computing strong \Groebner\ bases of right ideals over the polynomial ring Z[x1, ..., xn] is proposed; unlike earlier algorithms proposed by Baader as well as by Madlener and Reinert which work only for right admissible term orderings with the boundedness property, this algorithm works for any right admissible term ordering. The algorithms for some of these unification problems are expected to be integrated into Naval Research Lab.'s Protocol Analyzer (NPA), a tool developed by Catherine Meadows, which has been successfully used to analyze cryptographic protocols, particularly emerging standards such as the Internet Engineering Task Force's (IETF) Internet Key Exchange and Group Domain of Interpretation protocols. Techniques from several different fields -- particularly symbolic computation (ideal theory and Groebner basis algorithms) and unification theory --- are thus used to address problems arising in state-based cryptographic protocol analysis.
To view entire paper click here