Semi-Commutativity Sets of Morphisms over Finitely Generated Free Monoids.dvi

The notion of a semi-commutativity set for word mappings was defined in [3] as an abstraction of a problem in cryptography. The notion is of special interest in case the mappings are morphisms. Then rather surprising constructions become possible. We investigate such constructions, paying special attention to exceptional values of inverse mappings. Some of our results bear a close relation to certain important issues in the theory of formal languages. 1 Background of the problem Consider the following problem in the theory of cryptographic protocols. A seller, S, possesses a number of secrets. S has published a list of descriptions of the secrets and is offering them for sale at a price that is the same for each of the secrets. A buyer B wants to buy one of the secrets. However, he is not willing to disclose to anybody, not even to S, which of the secrets he wants. On the other hand, B should not learn more than one secret. These two seemingly contradictory requirements, S disclosing the secret B wants but no other secrets and S not knowing which secret he disclosed, can be fulfilled using cryptographic protocols based on one-way functions. The first solution suggested in [1] is very complicated. The solution in [7] is simple but requires several buyers. Recently a simple solution in the original set-up of one buyer has been given in [4]. The protocol for secret selling of secrets can also be used as a building block for more complicated protocols, [2]. The reader is referred to [6] as a general introduction to one-way functions and cryptographic protocols. The work reported here has been supported by the Project 11281 of the Academy of Finland Academy of Finland and Mathematics Department, University of Turku, 20500 Turku, Finland Institute of Mathematics,Str.Academiei 14, 70109 Bucuresti, Romania