Stream/block ciphers, difference equations and algebraic attacks

In this paper we model a class of stream and block ciphers as systems of (ordinary) explicit difference equations over a finite field. We call this class "difference ciphers" and we show that ciphers of application interest, as for example systems of LFSRs with a combiner, Trivium and Keeloq, belong to the class. By using Difference Algebra, that is, the formal theory of difference equations, we can properly define and study important properties of these ciphers, such as their invertibility and periodicity. We describe then general cryptanalytic methods for difference ciphers that follow from these properties and are useful to assess the security. We illustrate such algebraic attacks in practice by means of the ciphers Bivium and Keeloq.