ИСТИНА

Algebraic dynamics studies dynamics on algebraic structures, i.e., on sets endowed with operations.As a cryptographic primitive normally is constructed from some basic operations (instructions) and as it somehow transforms input information, to study these transformations various approaches from algebraic dynamics can be applied. Initially, the first application was the ergodic theory of T-functions: It turned out that bijectivity/single cycle property of T-functions are respectively measure-preservation/ergodicity of non-expanding transformations of the space of 2-adic integers which are treated as infinite binary sequences. Later, it was found that much more complicated primitives and ciphers can be considered as dynamical systems, namely, as wreath products of T-functions, LFSRs, etc. Moreover, it turned out that various crucial cryptographic properties (e.g., linear complexity, distribution, etc.) of the ciphers can be estimated by using powerful dynamical tools. Now an effective toolbox which include various methods from both Archimedean and non-Archimedean analysis and the respective dynamics is developed. By using the toolbox it is possible now to study rather complicated behavior of various automata representing ciphers, hash functions, etc. Although methods of algebraic dynamics have already been successfully applied to find vulnerabilities in some ciphers as well as to give some reasoning related to security of other cryptographic algorithms, the challenge now is to apply the dynamical approach to security proofs.