Groups of automorphisms of p-adic integers and the problem of the existence of fully homomorphic ciphers
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In this paper, we study groups of automorphisms of algebraic systems over a set of p-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo p^k, k> 1. The main result of this paper is the description of groups of automorphisms of p-adic integers with one or two arithmetic or coordinate-wise logical operations on p-adic integers. To describe groups of automorphisms, we use the apparatus of the p-adic analysis and p-adic dynamical systems. The motive for the study of groups of automorphism of algebraic systems over p-adic integers is the question of the existence of a fully homomorphic encryption in a given family of ciphers. The relationship between these problems is based on the possibility of constructing a "continuous" p-adic model for some families of ciphers (in this context, these ciphers can be considered as "discrete" systems). As a consequence, we can apply the "continuous" methods of p-adic analysis to solve the "discrete" problem of the existence of fully homomorphic ciphers.
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