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In this paper, we study groups of automorphisms of algebraic systems over aset of $p$-adic integers with different sets of arithmetic and coordinate-wiselogical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The mainresult of this paper is the description of groups of automorphisms of $p$-adicintegers with one or two arithmetic or coordinate-wise logical operations on$p$-adic integers. To describe groups of automorphisms, we use the apparatus ofthe $p$-adic analysis and $p$-adic dynamical systems. The motive for the studyof groups of automorphism of algebraic systems over $p$-adic integers is thequestion of the existence of a fully homomorphic encryption in a given familyof ciphers. The relationship between these problems is based on the possibilityof constructing a "continuous" $p$-adic model for some families of ciphers (inthis context, these ciphers can be considered as "discrete" systems). As aconsequence, we can apply the "continuous" methods of $p$-adic analysis tosolve the "discrete" problem of the existence of fully homomorphic ciphers.

Groups of automorphisms of p-adic integers and the problem of the existence of fully homomorphic ciphers