  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 $p^k,$ $k\ge 1.$ 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.
