1 Introduction
In this paper, we study groups of automorphisms of adic integers We consider the set as an algebraic system with a given set of binary operations and relations (or predicates). We recall that the algebraic system is a triple , where is a set (i.e., a carrier of system ), is a set of operations (in our case binary) on (i.e., an operator domain), and is a set of relations (in our case binary) on (i.e., a predicate domain), see, for example, Coh and Maltcev . A predicate on (in our case binary) is a mapping We denote a predicate as instead of
. In fact, the predicate is the characteristic function of some subset of
i.e. relations on Therefore, the concepts of relation and predicate are treated as synonyms.An automorphism of an algebraic system is a bijective mapping such that , for any operation Moreover, if , then for any predicate , (in other words, preserves all the operations and predicates (or relations)).
For adic integers, we consider the algebraic system of the following form where predicate domain is determined by the congruence relations modulo and operator domain consists of one or two operations from the set Operations "" and "" are arithmetic operations on Coordinatewise logical operations "" and "" are also given on Their meaning is to implement the logical operations of addition and multiplication on the set for each coordinate of the canonical representation of a adic integer (for more details, see Section 1.4). To denote the algebraic systems under consideration, we shall use the notation for one operation and for two operations, where
The main results are presented in Section 2. In Theorems 2.1 and 2.4, we give a description of groups of automorphisms of algebraic systems of adic integers where Here "" is one of the arithmetic ("" and "") or coordinatewise logical ("" and "") operations.
These results were obtained on the basis of the apparatus developed in our previous works on adic (and, especially, measurepreserving) dynamical systems MeraJNT , SOL , see also pioneering papers of V. Anashin Tfunc An_avt_1 and monograph ANKH . See also works V00 , V1 on the general theory of adic dynamical system and more generally interrelation between number theory and dynamical systems. In particular, in terms of adic dynamics, an automorphism is a 1Lipschitz measurepreserving function that is a homomorphism with respect to a given operation "*". Here the condition "1Lipschitz" corresponds to the preservation of the predicates that define the congruence relations modulo and the condition "preserves the measure" corresponds to the bijectivity (reversibility) of the function whereby the automorphism is determined.
In Theorem 2.5, we consider the case where any two operations from a set of arithmetic and coordinatewise logical operations are defined on It turned out that all groups of automorphisms of algebraic system of adic integers for are trivial. Due to the result of Theorem 2.5, there arises the question of the existence of an algebraic system of adic integers where and are "new" operations for which the group of automorphisms differs from the trivial group. In Proposition 2.6, we describe all the operations "" (here ) on for which the groups of automorphisms of the algebraic systems are not trivial (here operations "" are given by a convergent series on ).
We also consider the case where "new" operations are given as formulas in a basis of two arbitrary arithmetic and coordinatewise logical operations over In this case, the necessary condition for the nontriviality of the group of automorphisms is that the set of formulas in the basis of the operations does not coincide with the set of formulas in the chosen basis of arithmetic or coordinatewise logical operations over (see Proposition 2.9).
Our main reason to consider such groups of automorphisms of adic integers is the possibility of using the apparatus of adic analysis to introduce the transformations on which can be used to construct fully homomorphic ciphers. Recall that a ciher is a family of bijective mappings of a set of open texts into a set of ciphered texts , where the parameter is a key. Note that in the general case only required property of injectivity, but usually, it is considered bijective transformation. We consider ciphers for which the sets and coincide and consist of words of finite length in the alphabet for prime number In this case, if one operation (or two operations) on is given and for any the transformation is a homomorphism with respect to this operation (respectively, to these operations), then it is said that the cipher is homomorphic (respectively, fully homomorphic). The problem of constructing a fully homomorphic encryption is relevant for the secure cloud computing (for more details, see 3).
It turns out that algebraic systems of adic integers for are "continuous" adic models of the ciphers under consideration with operations that are discrete analogs of operations in The description of ciphers for which there exist "continuous" adic models, as well as the rationale for the choice of such models, are presented in Section 3. If there is a description of automorphism groups of adic integers , in the framework of a "continuous" adic model, then, choosing the corresponding "discrete" analogues of these automorphisms, we can construct homomorphic (fully homomorphic) ciphers from the family of ciphers under consideration.
We recall some definitions related to the adic analysis and we introduce the necessary notations.
1.1 adic numbers
For any prime number the adic norm is defined on in the following way. For every nonzero integer let be the highest power of which divides . Then we define , and .
The completion of with respect to the adic metric is called the field of adic numbers . The metric satisfies the socalled strong triangle inequality . The set is called the set of adic integers.
Hereinafter, we will consider only the adic integers. Every can be expanded in canonical form, namely, in the form of a series that converges for the adic norm:
Partial sums of this series, we denote as i.e.
If residues of the ring are set as minimal nonnegative integers, then for we can consider notation in the sense of
(1.1) 
Let and be positive integers. The set is a ball of radius with a center .
1.2 adic functions
In this paper, we consider functions , which satisfy the Lipschitz condition with a constant 1 (i.e., 1Lipschitz functions). Recall that is a 1Lipschitz function if for all . This condition is equivalent to the following: implies for all .
For all a 1Lipschitz transformation of the reduced mapping modulo is
(1.2) 
A mapping is well defined (i.e. the does not depend on the choice of representative in the ball ). We use the notation taking into account (1.1).
1.2.1 Van der Put series
Continuous adic functions can be represented in the form of the van der Put series. The van der Put series is defined in the following way. Let be a continuous function. Then there exists a unique sequence of adic coefficients such that
(1.3) 
for all Here the characteristic function is given by if and otherwise, where if , and is uniquely defined by the inequality otherwise (see Schikhof’s book Schikhof for a detailed presentation of the theory of the van der Put series).
The coefficients are related to the values of the function in the following way. Let , , and then if and otherwise.
1Lipschitz functions in terms of the van der Put series were described in Schikhof . We follow Theorem 3.1 Tfunc as a convenience for further study. In this theorem, the function presented via the van der Put series is 1Lipschitz if and only if for all Assuming , we find that the function is 1Lipschitz if and only if it can be represented as
(1.4) 
for suitable
1.2.2 Coordinate representation of 1Lipschitz functions
In this section we describe a coordinate representation of adic functions, see, for example, YuRecent .
Let functions be the th digit in a base expansion of the number i.e.
Any map can be represented in the form:
(1.5) 
According to Proposition 3.33 in ANKH , is a 1Lipschitz function if and only if for every the kth coordinate function does not depend on for all , i.e. for all .
Then any 1Lipschitz function can be represented as
(1.6) 
The function can be defined by its subfunctions obtained by fixing the first variables . Subfunction of the function which is obtained by fixing the variables is denoted by , where .
Thus, we can rewrite the 1Lipschitz function as
(1.7) 
where , if and otherwise.
We call the relation (1.7) the subcoordinate representation of a 1Lipschitz function see Subcoord_Hensel and Subcoord_Hensel_1 . Functions , can be considered as a function of valued logic and as a transformation of the ring .
Remark 1.1.
If residues of the ring are set as minimal nonnegative integers, then operations in the ring can be regarded as operations on the set In this article, it will be convenient to use a special notation for such operations. Namely, we denote these operations on the set by "" and "" given as and correspondingly.
1.3 adic dynamics
Dynamical system theory studies trajectories (orbits), i.e. sequences of iterations of the function : where .
We consider a adic autonomous dynamical system for more details see, for example, Tfunc V1 , as well as Jeong . The space
is equipped with a natural probability measure
, namely, the Haar measure ().A measurable mapping is called measurepreserving if
for each measurable subset
Criteria of measurepreserving for 1Lipschitz functions are presented in the following theorems.
Theorem 1.2.
1.4 Automorphisms of algebraic systems
Recall that an algebraic system is an object where is the carrier set, is the set of operations on and is the set of predicates on A predicate on the set is considered as a characteristic function of the relation on this set (that is, the predicate determines the relation and vice versa). Further, we consider only binary operations and predicates (relations).
We remind that an automorphism of the algebraic system is a bijective mapping such that

for any operation "" from the map is a homomorphism with respect to the operation "", that is for ;

for any predicate from it follows for (or in terms of relations, , where is a relation defined by a predicate ).
Hereinafter, we consider the algebraic system for which the carrier is a set of adic integers, namely,
The family of predicates determines the congruence relations modulo
A set of operations consists of one or two operations from the set given on Here operations "" and "" are arithmetical operations on , and coordinatewise logical operations "" and "" are defined in the following way. Let adic numbers be defined in the canonical form. Then, taking into account Remark 1.1, we have
In this paper, we consider algebraic systems of the form or , where . These algebraic systems differ only in the set of operations (the carrier and the set of predicates for these systems are fixed). Therefore, we shall specify only the operations under consideration to denote such algebraic systems. For example, through a we denote the algebraic system for
The set of all automorphisms of an algebraic system with respect to the operation of a composition of automorphisms forms a group which in our notation will be written in the form (and in the case of two operations). We denote an identity element of the group of automorphisms by
2 Groups of automorphisms of adic integers
In this section we give a description of the groups of automorphisms of the following algebraic systems:
As shown in Theorem 2.5, all groups of automorphisms , where are trivial groups (i.e., groups that have only one element). In this regard, in the section 2.3 we consider the question of the existence of algebraic systems of the form where are some "new" operations, for which the group of automorphisms differs from the identity.
We use the apparatus developed in our previous works on adic dynamical systems, see, for example, MeraJNT and SOL , to describe the groups of automorphisms of adic integers.
This possibility is explained by the following circumstances:

a function preserves all relations modulo if and only if is a 1Lipschitz function. Indeed, if follows from , then this is equivalent to ;

a composition of 1Lipschitz functions is a 1Lipschitz function. Indeed, ;

a 1Lipschitz function is bijective on if and only if preserves the measure, see Corollary 4.5. from Subcoord_Hensel ;

a composition of measurepreserving 1Lipschitz functions is a measurepreserving 1Lipschitz function.
In terms of dynamical systems, the problem of describing automorphisms of adic integers reduces to describing the measurepreserving 1Lipschitz functions, which preserve the operations of the considered algebraic system.
2.1 Groups of automorphisms on with one operation
In this section, we describe the groups of automorphisms of algebraic systems for each operation from the set
Note that the functions that define the homomorphisms with respect to arithmetic operations "" and "" on the adic analogue of the field of complex numbers were considered in Schikhof . In contrast to this case, we consider the functions that preserve the measure and define the homomorphism on for a wider set of binary operations. A full description of measurepreserving, 1Lipschitz functions, which define homomorphisms for specific operations on is presented in Theorem 2.1 (for arithmetic operations) and Theorem 2.4 (for logical operations).
Theorem 2.1 (Arithmetic operations).

The group of automorphisms of the algebraic system consists of functions of the form:
where and

The group of automorphisms of the algebraic system consists of functions of the form:
(2.1) where , and
Proof.
As we have already noted (see Section 1.4) elements of (or correspondingly) are 1Lipschitz functions (the condition of preserving the congruence relations modulo ). To prove this Theorem, we describe all 1Lipschitz functions, which define a homomorphism with respect to the considered operation, and then, using the results for measurepreserving functions, we find the functions that are bijective on As a result, we obtain a description of the elements of the groups and
Let defines a homomorphism with respect to the operation "". Then Let Since 1Lipschitz function is continuous on and is dense in , then . The function preserves the measure if and only if It is clear that the function defines a homomorphism with respect to addition on
Let defines a homomorphism with respect to multiplication on . Let be distinct from the identity function. In particular, Indeed, if this is not so, then from follows for any In addition, we assume that there exists such that (i.e. is a nonzero function).
We write each nonzero adic number with the aid of the Teichmüller representation (see, for example, p. 81 in Schikhof ), namely in the following form:
(2.2) 
and . Note that if then and any nonzero adic number is represented as
Let and be a set of all nonzero Teichmüller representatives (in other words, is the set of all solutions of equation in ). is a cyclic group (with respect to the operation of multiplication) generated, for example, by the element Notice, that Indeed, let Since is a homomorphism, then i.e. If for some , then In particular, the homomorphism induces a mapping of the form
As is a 1Lipschitz function, then i.e. The set forms a group with respect to the operation of multiplication. Indeed, for and is contained in the set of invertible elements of the ring This means that induces a homomorphism It is clear that is a 1Lipschitz function (as a restriction of the 1Lipschitz function to the set ).
Let Since is a homomorphism, then
As is a 1Lipschitz function, then , i.e. for some
Thus, the function , which defines a homomorphism with respect to multiplication on can be represented in the form (taking into account the representation from the relation (2.2)):
where and a 1Lipschitz function defines a homomorphism with respect to multiplication on
Let us find the representation of the function Let be the adic exponential function ( for ) and be the adic logarithm ( for ). We consider the function ( for ) such that Then, the function defines a homomorphism with respect to addition on (on for ):
Therefore, there exists such that Since then
Let (and for ). Then
Thus, the function can be represented in the form
Performing the corresponding calculations, we see that the function of this type defines a homomorphism on with respect to multiplication.
Let us find the values where the function of the form (2.1) preserves the measure. For this, we use the criterion of Theorem 1.3. Let us find the value of the van der Put coefficients of the function Let Then and
Since then coincides with the set of all nonzero residues modulo if and only if and The set coincides with the set of all nonzero residues modulo as Then, by Theorem 1.3 the function preserves the measure if and only if ∎
Remark 2.2.
If in (2.1) we set for some then That is, all such polynomials define a homomorphism with respect to multiplication on . Functions of the form for do not preserve the measure.
Remark 2.3.
We note that each element (or function) is uniquely determined by the set of parameters where and Here is the group of units of the ring and is the group of units of the ring . Let the elements (or functions) be defined by the parameters and Then the composition is determined by the parameters
where
Theorem 2.4 (Logical operations).

The group of automorphisms of the algebraic system consists of functions given in the coordinate form:
where are valued logical functions and
where and

The group of automorphisms of the algebraic system consists of functions given in the coordinate form:
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