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

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.

## Authors

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 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 Coordinate-wise 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 coordinate-wise logical ("" and "") operations.

These results were obtained on the basis of the apparatus developed in our previous works on -adic (and, especially, measure-preserving) 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 1-Lipschitz measure-preserving function that is a homomorphism with respect to a given operation "*". Here the condition "1-Lipschitz" 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 coordinate-wise 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 coordinate-wise logical operations over In this case, the necessary condition for the non-triviality 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 coordinate-wise 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.

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 so-called 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 non-negative integers, then for we can consider notation in the sense of

 x(modpk)=[x]korx≡[x]k(modpk). (1.1)

Let and be positive integers. The set is a ball of radius with a center .

In this paper, we consider functions , which satisfy the Lipschitz condition with a constant 1 (i.e., 1-Lipschitz functions). Recall that is a 1-Lipschitz function if for all . This condition is equivalent to the following: implies for all .

For all a 1-Lipschitz transformation of the reduced mapping modulo is

 fk−1:Z/pkZ→Z/pkZ,z↦f(z)(modpk). (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

 f(x)=∞∑m=0Bmχ(m,x) (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.

1-Lipschitz 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 1-Lipschitz if and only if for all Assuming , we find that the function is 1-Lipschitz if and only if it can be represented as

 f(x)=∞∑m=0p⌊logpm⌋bmχ(m,x) (1.4)

for suitable

#### 1.2.2 Coordinate representation of 1-Lipschitz 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:

 f(x)=δ0(f(x))+pδ1(f(x))+…+pkδk(f(x))+…. (1.5)

According to Proposition 3.33 in ANKH , is a 1-Lipschitz function if and only if for every the k-th coordinate function does not depend on for all , i.e. for all .

Taking into account notation (1.1) for , we consider the following functions of -valued logic

 φk:{0,…,p−1}×…×{0,…,p−1}k+1→{0,…,p−1},

and

Then any 1-Lipschitz function can be represented as

 f(x)=f(x0+…+pkxk+…)=∞∑k=0pkφk(x0,…,xk). (1.6)

The function can be defined by its sub-functions obtained by fixing the first variables . Sub-function of the function which is obtained by fixing the variables is denoted by , where .

Thus, we can rewrite the 1-Lipschitz function as

 f(x)=f(x0+px1+…+pkxk+…)==φ0(x0)+∞∑k=1pkpk−1∑a=0Ia([x]k)φk,a(xk), (1.7)

where , if and otherwise.

We call the relation (1.7) the sub-coordinate representation of a 1-Lipschitz 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 non-negative 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.

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 measure-preserving if
for each measurable subset

Criteria of measure-preserving for 1-Lipschitz functions are presented in the following theorems.

###### Theorem 1.2.

(Unif0 , ANKH ) A 1-Lipschitz functions preserves the measure if and only if is bijective on for any

###### Theorem 1.3 (Theorem 2.1, MeraJNT ).

A 1-Lipschitz function represented by the van der Put series (1.4) preserves the measure if and only if

1. constitutes a complete set of residues modulo (i.e. is bijective modulo );

2. the elements in the set are all nonzero residues modulo for any

###### Theorem 1.4 (Theorem 3.1, Sol ).

A 1-Lipschitz function represented in the coordinate form (1.7) preserves the measure if and only if all functions and are bijective on .

### 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

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

2. 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 coordinate-wise 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

 xXORy =(x0⊕py0)+(x1⊕py1)p+… xANDy =(x0⊙py0)+(x1⊙py1)p+….

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 p-adic integers

In this section we give a description of the groups of automorphisms of the following algebraic systems:

1. where see subsection 2.1;

2. where see subsection 2.2.

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:

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

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

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

4. a composition of measure-preserving 1-Lipschitz functions is a measure-preserving 1-Lipschitz function.

In terms of dynamical systems, the problem of describing automorphisms of -adic integers reduces to describing the measure-preserving 1-Lipschitz functions, which preserve the operations of the considered algebraic system.

### 2.1 Groups of automorphisms on Zp 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 measure-preserving, 1-Lipschitz 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).

1. The group of automorphisms of the algebraic system consists of functions of the form:

 f(x)=Ax,

where and

2. The group of automorphisms of the algebraic system consists of functions of the form:

 f(x)={pkAkθs(1+pt)a,ifx=pkθ(1+tp),0,ifx=0 (2.1)

where , and

 A≢0(modp),a≢0(modp),GCD(s,p−1)=1.
###### Proof.

As we have already noted (see Section 1.4) elements of (or correspondingly) are 1-Lipschitz functions (the condition of preserving the congruence relations modulo ). To prove this Theorem, we describe all 1-Lipschitz functions, which define a homomorphism with respect to the considered operation, and then, using the results for measure-preserving 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 1-Lipschitz 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 non-zero function).

We write each non-zero -adic number with the aid of the Teichmüller representation (see, for example, p. 81 in Schikhof ), namely in the following form:

 x=pkθ(1+pt),k≥0,t∈Zp, (2.2)

and . Note that if then and any non-zero -adic number is represented as

Let and be a set of all non-zero 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 1-Lipschitz 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 1-Lipschitz function (as a restriction of the 1-Lipschitz function to the set ).

Let Since is a homomorphism, then

 f(P)={1,f(p),f(p2),…}.

As is a 1-Lipschitz 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)):

 f(x)={pk⋅Ak⋅θs⋅ϕ(1+pt),ifx=pkθ(1+tp),k≥0,0,ifx=0,

where and a 1-Lipschitz 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 ):

 g(τ1+τ2)= LNp(ϕ(EXPp(τ1+τ2)))= LNp(ϕ(EXPp(τ1)⋅EXPp(τ2)))= LNp(ϕ(EXPp(τ1)))+LNp(ϕ(EXPp(τ2)))=g(τ1)+g(τ2).

Therefore, there exists such that Since then

 EXPp(g(τ))=EXPp(a⋅τ)=EXPp(τ)a=ϕ(EXPp(τ)).

Let (and for ). Then

Thus, the function can be represented in the form

 f(x)=f(pkθ(1+pt))=pk⋅Ak⋅θs⋅(1+pt)a.

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

 bpkθ(1+p(t+prh))(modpk+r+1)=1pk+rBpkθ(1+p(t+prh))(modpk+r+1)≡≡1pk+r(f(pkθ(1+p(t+prh)))−f(pkθ(1+p(t)))≡≡aAkθsh(modp),r≥1,k≥1,
 bpkθ(modpk+1)=1pkBpkθ(modpk+1≡1pk(f(pkθ)−f(0))≡≡Akθs(modp),r=0,k≥1,
 bθ(modp)=Bθ(modp)≡f(θ)≡θs(modp),k=0.

Since then coincides with the set of all non-zero residues modulo if and only if and The set coincides with the set of all non-zero 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

 (s⋅d,a⋅b,A⋅(θB)s(1+pB1)a),

where

###### Theorem 2.4 (Logical operations).

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

 f(x)=f(x0+…+pkxk+…)=∞∑k=0pkφk(x0,…,xk),

where are -valued logical functions and

 φk(x0,…,xk)=α(k)0x0⊕pα(k)1x1⊕p…⊕pα(k)kxk,

where and

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

 f(x)=f(x0+px1+…+pkxk+…