Some analysis on conjugacy search problem for Diffie-Hellman protocol

– The field in nonabelian group-based cryptosystem have gain attention of the researchers as it expected to offers higher security when confronted with quantum computational due to more complex algebraic structures. Hence, this paper intents to give an overview on Diffie-Hellman protocol considering the mathematically


INTRODUCTION
The public key cryptography was firstly introduced by Diffie and Hellman in 1976 namely Diffie-Hellman key exchange [1]. The scheme is one of the most common public keys currently in use along with the RSA cryptosystem, the ElGamal cryptosystem and the elliptic curve cryptosystem. These public keys mainly based on the number theory and hence depend on the structure of abelian groups. With the increasing power of computing machinery and the realisation of quantum computers, the cryptosystem for both public and classical key becoming less secure and that means abelian groups are too easy to understand and becoming vulnerable for quantum computing. Hence the needs to improve the security and much attention to be put on the nonabelian groups from the algebraic point of view [2].
The classical Diffie-Hellman protocol as mentioned in [1], where such simplest and original implementation of this protocol uses * , p Z the multiplicative group of integers modulo p, where p is prime and g is the generator with modulo p.
The Diffie-Hellman (DH) protocol as presented in [3] as follows: Let G be a cyclic group with g as the generating element in G.
1) Alice and Bob agree on a group G of order q and an element g in G.
2) Alice picks a random natural number m q < and sends m g to Bob.
3) Bob picks a random natural number n q < and sends n g to Alice. 4) Alice computes secret key, ( ) . The property of the multiplicative group of integers is commutative as in , mn nm = thus both Alice and Bob are now in possession of the same group element , where it came out as the shared secret key. For the protocol to be considered secure, G and g are needed to be chosen properly. The difficulty of such problem lies on the recovery of mn g from , , m g g and n g (publicly known) whereby it means to recover the shared secret key, K. Essentially, the hardness of the well-known mathematical problem in number theory namely Integer Factorization Problem and Discrete Logarithm Problem are the ground to the problem in cryptography and the security of Diffie-Hellman protocol relies on the Diffie-Hellman problem or the Discrete Logarithm Problem which are defined as follows, respectively [4]: Diffie-Hellman Problem: Let G be a group. If , , x y g g g G ∈ are known, find the value of . ABSTRACT -The field in nonabelian group-based cryptosystem have gain attention of the researchers as it expected to offers higher security when confronted with quantum computational due to more complex algebraic structures. Hence, this paper intents to give an overview on Diffie-Hellman protocol considering the mathematically hard problem such as the conjugacy search problem in a group G. In this paper we provide examples for G of non abelian group particularly the group of SL (2,3).
However, both Integer Factorization Problem and Discrete Logarithm Problem would be efficiently solved on the realisation of the quantum computer, hence the emergence of numerous group-based cryptosystem in recent times has raised the attention of researchers. In other words, some mathematical problems involving non-commutative groups are substantially harder to solve when the quantum computation algorithm applied [5]. Thus, this paper mainly concerned with group-based cryptography particularly the attention is directed towards the nonabelian groups [6]. Some examples on the matrix groups are provided as well in the preliminaries section.
The rest of the paper is structured as follows. In the preliminary section, the basic notions used for the research are provided. The next section is the main part where the proofs for the results obtained are presented and then followed by concluding remark in the last section.

PRELIMINARIES
In this section, some definitions that are important in the research are stated. The definiton of Conjugacy Search Problem is given in the Definition 2.1 as follows: then the relation of the conjugate of g by x, that is 1 .
The task is to find some x G ∈ in the above relation, thus it is known as Conjugacy Search Problem (CSP). The hardness of the Conjugacy Search Problem upon the group is taken into consideration other than assuming that the group's elements are easily stored and manipulated. In Definition 2.2 and 2.3, the definition of the conjugacy class and center of a group are defined respectively as follows.

Definition 2.2 [8] (Conjugacy Class):
If G is a group, then the equivalence class of a G ∈ under the relation "y is conjugate of x in G" is called the conjugacy class of a; it is denoted by .

G a
The conjugacy class G a is the set of all the conjugates of a in G. [9] Definition 2.3 [10] (Center): The set of all elements of G which commute with every element of G is a center of a group G, that is Next, proposition with the condition for the element to be the center of the group is provided.
Proposition 2.4 [10]: If a is the only element of order k in G, then a is in the center of G.
In this paper, the group of SL(2,3) is studied and the elements of the group are classified according to its conjugacy classes. In particular, SL(2,3) is a nonabelian group of order 24 is given as Example 2.5 as follows.

Example 2.5:
The group SL (2,3) is the multiplicative group of two-by-two invertible matrices of determinant 1 with entries from the field 3 {0,1, 1} Ζ = − under addition and multiplication modulo 3 [8]. More detail of the elements with the order in the respective conjugacy class provided in the table in Table 1. By Definition 2.1, there are seven conjugacy classes in SL(2,3) that has been identified as provided in Table 1.
Likewise, the elements of SL(2,3) with respect to Diffie-Hellman protocol are provided and it is identified that such elements in the same conjugacy classes will not necessarily generate the same shared secret key. Example 2.6 shows an example for this case.

RESULTS AND ANALYSIS
The public key exchange protocol in the spirit of Diffie-Hellman protocol are provided and examined on any nonabelian group.
In this case, general protocol is specialised as follows: for any element , . g x G ∈ 1) Alice and Bob agree on a group G and public elements , , .
x y g g g G ∈ 2) Alice selects private elements x and Bob selects y as the private elements. and send the element to Alice. 4) Alice compute secret key, and Bob compute secret key, 1 .
be the common shared secret key.
In the subsequent propositions, some conditions given to be satisfied for nonabelian group in Diffie-Hellman protocol namely Proposition 3.1 and 3.2.

Proposition 3.1: Suppose ,
x y G ∈ be the private keys where G is nonabelian group in Diffie-Hellman protocol using Conjugacy Search Problem. If it holds true for xy yx = , then the shared secret key exist.
Proof: Let the public key generated by Alice is Such element from previous example with an order of 2 is shown to be the center of the group in the following Corollary 3.3. Then, in Proposition 3.4, such shared secret key achieved given the stated condition.

Corollary 3.3:
If a is the only element in G of order 2, then the element is in the center of the group.
Proof: By Proposition 2.4, a is the center of the group and will commute with any element in G.

Proposition 3.4:
Let g G ∈ be generator and the only element of order 2, then it will generate the shared secret key the same element as g in G.
Proof: In order to prove the shared secret key is exist, we need to prove that ( ) ( ) Suppose that g G ∈ be generator of order 2. Thus, g will commute with any , x y G ∈ ,i.e . xy yx = By Proposition 3.1, the secret shared key is exist and the key is ( )

CONCLUSION
The Diffie-Hellman key exchange protocol is presented based on non-abelian group in particular the mathematically hard problem that is Conjugacy Search Problem is considered for the nonabelian group. Some of the proofs for some cases in nonabelian groups that will work for the Diffie-Hellman key exchange are provided as well. Our protocol can be based on any nonabelian group though in this paper the group SL(2,3) is given as an example.