Group-Based Key Exchange Protocol Based on Complete Decomposition Search Problem

Abstract

Let G be a finite non-abelian group. Let \(A_1,\cdots , A_k\) be non-empty subsets of G, where \(k\ge 2\) is an integer such that \(A_i\cap A_j = \emptyset \) for integers \(i,j= 1,\cdots , k\) \((i \ne j)\). We say that \((A_1, \cdots , A_k)\) is a complete decomposition of G if the product of subsets \(A_{i_1} \cdots A_{i_k} = \{a_{i_1}...a_{i_k} | a_{i_j}\in A_{i_j}; j=1,\cdots , k\}\) coincides with G where the \(A_{i_j}\) are all distinct and \(\{A_{i_1},\cdots , A_{i_k}\}= \{A_1,\cdots , A_k\}\). The complete decomposition search problem in G is defined as recovering \(B \subseteq G\) from given A and G such that \(AB=G\). The aim of this paper is twofold. The first aim is to propose the complete decomposition search problem in G. The other objective is to provide a key exchange protocol based on the complete decomposition search problem using generalized quaternion group \(Q_{2^n}\) as the platform group for integer \(n \ge 3\). In addition, we show some constructions of complete decomposition of generalized quaternion group \(Q_{2^n}\). Further, we propose an algorithm that can solve computational complete decomposition search problem and show that the algorithm takes exponential time to break the scheme.