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.