Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem

Abstract

Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC, and ElGamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found that are able to solve certain mathematical problems on noncommutative algebraic structures. Against this background, Raulynaitis et al. have proposed a novel asymmetric cipher protocol using a matrix decomposition problem. Their proposed scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we show that the asymmetric cipher of the non-commutative cryptography scheme is vulnerable to a linear algebra attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. We also propose an improvement to enhance the scheme of Raulynaitis et al.

摘要

创新点

量子计算技术的发展对基于大整数因子分解,离散对数等问题具有交换代数结构的密码体制 (如 RSA,ECC 和 ElGamal 密码)构成威胁, 因此研究具有非交换代数结构的密码体制是一项富有挑战性的课题.针对该课题, Raulynaitis 等人基于矩阵分解构造了一个非对称密码协议. 本文对基于有限域上的非对称密码协议,提出了一种结构攻击方法并且给出了对应的算法描述和有效性分析.通过分析可知, 该结构攻击算法能够在多项式计算复杂度内从相关的公钥获得等价密钥. 最后,本文在给出攻击算法的基础上对该非对称密码协议给出一个修正方案.