Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem

Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC, and E1Gamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found to be able to solve certain mathematical problems on non-commutative 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.