Practical Cryptanalysis of a Public Key Cryptosystem Based on the Morphism of Polynomials Problem

Multivariate Public Key Cryptography (MPKC) has intensively and rapidly developed during the past three decades. MPKC is a promising candidate for post-quantum cryptography. However, designing it is universally regarded as a difficult task to design a secure MPKC foundation scheme, such as an encryption scheme and key exchange scheme. In this work, we investigate the security of a new public key cryptosystem that is based on the Morphism of Polynomials (MP). The public key cryptosystem proposed by Wang et al. (Wuhan University, China) comprises a key exchange scheme and encryption scheme. Its security can be provably reduced to the hardness of solving a new difficult problem, namely, the Decisional Multivariate Diffie Hellman (DMDH) problem. This problem is a variant of the MP problem, which is difficult to solve by random systems. We present a proposition that reduces the DMDH problem to an easy example of the MP problem. Then, we propose an efficient algorithm for the Key Recover Attack (KRA) on the schemes of the public key cryptosystem. In practice, we are able to entirely break the cryptosystem's claimed parameter of 96 security levels in less than 17.252 s. Furthermore, we show that finding parameters that yield a secure and practical scheme is impossible.