Novel Weakness Multivariate Quadratic Structures Detected within Macaulay Matrix

The security of a Multivariate Public-Key Cryptosystem (MPKC) is based on the hard mathematical problem of solving Multivariate Quadratic (MQ) equations over finite fields, also known as the MQ problem. An MPKC has the potential to be a post-quantum cryptosystem. In this paper, we identify new weaknesses in the Macaulay matrix identified via Wang's technique, which was initially designed for solving multivariate quadratic equation systems. This new weakness occurs in the case of random coefficients in any column vector for different variables of monomials and random coefficients are assigned to other monomials. The weakness is exposed through the use of Gaussian elimination to obtain a univariate equation. We illustrate our findings using a random example.