Two-sided multiplications are reduced to one-sided multiplication in linear piece in hand matrix methods

The linear Piece In Hand (PH, for short) matrix methods are general prescriptions which can be applicable to any type of multivariate public key cryptosystems (MPKCs, for short) for the purpose of enhancing their security. Among them, the primitive linear PH matrix method was introduced in our previous work [S. Tsujii, K. Tadaki, and R. Fujita, Cryptology ePrint Archive, Report 2004/366, December 2004] to explain the notion of the PH matrix methods in general in an illustrative manner and not for a practical use to enhance the security of any given MPKC. In 2005, for the purpose of enhancing the security of the primitive linear PH matrix method to a practical level, Akasaki proposed a variant of the primitive linear PH matrix method, called the two-sided linear PH matrix method. In this paper we show that the two-sided linear PH matrix method is reduced to the primitive linear PH matrix method. Based on this, we show that the two-sided linear PH matrix method cannot be more secure than the primitive linear PH matrix method.