Bypassing the decomposition attacks on two-round multivariate schemes by a practical cubic round

It was reported that a multivariate public key cryptosystem (MPKC) could be strengthened if its public key is generated by composition of two original public keys. In fact, two existing keys are used to constitute two quadratic rounds of the new key. But such a two-round scheme, called 2R, was claimed to have been decomposed, and even a further improved 2R, named 2R-, was shown to be similarly vulnerable. The result casts doubts on the principle of using a two-round structure to improve the security. However, this study clearly states that the decomposition attacks depend on the prerequisite that either of the rounds is quadratic. It shows that these attacks, even the conceivable extended ones, do not work in theory or in practice if the first round is of higher degree, although the threat still remains when only the degree of the second round is changed. Therefore adopting a cubic first round becomes a rule in the design of the 2-round schemes. The analysis and experiments in this study also demonstrate that the new schemes with such an indecomposable two-round public key can provide the desired security against many other known and potential attacks on MPKCs and that the key size can be practically controlled.