On the Equivalent Keys in Multivariate Cryptosystems

The number of equivalent keys in multivariate cryptosystem is closely related to the scheme security. This study analyzes the structure of the private key space in some multivariate schemes. The result gives the lower bounds on the number of equivalent keys of some variants of the hidden field equation (HFE) scheme including plus, minus-plus, embedding, and internal perturbation. This method estimates the number of invertible transformations which maintain the form of the central map invariant. Furthermore, a formal proof shows that the two modifications of fixing and embedding are equivalent in security analyses of multivariate schemes. Also this paper corrects previous proofs in Wolf’s work on the number of equivalent keys in HFEv, the unbalanced oil and vinegar (UOV) scheme, and the stepwise triangular systems (STS).