Equivalent Keys in HFE, C*, and Variations

In this article, we investigate the question of equivalent keys for two Multivariate Quadratic public key schemes HFE and C∗−− and improve over a previously known result, to appear at PKC 2005. Moreover, we show a new non-trivial extension of these results to the classes HFE, HFEv, HFEv-, and C∗−−, which are cryptographically stronger variants of the original HFE and C∗ / MIA schemes. In particular, we are able to reduce the size of the private — and hence the public — key space by at least one order of magnitude. While the results are of independent interest themselves, we also see applications both in cryptanalysis and in memory efficient implementations.