Equivalent keys in HFE, C*, and variations

Abstract

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, which appeared 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* 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 and several orders of magnitude on average. While the results are of independent interest themselves as they broaden our understanding of Multivariate Quadratic schemes, we also see applications both in cryptanalysis and in memory efficient implementations. © Springer-Verlag Berlin Heidelberg 2005.