Cryptanalysis of the Square Cryptosystems

Following the cryptanalyses of the encryption scheme HFE and of the signature scheme SFLASH, no serious alternative multivariate cryptosystems remained, except maybe the signature schemes UOV and HFE--. Recently, two proposals have been made to build highly efficient multivariate cryptosystems around a quadratic internal transformation: the first one is a signature scheme called square-vinegar and the second one is an encryption scheme called square introduced at CT-RSA 2009. In this paper, we present a total break of both the square-vinegar signature scheme and the square encryption scheme. For the practical parameters proposed by the authors of these cryptosystems, the complexity of our attack is about 235 operations. All the steps of the attack have been implemented in the Magma computer algebra system and allowed to experimentally assess the results presented in this paper.