International Association
for Cryptologic Research

We propose a new multivariate probabilistic encryption scheme with decryption errors MQQ-ENC that belongs to the family of MQQ-based public key schemes. Similarly to MQQ-SIG, the trapdoor is constructed using quasigroup string transformations with multivariate quadratic quasigroups, and a minus modifier with relatively small and fixed number of removed equations. To make the decryption possible and also efficient, we use a universal hash function to eliminate possibly wrong plaintext candidates. We show that, in this way, the probability of erroneous decryption becomes negligible.

MQQ-ENC is defined over the fields $\\mathbb{F}_{2^k}$ for any $k \\geq 1$, and can easily be extended to any $\\mathbb{F}_{p^k}$, for prime $p$. One important difference from MQQ-SIG is that in MQQ-ENC we use left MQQs (LMQQs) instead of bilinear MQQs. Our choice can be justified by our extensive experimental analysis that showed the superiority of the LMQQs over the bilinear MQQs for the design of MQQ-ENC.

We apply the standard cryptanalytic techniques on MQQ-ENC, and from the results, we pose a plausible conjecture that the instances of the MQQ-ENC trapdoor are hard instances with respect to the MQ problem. Under this assumption, we adapt the Kobara-Imai conversion of the McEliece scheme for MQQ-ENC and prove that it provides $\\mathsf{IND-CCA}$ security despite the negligible probability of decryption errors.

We also recommend concrete parameters for MQQ-ENC for encryption of blocks of 128 bits for a security level of $\\mathcal{O}(2^{128})$.