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In this paper, we investigate the security of a public-key encryption scheme introduced by Huang, Liu and Yang HLY at PKC'12. This new scheme can be provably reduced to the hardness of solving a set of quadratic equations whose coefficients of highest degree are chosen according to a discrete Gaussian distributions. The other terms being chosen uniformly at random. Such a problem is a variant of the classical problem of solving a system of non-linear equations PoSSo, which is known to be hard for random systems. The main hypothesis of Huang, Liu and Yang is that their variant is not easier than solving PoSSo for random instances. In this paper, we disprove this hypothesis. To this end, we exploit the fact that the new problem proposed by Huang, Liu and Yang reduces to an easy instance of the Learning With Errors LWE problem. The main contribution of this paper is to show that security and efficiency are essentially incompatible for the HLY proposal. That is, one cannot find parameters which yield a secure and a practical scheme. For instance, we estimate that a public-key of at least $1.03~\textnormal{GB}$ is required to achieve 80-bit security against the simplest of our attacks. As a proof of concept, we present 3 practical attacks against all the parameters proposed by Huang, Liu and Yang. With the most efficient attack, we have been able to recover the private-key in roughly 5 minutes for the first challenge i.e. Case 1 proposed by HLY and less than 30 minutes for the second challenge i.e. Case2.

Practical Cryptanalysis of a Public-Key Encryption Scheme Based on New Multivariate Quadratic Assumptions