In this paper, we consider the Polly Cracker with Noise (PCN) cryptosystem by Albrecht, Farshim, Faugere, and Perret (Asiacrypt 2011), which is a public-key cryptosystem based on the hardness of computing Grobner bases for noisy random systems of multivariate equations. We examine four settings, covering all possible parameter ranges of PCN with zero-degree noise. In the first setting, the PCN cryptosystem is known to be equivalent to Regev's LWE-based scheme. In the second, it is known to be at most as secure as Regev's scheme. We show that for one other settings it is equivalent to a variants of Regev's with less efficiency and in the last setting it is completely insecure and we give an efficient key-recovery attack. Unrelated to the attack, we also fix some flaws in the security proofs of PCN.
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