The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the RingLWE problem for general number rings and demonstrate provably weak instances of the Decision Ring-LWE problem. We construct an explicit family of number \ufb01elds for which we have an e\ufb03cient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus. Our attack is based on the attack on Poly-LWE which was presented in [EHL]. We extend the EHL-attack to apply to a larger class of number \ufb01elds, and show how it applies to attack Ring-LWE for a heuristically large class of \ufb01elds. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the \ufb01rst weak instances of the Ring-LWE problem. We also provide additional examples of \ufb01elds which are vulnerable to our attacks on Poly-LWE, including power-of-2 cyclotomic \ufb01elds, presented using the minimal polynomial of \u03b62n\u00b11.<\/p>\n","protected":false},"excerpt":{"rendered":"
The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. 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