On Dual Lattice Attacks Against Small-Secret LWE and Parameter Choices in HElib and SEAL

We present novel variants of the dual-lattice attack against LWE in the presence of an unusually short secret. These variants are informed by recent progress in BKW-style algorithms for solving LWE. Applying them to parameter sets suggested by the homomorphic encryption libraries HElib and SEAL yields revised security estimates. Our techniques scale the exponent of the dual-lattice attack by a factor of (2L)/(2L+1)(2L)/(2L+1) when logq=Θ(Llogn)log⁡q=Θ(Llog⁡n), when the secret has constant hamming weight hh and where LL is the maximum depth of supported circuits. They also allow to half the dimension of the lattice under consideration at a multiplicative cost of 2h2h operations. Moreover, our techniques yield revised concrete security estimates. For example, both libraries promise 80 bits of security for LWE instances with n=1024n=1024 and log2q≈47log2⁡q≈47, while the techniques described in this work lead to estimated costs of 68 bits (SEAL) and 62 bits (HElib).