The Complexity of Algebraic Algorithms for LWE

Abstract: Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner basis computations on LWE polynomial systems under the assumption of semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove that it satisfies a genericity condition recently introduced by Caminata & Gorla, called being in generic coordinates. For polynomial systems in generic coordinates one can always estimate the complexity of DRL Gr\"obner basis computations in terms of the Castelnuovo-Mumford regularity and henceforth also via the Macaulay bound. Moreover, we generalize the Gr\"obner basis algorithm of Semaev & Tenti to arbitrary polynomial systems with a finite degree of regularity. In particular, existence of this algorithm yields another approach to estimate the complexity of DRL Gr\"obner basis computations in terms of the degree of regularity. In practice, the degree of regularity of LWE polynomial systems is not known, though one can always estimate the lowest achievable degree of regularity. Consequently, from a designer's worst case perspective this approach yields sub-exponential complexity estimates for general, binary secret and binary error LWE. In recent works by Dachman-Soled et al. the hardness of LWE in the presence of side information was analyzed. Utilizing their framework we discuss how hints can be incorporated into LWE polynomial systems and how they affect the complexity of Gr\"obner basis computations.