Algebraic Algorithms for LWE Problems.

Abstract

Abstract. We analyse the complexity of algebraic algorithms for solving systems of linear equations with noise. Such systems arise naturally in the theory of error-correcting codes as well as in computational learning theory. More recently, linear systems with noise have found application in cryptography. The Learning with Errors (LWE) problem has proven to be a rich and versatile source of innovative cryptosystems, such as fully homomorphic encryption schemes. Despite the popularity of the LWE problem, the complexity of algorithms for solving it is not very well understood, particularly when variants of the original problem are considered. Here, we focus on and generalise a particular method for solving these systems, due to Arora & Ge, which reduces the problem to non-linear but noise-free system solving. Firstly, we provide a refined complexity analysis for the original Arora-Ge algorithm for LWE. Secondly, we study the complexity of applying algorithms for computing Gröbner basis, a fundamental tool in computational commutative algebra, to solving Arora-Ge-style systems of non-linear equations. We show positive and negative results. On the one hand, we show that the use of Gröbner bases yields an exponential speed-up over the basic Arora-Ge approach. On the other hand, we give a negative answer to the natural question whether the use of such techniques can yield a subexponential algorithm for the LWE problem. Under a mild algebraic assumption, we show that it is highly unlikely that such an improvement exists. We also consider a variant of LWE known as BinaryError-LWE introduced by Micciancio and Peikert recently. By