On the Asymptotics of Solving the LWE Problem Using Coded-BKW With Sieving

Abstract

The learning with errors problem (LWE) has become a central topic in recent cryptographic research. In this paper, we present a new solving algorithm combining important ideas from previous work on improving the Blum–Kalai–Wasserman (BKW) algorithm and ideas from sieving in lattices. The new algorithm is analyzed and demonstrates an improved asymptotic performance. For the Regev parameters $q=n^{2}$ and noise level $\sigma = n^{1.5}/(\sqrt {2\pi }\log _{2}^{2}n)$ , the asymptotic complexity is $2^{0.893n}$ in the standard setting, improving the previously best known complexity of roughly $2^{0.930n}$ . The newly proposed algorithm also provides asymptotic improvements when a quantum computer is assumed or when the number of samples is limited.