Universal product learning with errors: A new variant of LWE for lattice-based cryptography

Abstract

The middle product learning with errors problem (MP-LWE) is a variant of the learning with errors problem (LWE). The public-key cryptography under MP-LWE assumption gets an equilibrium between “efficiency” and “security”. However, the hardness of MP-LWE has not been sufficiently explored since its introduction. So in this paper, we focus on the hardness of MP-LWE.By the axiomatic method, we first investigate the essential attributes about the middle product in MP-LWE. Then we present the formal definition of universal product and the corresponding universal product learning with errors problem (UP-LWE). In this sense, MP-LWE can be viewed as an instance of UP-LWE. Then we proved that for more polynomials than those have been proved in the literature, PLWE(f) can be reduced to MP-LWE. In other words, we improve the result about the hardness of MP-LWE.Furthermore, we define the asymmetric middle product which can be viewed as a construction of universal product. The corresponding UP-LWE is discussed. It provides much more flexible parameter setting than MP-LWE.