Non-commutative Ring Learning with Errors from Cyclic Algebras

Abstract

The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice-based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of ‘structured’ LWE, trading off a hard to quantify loss of security for an increase in efficiency by working over a well-chosen ring. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a ring. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE) to replicate the addition of the ring structure taking LWE to Ring LWE by adding cyclic structure to Module LWE. We show that the security reductions expected for an LWE problem hold, namely a reduction from certain structured lattice problems to the hardness of the decision variant of the CLWE problem (under the condition of constant rank d). As a contribution of theoretic interest, we view CLWE as the first variant of Ring LWE which supports non-commutative multiplication operations. This ring structure compares favorably with Module LWE, and naturally allows a larger message space for error correction coding.

New trend for LWE application in different computer science fields

“…It used to construct Linkable ring signatures, the reference [14] is based on the homomorphic inheritance on ideal lattices Nuo "scheme, ∑-protocol with Fiat-Shamir" grouping in [24] transform method to construct Chainable ring signatures.…”

“…Application in different computer science fields such as in [23] author proposed simultaneously can encrypt multiple plaintext bits with smaller public parameters. also various non-commutative multiplication operations RLWE support error correction coding by larger message space and compares with normal LWE, In [24], finally The supply procedure with output of RLWE hardware implementation as accelerator yields good performance rather than other state of art in [25].…”

From learning with errors (LWE) problem to CLWE problem

“…Now the theory on cyclic algebra has been emphasized in the coding theory filed [ 22,23] , therefore based on some previous work, the LWE problem on cyclic algebra over noncommutative rings was considered. Consider restating the LWE problem like Definition 2.…”