A Non-commutative Cryptosystem Based on Quaternion Algebras

TL;DR

This paper introduces BQTRU, a non-commutative cryptosystem based on quaternion algebras, which offers faster key generation and encryption, enhanced resistance to attacks, and comparable security with smaller keys compared to NTRU.

Contribution

The paper presents BQTRU, a novel quaternion algebra-based cryptosystem that improves speed and security over traditional NTRU by leveraging non-commutative structures and hybrid lattice properties.

Findings

Key generation and encryption are approximately 16/7 times faster than NTRU.

BQTRU's lattice structure increases resistance to standard lattice attacks.

Allows smaller key sizes for equivalent security levels.

Abstract

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.