Efficient quantum algorithms to break group ring cryptosystems

Abstract

The security of widely-used public-key cryptographic protocols like RSA, Diffie–Hellman key exchange and the Digital Signature Algorithm (DSA) is under threat due to the emergence of quantum computers. Shor’s groundbreaking quantum algorithm poses a significant risk by efficiently factoring large integers into their prime factors, compromising RSA security. Additionally, it solves the Discrete Logarithm Problem, impacting certain Diffie–Hellman-based cryptosystems and digital signatures. Given this, it is imperative to enhance our current cryptographic tools for the post-quantum era, aiming to make it impractical, even with quantum algorithms, to breach the security of new cryptosystems. Prominent alternatives include elliptic curve and lattice-based cryptography, with exploration into other algebraic systems featuring difficult problems to ensure security. This paper establishes that systems based on the difficulty of inverting group ring elements are not quantum-resistant.