[Paper Review] Group ring cryptography

Barry Hurley, Ted Hurley|arXiv (Cornell University)|Apr 9, 2011

Coding theory and cryptography4 references3 citations

TL;DR

This paper proposes a novel public-key cryptography system based on units in group rings, leveraging the algebraic structure of group rings to combine encryption and error-correcting coding. By combining non-commutative units and introducing the discrete logarithm problem via unit powers, the method achieves strong security and efficient O(n log n) encryption/decryption, with demonstrated integration of Hamming codes for error resilience and compatibility with RSA.

ABSTRACT

Cryptographic systems are derived using units in group rings. Combinations of types of units in group rings give units not of any particular type. This includes cases of taking powers of units and products of such powers and adds the complexity of the {\em discrete logarithm} problem to the system. The method enables encryption and (error-correcting) coding to be combined within one system. These group ring cryptographic systems may be combined in a neat way with existing cryptographic systems, such as RSA, and a combination has the combined strength of both systems. Examples are given.

Motivation & Objective

• To develop a new public-key cryptographic system based on the algebraic properties of group rings and their units.
• To integrate error-correcting coding (e.g., Hamming codes) directly into the encryption process within the same algebraic framework.
• To enhance security by combining non-commutative units and powers of units, introducing the discrete logarithm problem into the system.
• To enable compatibility with existing systems like RSA, allowing hybrid cryptosystems with combined strength.
• To demonstrate practical feasibility through computational examples with measurable performance (e.g., 2.5 seconds for 4096-length data).

Proposed method

• Represent plaintext as a group ring element by mapping digits to coefficients in a ring R over a finite group G.
• Use invertible elements (units) in the group ring RG as encryption keys, with encryption defined as w ↦ w*u or w ↦ u*w for unit u.
• Construct units from combinations of non-commutative elements (e.g., uab, bau) and cyclic units (h), ensuring non-commutativity increases complexity.
• Apply powers of units (e.g., u^127) to introduce the discrete logarithm problem, strengthening security.
• Integrate error-correcting codes (e.g., Hamming codes) by encoding coefficients of the encrypted group ring element before transmission.
• Decompose large messages into blocks when the number of digits exceeds the group order, applying block encryption recursively.

Experimental results

Research questions

• RQ1 Can group ring units be used to construct a secure public-key cryptosystem that integrates both encryption and error correction?
• RQ2 How does combining non-commutative units and unit powers enhance security beyond traditional discrete logarithm or factoring-based systems?
• RQ3 To what extent can the group ring framework support efficient O(n log n) encryption and decryption without sacrificing security?
• RQ4 Can error-correcting codes be seamlessly embedded in the group ring structure to enable reliable transmission and automatic error recovery?
• RQ5 How can existing systems like RSA be combined with group ring cryptography to create hybrid cryptosystems with enhanced security?

Key findings

• The method achieves O(n log n) time complexity for encryption and decryption, demonstrated in examples with 4096-length data processed in 2.5 seconds.
• A unit constructed as a high power (e.g., 127th power) of a base unit introduces the discrete logarithm problem, significantly increasing security.
• The system successfully integrates Hamming code-based error correction, correcting a single-bit error in a transmitted coefficient sequence.
• Non-commutative units (e.g., uab and bau) do not commute, and their product yields a new unit not of any known type, increasing cryptographic complexity.
• The inverse of a product of units is not the product of inverses in reverse order, which adds to the system's non-trivial structure and security.
• The method supports hybrid integration with RSA, allowing the combined strength of both systems in a single framework.