[Paper Review] On some quasigroup cryptographical primitives

Piroska Csörgő, Victor Shcherbacov|arXiv (Cornell University)|Oct 30, 2011

graph theory and CDMA systems14 references3 citations

TL;DR

This paper analyzes the security of quasigroup-based stream ciphers, demonstrating their vulnerability to chosen plaintext and chosen ciphertext attacks under Vojvoda's framework. It proposes enhanced cryptographic primitives using systems of orthogonal n-ary groupoids and modified encryption procedures to improve resistance against these attacks, particularly by integrating n-ary quasigroup operations and dynamic application sequences of encryption steps.

ABSTRACT

We propose modifications of known quasigroup based stream ciphers. Systems of orthogonal n-ary groupoids are used.

Motivation & Objective

• To identify and analyze the vulnerabilities of existing quasigroup-based cryptographical primitives to chosen plaintext and chosen ciphertext attacks.
• To improve the security of n-ary quasigroup-based stream ciphers by proposing new constructions and modifications.
• To develop a framework for constructing secure stream ciphers using systems of orthogonal n-ary groupoids.
• To explore the use of T-quasigroups over finite abelian groups to generate orthogonal quasigroups and their parastrophes.
• To propose dynamic, non-periodic application sequences of encryption procedures to increase resistance against cryptanalytic techniques.

Proposed method

• Adopting Vojvoda’s framework to assess the security of quasigroup-based primitives against chosen plaintext and chosen ciphertext attacks.
• Introducing a modified encryption procedure (Algorithm 2) that uses n-ary quasigroups with multiple leaders and recursive dependency on previous cipher symbols.
• Proposing a new procedure (Procedure 1) that maps each plaintext symbol to n cipher symbols using n-ary quasigroup operations.
• Combining Algorithm 2 and Procedure 1 in a non-periodic, sequence-based manner using irrational or transcendental number representations for key scheduling.
• Utilizing T-quasigroups of the form $ x \circ y = \varphi x + \psi y + a $ over finite abelian groups to construct orthogonal quasigroups and their parastrophes.
• Applying theoretical conditions from Theorem 2 to ensure orthogonality between quasigroups and their parastrophes, thereby enhancing cryptographic strength.

Experimental results

Research questions

• RQ1 Are the quasigroup-based cryptographical primitives proposed in [26] vulnerable to chosen plaintext and chosen ciphertext attacks?
• RQ2 Can the security of n-ary quasigroup-based stream ciphers be improved through modifications of the encryption process?
• RQ3 What role do systems of orthogonal n-ary groupoids play in enhancing the resistance of quasigroup-based ciphers to cryptanalysis?
• RQ4 How can T-quasigroups over finite abelian groups be used to generate quasigroups that are orthogonal to their parastrophes?
• RQ5 Can non-periodic application sequences of encryption procedures significantly increase the difficulty of chosen plaintext and chosen ciphertext attacks?

Key findings

• The quasigroup-based primitives from [26] are vulnerable to both chosen plaintext and chosen ciphertext attacks, as demonstrated using Vojvoda’s framework.
• The proposed modifications, including the use of multiple leaders and recursive ciphering in Algorithm 2, increase the complexity of cryptanalysis.
• Procedure 1, which maps each plaintext symbol to n cipher symbols using n-ary quasigroups, enhances diffusion and increases resistance to statistical attacks.
• The integration of non-periodic application sequences—such as those derived from decimal expansions of irrational numbers—significantly complicates chosen plaintext and chosen ciphertext attacks.
• T-quasigroups of the form $ x \circ y = kx + my + a $ over $ \mathbb{Z}_p $, where $ p $ is prime, can be constructed to be orthogonal to all their parastrophes if the coefficients satisfy specific non-degeneracy conditions.
• Theoretical conditions in Theorem 2 ensure that orthogonality between a T-quasigroup and its parastrophes is achieved if certain combinations of automorphisms are permutations of the underlying group.