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An encryption algorithm using a generalization of the Markovski algorithm and a system of orthogonal operations based on T-quasigroups

Nadezhda Malyutina, Alexander Popov, Victor Shcherbacov

TL;DR

The paper addresses designing a cryptographic encoder that combines a generalized Markovski algorithm with a system of orthogonal operations built from T-quasigroups. It develops Algorithm 1* which alternates left/right translations of T-quasigroups and a suite of orthogonal n-ary quasigroups controlled by cryptofunctions F1, F2, and F3 to transform plaintext into ciphertext. A detailed Z_313-based example demonstrates the construction of multiple parastrophe-orthogonal pairs and a step-by-step encryption/decryption procedure, illustrating feasibility and correctness. The work shows that the Markovski algorithm is a special case of Algorithm 1* and discusses security considerations, including the need for varying exponents to deter chosen-plaintext and ciphertext attacks, as well as practical implementation choices. Overall, it offers a flexible, higher-arity cipher framework with potential for extension to n-ary plaintexts and fosters further cryptanalytic and optimization work.

Abstract

Here is a more detailed description of the algorithm proposed in [1]. This algorithm simultaneously uses two cryptographic procedures: encryption using a generalization of the Markovski algorithm [2] and encryption using a system of orthogonal operations. In this paper, we present an implementation of this algorithm based on T-quasigroups, more precisely, based on medial quasigroups.

An encryption algorithm using a generalization of the Markovski algorithm and a system of orthogonal operations based on T-quasigroups

TL;DR

The paper addresses designing a cryptographic encoder that combines a generalized Markovski algorithm with a system of orthogonal operations built from T-quasigroups. It develops Algorithm 1* which alternates left/right translations of T-quasigroups and a suite of orthogonal n-ary quasigroups controlled by cryptofunctions F1, F2, and F3 to transform plaintext into ciphertext. A detailed Z_313-based example demonstrates the construction of multiple parastrophe-orthogonal pairs and a step-by-step encryption/decryption procedure, illustrating feasibility and correctness. The work shows that the Markovski algorithm is a special case of Algorithm 1* and discusses security considerations, including the need for varying exponents to deter chosen-plaintext and ciphertext attacks, as well as practical implementation choices. Overall, it offers a flexible, higher-arity cipher framework with potential for extension to n-ary plaintexts and fosters further cryptanalytic and optimization work.

Abstract

Here is a more detailed description of the algorithm proposed in [1]. This algorithm simultaneously uses two cryptographic procedures: encryption using a generalization of the Markovski algorithm [2] and encryption using a system of orthogonal operations. In this paper, we present an implementation of this algorithm based on T-quasigroups, more precisely, based on medial quasigroups.
Paper Structure (3 sections, 4 theorems, 10 equations, 1 algorithm)

This paper contains 3 sections, 4 theorems, 10 equations, 1 algorithm.

Table of Contents

  1. Basic concepts and definitions
  2. Results
  3. Conclusion

Key Result

Theorem 1.1

For a finite quasigroup $(Q,A)$, the following equivalences hold: $(i) A \perp {}^{(12)}A \Leftrightarrow ((x\backslash z) \cdot x=(y \backslash z) \cdot y \Rightarrow x=y)$; $(ii) A \perp {}^{(13)}A \Leftrightarrow (zx\cdot x=zy \cdot y \Rightarrow x=y)$; $(iii) A \perp {}^{(23)}A \Leftrightarro

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 8 more