Interactions Between Brauer Configuration Algebras and Classical Cryptanalysis to Analyze Bach's Canons

TL;DR

This work uses Brauer configuration algebras (BCAs) as a unifying algebraic framework to study both classical cryptanalysis and Bach’s canons. It proves that certain block ciphers induce labeled BCAs and derives explicit dimension and center formulas for BCAs arising from the Vigenère and permutation schemes, while also modeling musical content as Brauer messages and applying this to Bach's canons (BWV 1076, 1073, 1079). The results reveal invariant algebraic descriptors that connect cryptographic structure with musical symbolism, enabling algebraic insights to inform cryptanalysis and musicology alike. By linking ciphers, music, and Brauer configurations, the paper opens avenues for cross-disciplinary analysis and future exploration of Bach’s symbols within an algebraic combinatorial setting.

Abstract

Since their introduction, Brauer configuration algebras (BCAs) and their specialized messages have helped research in several fields of mathematics and sciences. This paper deals with a new perspective on using such algebras as a theoretical framework in classical cryptography and music theory. It is proved that some block cyphers define labeled Brauer configuration algebras. Particularly, the dimension of the BCA associated with a ciphertext-only attack of the Vigenere cryptosystem is given by the corresponding key's length and the captured ciphertext's coincidence index. On the other hand, historically, Bach's canons have been considered solved music puzzles. However, due to how Bach posed such canons, the question remains whether their solutions are only limited to musical issues. This paper gives alternative solutions based on the theory of Brauer configuration algebras to some of the puzzle canons proposed by Bach in his Musical Offering (BWV 1079) and the canon â 4 Voc: Perpetuus (BWV 1073). Specifically to the canon â 6 Voc (BWV 1076), canon 1 â2 (also known as the crab canon), and canon â4 Quaerendo Invenietis. These solutions are obtained by interpreting such canons as ciphertexts (via route and transposition cyphers) of some specialized Brauer messages. In particular, it is noted that the structure or form of the notes used in such canons can be described via the shape of the most used symbols in Bach's works.

Figures (13)

• Figure 1: The route (path) used to decrypt the message (\ref{['decrypt']}) obtained via permutation.
• Figure 2: Brauer quiver of a Vigenere ciphertext.
• Figure 3: Ciphertext defined by the Brauer configuration (\ref{['SLYm']}).
• Figure 4: Brauer quiver associated with the $\mathfrak{M}$-reduced Brauer configuration (\ref{['SLYm']}).
• Figure 5: Symbols that are constructed using points and lines. And examples of symbols written or mentioned in Bach's manuscripts. Bach's monogram on his wax seal based on his initials and different versions of the Greek letters $\alpha$ and $\omega$SealChris.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Theorem 4