Classification of small binary bibraces via bilinear maps

2025/10/07 by Roberto Civino, Civino, Roberto, Valerio Fedele +1
Engineering · Mathematics · Physics and Astronomy · #08A35 #16T25 #94A60 #Advanced Numerical Analysis Techniques #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2510.05848

openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify small binary bibraces, using the correspondence with alternating algebras over the field F2, up to dimension eight, also determining their isomorphism classes. These finite-dimensional algebras, defined by an alternating bilinear multiplication and nilpotency of class two, can be represented by subspaces of skew-symmetric matrices, with classification corresponding to GL(m, F₂)-orbits under congruence. Our approach combines theoretical invariants, such as rank sequences and the identification of primitive algebras, with computational methods implemented in Magma. These results also count the number of possible alternative operations that can be used in differential cryptanalysis.

Citations

• An Algorithm for Computing Hopf--Galois Structures and Skew Bracoids of Low Degree
• Hopf-Galois structures on parallel extensions
• Binary bi-braces and applications to cryptography
• Optimal s-boxes against alternative operations and linear propagation
• Differential experiments using parallel alternative operations
• Module braces: relations between the additive and the multiplicative groups
• Computation of Hopf Galois structures on low degree separable extensions\n and classification of those for degrees p² and 2p
• On Hidden Sums Compatible with A Given Block Cipher Diffusion Layer

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