Classification of small binary bibraces via bilinear maps
Roberto Civino, Valerio Fedele
TL;DR
The paper resolves the classification of small binary bibraces by recasting them as binary alternating algebras over $\mathbb{F}_2$ and encoding their defining data as subspaces of skew-symmetric matrices, up to $\mathrm{GL}(m,\mathbb{F}_2)$-congruence. It develops invariants such as rank sequences and the notion of primitivity to separate isomorphism classes, and provides comprehensive counts for dimensions up to eight, including a detailed analysis of the challenging case with a $2$-dimensional annihilator in dimension $8$. The results yield explicit enumerations of admissible alternative operations relevant to differential cryptanalysis and illuminate the landscape of possible algebraic structures underpinning binary bibraces. The approach combines theoretical invariants with computational tools (Magma) to achieve complete classifications in small dimensions and to quantify the cryptanalytic flexibility offered by these algebras.
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, F_2)-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.