Classification of small binary bibraces via bilinear maps

Abstract We classify small binary bibraces, using the correspondence with alternating algebras over the field $$\mathbb F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , 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 $${{\,\textrm{GL}\,}}(m, \mathbb F_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>GL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -orbits under congruence. Our approach combines theoretical invariants, such as rank sequences and the identification of primitive algebras, with computational methods implemented in . These results also count the number of possible alternative operations that can be used in differential cryptanalysis.