Binary bi-braces from the point of view of nilpotent F2-algebras: explicit construction, automorphisms and classification

Abstract

Let (V, +, ·) be a commutative associative radical F2-algebra of F2-dimension n, such that V^3 = 0 (i.e. for every u, v, w ∈ V , u · v · w = 0). Define the circle operation on V by u ◦ v = u + v + u · v. It is known that (V, +, ◦) is a bi-skew brace, which we call binary bi-brace.
Motivated by the interest of the automorphism groups of these structures in the context of symmetric cryptography, we provide a characterization of Aut(V, +, ◦) and a description of the isomorphism classes in terms of congruences of suitable subspaces of alternating bilinear forms over F2 (i.e. whose associated matrices are symmetric and 0-diagonal).
In particular, a complete classification is provided for n ⩽ 7 and for some subcases when n > 7.