On the Lower Block Triangular Nature of the Incidence Matrices to Compute the Algebraic Immunity of Boolean Functions

Abstract

The incidence matrix between two sets of vectors in ${\mathbb F}_2$ has a great importance in different areas of mathematics and sciences. The rank of these matrices are very useful while computing the algebraic immunity($\mathsf{AI}$) of Boolean functions in cryptography literature [3, 7]. With a proper ordering of monomial (exponent) vectors and support vectors, some interesting algebraic structures in the incidence matrices can be observed. We have exploited the lower-block triangular structure of these matrices to find their rank. This structure is used for faster computation of the $\mathsf{AI}$ and the low degree annihilators of an n-variable Boolean functions than the known algorithms. On the basis of experiments on at least 20 variable Boolean functions, we conjecture about the characterization of power functions of algebraic immunity 1, could verify the result on the $\mathsf{AI}$ of n-variable inverse S-box presented in [6](i.e., $\lceil 2\sqrt{n}\rceil -2$), and presented some results on the $\mathsf{AI}$ of some important power S-boxes.