Fast computing the algebraic degree of boolean functions

Abstract

Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the vectors of the n-dimensional Boolean cube. Byte-wise and bitwise versions of a search based on the WLO and their implementations are discussed. They are compared with the usual exhaustive search applied in computing the algebraic degree. For Boolean functions of n variables, the bitwise implementation of the search by WLO has total time complexity (Formula Presented). When such a function is given by its truth table vector and its algebraic degree is computed by the bitwise versions of the algorithms discussed, the total time complexity is (Formula Presented). All algorithms discussed have time complexities of the same type, but with big differences in the constants hidden in the (Formula Presented) -notation. The experimental results after numerous tests confirm the theoretical results—the running times of the bitwise implementation are dozens of times better than the running times of the byte-wise algorithms.