# Fast Computing the Algebraic Degree of Boolean Functions

## TL;DR

This paper presents a fast method for computing the algebraic degree of Boolean functions by combining ANF transform with a weight-lexicographic order search, significantly improving efficiency over traditional exhaustive methods.

## Contribution

It introduces a novel bitwise implementation of the WLO-based search for algebraic degree, achieving substantial speedups over byte-wise algorithms and traditional exhaustive search.

## Key findings

- Bitwise implementation has complexity O(n.2^n).
- Experimental results show bitwise algorithms are dozens of times faster.
- Theoretical analysis confirms the efficiency improvements.

## 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 $O(n.2^n)$. 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 $\Theta((9n-2).2^{n-7})=\Theta(n.2^n)$. All algorithms discussed have time complexities of the same type, but with big differences in the constants hidden in the $\Theta$-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.