A Method for Fast Computing the Algebraic Degree of Boolean Functions

TL;DR

This paper introduces a new, faster method for computing the algebraic degree of Boolean functions by combining existing approaches and optimizing for bitwise implementation, significantly improving computational efficiency.

Contribution

It presents the fastest known algorithm for algebraic degree computation, integrating two main methods and analyzing their theoretical and practical performance.

Findings

Bitwise approach is dozens of times faster than byte-wise methods.

Theoretical complexity is Θ(n·2^n) for both approaches, with significant constant factor improvements.

Experimental results confirm the efficiency of the proposed method.

Abstract

The algebraic degree of Boolean functions (or vectorial Boolean functions) is an important cryptographic parameter that should be computed by fast algorithms. They work in two main ways: (1) by computing the algebraic normal form and then searching the monomial of the highest degree in it, or (2) by examination the algebraic properties of the true table vector of a given function. We have already done four basic steps in the study of the first way, and the second one has been studied by other authors. Here we represent a method for fast computing (the fastest way we know) the algebraic degree of Boolean functions. It is a combination of the most efficient components of these two ways and the corresponding algorithms. The theoretical time complexities of the method are derived in each of the cases when the Boolean function is represented in a byte-wise or in a bitwise manner. They are of the same type $\Theta(n.2^n)$ for a Boolean function of $n$ variables, but they have big differences between the constants in $\Theta$-notation. The theoretical and experimental results shown here demonstrate the advantages of the bitwise approach in computing the algebraic degree - they are dozens of times faster than the byte-wise approaches.