Nonlinearity of quartic rotation symmetric Boolean functions

TL;DR

This paper proves that for certain quartic rotation symmetric Boolean functions, their nonlinearity equals their weight, using new recursive formulas and sub-functions, and conjectures this property extends to higher degrees.

Contribution

The paper introduces new techniques and recursive formulas to establish that the nonlinearity of specific quartic rotation symmetric Boolean functions equals their weight, confirming this for a class of functions.

Findings

Nonlinearity equals weight for the studied quartic functions

New recursive formulas developed for analysis

Conjecture extended to higher-degree rotation symmetric functions

Abstract

Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ is the same as its weight. So we answer the above question affirmatively. Finally, we conjecture that if $l\ge 4$ is an integer, then the nonlinearity of the rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}...x_{le}$ equals its weight.