Existence of balanced functions that are not derivative of bent functions

TL;DR

The paper tackles Tokareva's conjecture proposing that every balanced Boolean function of even $n$ and degree at most $n/2-1, with $f(x)=f(x\oplus y)$ for some nonzero $y$, must be a derivative of a bent function. It leverages asymptotic upper bounds for the numbers of bent and $1$-plateaued functions, particularly those arising from restricting bent functions to hyperplanes, to construct a counterexample for sufficiently large even $n$. The authors prove that there exists an $(n-1)$-variable balanced function of degree ≤ $n/2-1$ that is not a derivative of any bent function, thereby refuting the conjecture (Theorem 1). The approach combines counting arguments on plateaued function restrictions with lower bounds on balanced degree-bounded functions and suggests utility for verifying similar hypotheses in the study of Walsh–Hadamard transform supports and bent-function derivatives.

Abstract

It is disproved the Tokareva's conjecture that any balanced boolean function of appropriate degree is a derivative of some bent function. This result is based on new upper bounds for the numbers of bent and plateaued functions.

Theorems & Definitions (9)

• Conjecture 1: Tok16, p. 1
• Proposition 1: Shap, Theorem 4
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5: Pot23+,Corollary 4
• Proposition 5: Pot23+,Corollary 4
• Theorem 1
• Theorem 2