[Paper Review] Construction and Count of Boolean Functions of an Odd Number of Variables with Maximum Algebraic Immunity

Na Li, Wen‐Feng Qi|ArXiv.org|May 30, 2006

Coding theory and cryptography12 references9 citations

TL;DR

This paper presents a constructive method to generate all $ n $-variable Boolean functions with maximum algebraic immunity for odd $ n $, by transforming the problem into finding invertible submatrices in a structured $ 2^{n-1} \times 2^{n-1} $ matrix. It proves that the number of such functions exceeds $ 2^{2^{n-1}} $, establishing their abundance and providing a complete characterization of this class.

ABSTRACT

Algebraic immunity has been proposed as an important property of Boolean functions. To resist algebraic attack, a Boolean function should possess high algebraic immunity. It is well known now that the algebraic immunity of an $n$-variable Boolean function is upper bounded by $\left\lceil {\frac{n}{2}} ight ceil $. In this paper, for an odd integer $n$, we present a construction method which can efficiently generate a Boolean function of $n$ variables with maximum algebraic immunity, and we also show that any such function can be generated by this method. Moreover, the number of such Boolean functions is greater than $2^{2^{n-1}}$.

Motivation & Objective

• To develop a complete and efficient construction method for $ n $-variable Boolean functions with maximum algebraic immunity when $ n $ is odd.
• To determine the exact number of such Boolean functions, resolving a gap in prior literature on their count.
• To show that every such function with maximum algebraic immunity can be generated by the proposed method, establishing completeness.
• To provide a framework that enables systematic generation and enumeration of these functions, useful for cryptographic design.

Proposed method

• The method reduces the problem of finding maximum algebraic immunity functions to identifying $ k \times k $ invertible submatrices within a $ 2^{n-1} \times 2^{n-1} $ matrix $ W(G_n) $ derived from a base function $ G_n $.
• It uses the algebraic normal form (ANF) and truth table representation of Boolean functions to define vectors $ v(X) $ corresponding to monomials up to degree $ \left\lceil \frac{n}{2} \right\rceil - 1 $.
• The construction involves selecting $ k $ onset and $ k $ offset points from the truth table, then using Gauss elimination to ensure linear independence of selected column vectors in $ W(G_n) $.
• A new function $ f_{(i_1,\ldots,i_k; j_1,\ldots,j_k)} $ is defined by flipping the output of $ G_n $ at $ k $ onset and $ k $ offset points, based on the invertible submatrix condition.
• The method leverages the duality between $ f $ and $ f \oplus 1 $, allowing the range of $ k $ to be restricted to $ 1 \leq k \leq 2^{n-2} $ to avoid redundancy.
• The number of such functions is shown to equal the number of $ k \times k $ invertible submatrices in $ W(G_n) $, which is proven to exceed $ 2^{2^{n-1}} $.

Experimental results

Research questions

• RQ1 How can all $ n $-variable Boolean functions with maximum algebraic immunity be systematically constructed when $ n $ is odd?
• RQ2 What is the exact number of such functions, and how does it scale with $ n $?
• RQ3 Can every Boolean function with maximum algebraic immunity be generated by a single, unified construction method?
• RQ4 What structural properties of the truth table and associated matrix $ W(G_n) $ determine maximum algebraic immunity?

Key findings

• The number of $ n $-variable Boolean functions with maximum algebraic immunity for odd $ n $ is greater than $ 2^{2^{n-1}} $, indicating a vast number of such functions.
• The construction method is complete: every $ n $-variable Boolean function with maximum algebraic immunity can be generated by the proposed algorithm.
• The method reduces the problem to finding $ k \times k $ invertible submatrices in a $ 2^{n-1} \times 2^{n-1} $ matrix $ W(G_n) $, which is computationally feasible for small $ k $.
• For $ k = 1 $, the construction simplifies to selecting a single onset and a single offset point where the corresponding column vector has a 1, enabling efficient generation.
• The method preserves the algebraic immunity property under complementation, so $ f $ and $ f \oplus 1 $ are treated equivalently in the construction.
• The result establishes a direct correspondence between the existence of invertible submatrices and the existence of functions with maximum algebraic immunity.