[Paper Review] Nonlinearity Computation for Sparse Boolean Functions

Çağdaş Çalık|arXiv (Cornell University)|May 3, 2013

Coding theory and cryptography3 references3 citations

TL;DR

This paper proposes a novel algorithm to compute the nonlinearity of sparse Boolean functions by reformulating the distance to linear functions using their algebraic normal form (ANF) coefficients, reducing the problem to binary integer programming. The method enables nonlinearity computation for functions with over 40 variables, where Fast Walsh Transform becomes infeasible, offering a scalable alternative for cryptographic applications.

ABSTRACT

An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the distances to linear functions is obtained. The special structure of these distances can be exploited to reduce the task of nonlinearity computation to solving an associated binary integer programming problem. The proposed algorithm can be used in cases where applying the Fast Walsh transform is infeasible, typically when the number of input variables exceeds 40.

Motivation & Objective

• To address the computational infeasibility of nonlinearity computation for Boolean functions with more than 40 variables.
• To develop a method that bypasses the Fast Walsh Transform, which becomes impractical at high variable counts.
• To exploit the structural properties of ANF coefficients to formulate nonlinearity computation as a binary integer programming problem.
• To provide a scalable solution for nonlinearity evaluation in cryptographic design, particularly for sparse functions.

Proposed method

• The nonlinearity is computed by expressing the Hamming distance from a Boolean function to each linear function in terms of its ANF coefficients.
• The formulation reveals a special structure in the distance expressions, enabling the use of optimization techniques.
• The problem is reduced to solving a binary integer programming problem that minimizes the distance to the nearest linear function.
• The algorithm leverages sparsity in the ANF to limit the number of variables and constraints in the integer program.
• The method avoids full enumeration by focusing only on relevant monomials in the ANF.
• The solution to the integer program yields the minimum distance, which directly gives the nonlinearity of the function.

Experimental results

Research questions

• RQ1 Can nonlinearity be computed efficiently for Boolean functions with more than 40 variables where Fast Walsh Transform is infeasible?
• RQ2 How can the algebraic normal form (ANF) coefficients be used to express distances to linear functions in a way that reveals exploitable structure?
• RQ3 Is it possible to reformulate nonlinearity computation as a binary integer programming problem?
• RQ4 What structural properties of ANF coefficients allow for a reduction in computational complexity for sparse functions?
• RQ5 Can the proposed method outperform existing approaches in terms of scalability for high-dimensional sparse Boolean functions?

Key findings

• The nonlinearity computation problem is successfully reformulated using ANF coefficients, enabling a new computational pathway.
• The method reduces nonlinearity computation to a binary integer programming problem, which is solvable even for functions with over 40 variables.
• The algorithm is particularly effective for sparse Boolean functions due to reduced problem size from limited monomial terms.
• The approach avoids the O(n·2^n) complexity of Fast Walsh Transform, making it viable for high-dimensional cases.
• The method provides an exact computation of nonlinearity, unlike heuristic or approximation-based alternatives.
• The algorithm demonstrates practical feasibility for cryptographic applications requiring nonlinearity evaluation of large, sparse functions.