[Paper Review] Generalized Maiorana-McFarland Constructions for Almost Optimal Resilient Functions

Weiguo Zhang, Xiao Guo-zhen|arXiv (Cornell University)|Mar 17, 2010

Coding theory and cryptography20 references3 citations

TL;DR

This paper introduces a generalized Maiorana-McFarland (GMM) construction method to generate almost optimal resilient Boolean functions with high nonlinearity, particularly for even and odd numbers of variables. By extending the classical M-M construction using disjoint linear codes and injective mappings, the authors achieve nonlinearity values of $2^{n-1} - 2^{n/2-1} - 2^{k-1}$ for even $n$, with $k < n/2$, and higher nonlinearity for odd $n$ using Patterson-Wiedemann or Kavut-Y"ucel functions. The method also supports multiple-output resilient functions with nonlinearity exceeding $2^{n-1} - 2^{n/2}$, and ensures strict avalanche criterion (SAC) and degree optimization.

ABSTRACT

In a recent paper \cite{Zhang-Xiao}, Zhang and Xiao describe a technique on constructing almost optimal resilient functions on even number of variables. In this paper, we will present an extensive study of the constructions of almost optimal resilient functions by using the generalized Maiorana-McFarland (GMM) construction technique. It is shown that for any given $m$, it is possible to construct infinitely many $n$-variable ($n$ even), $m$-resilient Boolean functions with nonlinearity equal to $2^{n-1}-2^{n/2-1}-2^{k-1}$ where $k2^{n-2}-2^{(n-1)/2}$ ($n$ odd) by using Patterson-Wiedemann functions or Kavut-Y$\ddot{u}$cel functions. Finally, we provide a GMM construction technique for multiple-output almost optimal $m$-resilient functions $F: \mathbb{F}_2^n\mapsto \mathbb{F}_2^r$ ($n$ even) with nonlinearity $&gt;2^{n-1}-2^{n/2}$. Using the methods proposed in this paper, a large class of previously unknown cryptographic resilient functions are obtained.

Motivation & Objective

• To develop a generalized construction technique for almost optimal resilient Boolean functions with improved nonlinearity beyond existing methods.
• To extend the classical Maiorana-McFarland class by incorporating disjoint linear codes and injective mappings to increase nonlinearity and resilience.
• To construct resilient functions satisfying the strict avalanche criterion (SAC) while maintaining high nonlinearity and optimal algebraic degree.
• To provide a framework for generating multiple-output resilient functions with nonlinearity exceeding $2^{n-1} - 2^{n/2}$ for even $n$.
• To explore the theoretical bounds on nonlinearity for resilient functions and propose conjectures on tightness of these bounds.

Proposed method

• A generalized Maiorana-McFarland (GMM) construction is proposed using disjoint linear codes $\{C_1, \dots, C_u\}$ and $\{C'_1, \dots, C'_v\}$ of dimension $r$ and minimum weight at least $m+1$, with $u$ and $v$ maximized.
• Injective mappings $\psi_i$ and $\phi_i$ are defined from subsets $E_0$ and $E_1$ of $\mathbb{F}_2^{n/2}$ to the code spaces $T_0$ and $T_1$, respectively, using matrix representations derived from basis mappings.
• The function $F: \mathbb{F}_2^n \to \mathbb{F}_2^r$ is constructed as $f_i(X_n) = \psi_i(X'_{n/2}) \cdot X''_{n/2}$ if $X'_{n/2} \in E_0$, and $\phi_i(X'_{n-k}) \cdot X''_k$ if $X'_{n-k} \in E_1$, with $k$ determined by a cardinality condition.
• The Walsh transform of the component functions is analyzed to bound the maximum absolute Walsh coefficient, leading to the nonlinearity expression $N_f = 2^{n-1} - 2^{n/2-1} - 2^{k-1}$.
• For odd $n$, the construction leverages Patterson-Wiedemann or Kavut-Y"ucel functions to achieve nonlinearity $> 2^{n-2} - 2^{(n-1)/2}$.
• The method ensures SAC by guaranteeing that the autocorrelation function vanishes for all $\alpha$ of Hamming weight 1, via construction of the mappings and code structure.

Experimental results

Research questions

• RQ1 Can the generalized Maiorana-McFarland construction produce infinitely many $n$-variable, $m$-resilient Boolean functions with nonlinearity $2^{n-1} - 2^{n/2-1} - 2^{k-1}$ for even $n$ and $k < n/2$?
• RQ2 Can the GMM construction be modified to ensure that the resulting resilient functions satisfy the strict avalanche criterion (SAC)?
• RQ3 For odd $n$, can nonlinearity exceeding $2^{n-2} - 2^{(n-1)/2}$ be achieved using known high-nonlinear functions like Patterson-Wiedemann or Kavut-Y"ucel?
• RQ4 Can the GMM framework be extended to multiple-output resilient functions $F: \mathbb{F}_2^n \to \mathbb{F}_2^r$ with nonlinearity $> 2^{n-1} - 2^{n/2}$ for even $n$?
• RQ5 What are the theoretical upper bounds on nonlinearity for $m$-resilient functions, and how tight are they?

Key findings

• The GMM construction generates infinitely many $n$-variable, $m$-resilient Boolean functions with nonlinearity $2^{n-1} - 2^{n/2-1} - 2^{k-1}$ for even $n$, where $k < n/2$, achieving almost optimal nonlinearity.
• The method ensures that the constructed functions satisfy the strict avalanche criterion (SAC) through careful design of injective mappings and code structures.
• For odd $n \geq 12$, the use of Patterson-Wiedemann or Kavut-Y"ucel functions enables construction of resilient functions with nonlinearity $> 2^{n-2} - 2^{(n-1)/2}$, surpassing the known bound for odd $n$.
• The construction is extended to multiple-output functions $F: \mathbb{F}_2^n \to \mathbb{F}_2^r$ for even $n$, achieving nonlinearity $> 2^{n-1} - 2^{n/2}$, with $NF = 2^{n-1} - 2^{n/2-1} - 2^{k-1}$.
• The algebraic degree of the constructed functions can be optimized, and the method allows for degree-optimized resilient functions.
• The paper proposes three conjectures on the tightness of nonlinearity bounds for $m$-resilient functions, suggesting $N_f \leq 2^{n-1} - 2^{n/2-1} - 2^{\lfloor n/4 \rfloor + m - 1}$ for $m < \lceil n/4 \rceil$, and $N_f < 2^{n-1} - 2^{n/2-1} - 2^{m+1}$ for $\lceil n/4 \rceil \leq m \leq n/2 - 2$, with a bound on the sum $m + r \leq n/2 - 1$ for multiple-output functions.