[Paper Review] Generalized bent functions - sufficient conditions and related constructions

Samir Hodžić, Enes Pašalić|arXiv (Cornell University)|Jan 29, 2016

Coding theory and cryptography13 references3 citations

TL;DR

This paper derives sufficient conditions for generalized bent (gbent) functions $ f: \mathbb{Z}_2^n \to \mathbb{Z}_q $ with even $ q $, using an alternative characterization of the generalized Walsh-Hadamard transform via component Boolean functions' spectra. The key contribution is a compact spectral formula enabling generic constructions of gbent functions, with sufficiency confirmed for $ q=4,8 $ and strong indications of necessity, especially in the generalized Maiorana-McFarland class.

ABSTRACT

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n ightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.

Motivation & Objective

• To identify sufficient conditions for a function $ f: \mathbb{Z}_2^n \to \mathbb{Z}_q $ to be generalized bent (gbent) when $ q $ is even.
• To provide a new spectral characterization of the generalized Walsh-Hadamard transform using the Walsh spectra of component Boolean functions.
• To develop generic construction methods for gbent functions applicable to any even $ q $, including non-power-of-two values.
• To investigate whether the derived sufficient conditions are also necessary, particularly in structured classes like generalized Maiorana-McFarland functions.

Proposed method

• Represent $ f(x) = a_0(x) + 2a_1(x) + \cdots + 2^{h-1}a_{h-1}(x) $ for $ q = 2^h $, and use a modified form $ f(x) = \frac{q}{2}a(x) + a_0(x) + \cdots + 2^{h-2}a_{h-2}(x) $ for $ 2^{h-1} < q < 2^h $ to simplify spectral analysis.
• Derive a compact formula for the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component functions $ a_i $.
• Use the spectral formula to establish sufficient conditions under which the transform matrix $ W^T $ equals a signed Sylvester-Hadamard matrix $ \pm H^{(r)}_{2^p} $.
• Apply the method to the generalized Maiorana-McFarland class (GMMF), showing that GMMF functions satisfy the derived sufficient conditions.
• Employ properties of Walsh spectra under complementation and translation, such as $ W_g(u) = -W_{g \oplus 1}(u) $ and $ W_{g \oplus \alpha}(u) = (-1)^{u \cdot \alpha} W_g(u) $, to construct examples satisfying the spectral condition.

Experimental results

Research questions

• RQ1 Can a general sufficient condition for gbentness be derived for all even $ q $, including non-powers of two?
• RQ2 Is the derived sufficient condition also necessary, particularly in structured classes like the generalized Maiorana-McFarland functions?
• RQ3 How can the generalized Walsh-Hadamard transform be efficiently computed and characterized using the Walsh spectra of component Boolean functions?
• RQ4 What are the implications of the spectral condition $ W^T = \pm H^{(r)}_{2^p} $ for constructing gbent functions?

Key findings

• A compact and efficient formula is derived for computing the generalized Walsh-Hadamard spectra of $ f: \mathbb{Z}_2^n \to \mathbb{Z}_q $ in terms of the Walsh spectra of its component Boolean functions.
• The sufficient conditions derived for gbentness coincide with the known necessary and sufficient conditions for $ q=4 $ and $ q=8 $, indicating they may be necessary in general.
• For $ q=16 $, a construction method is demonstrated using component functions such that the spectral matrix $ W^T $ matches $ \pm H^{(2)}_{8} $ or $ \pm H^{(3)}_{8} $, confirming the sufficiency of the conditions.
• The generalized Maiorana-McFarland class (GMMF) of gbent functions satisfies the derived sufficient conditions, supporting the conjecture that these conditions may be necessary in certain cases.
• The method enables generic constructions of gbent functions by selecting component functions such that their spectral combinations satisfy the required signed Hadamard matrix condition.