[Paper Review] Niho Bent Functions and Subiaco/Adelaide Hyperovals

Tor Helleseth, Alexander Kholosha+1 more|arXiv (Cornell University)|Oct 17, 2012

Advanced Topics in Algebra3 references3 citations

TL;DR

This paper establishes a direct correspondence between binomial Niho bent functions constructed by Dobbertin et al. and the o-polynomials generating Subiaco and Adelaide hyperovals in finite geometry. By linking these two mathematical structures, the authors expand the known class of bent functions associated with Subiaco hyperovals when $ m \equiv 2 \pmod{4} $, providing a unified framework that connects finite field theory, bent functions, and hyperoval classification in projective planes.

ABSTRACT

In this paper, the relation between binomial Niho bent functions discovered by Dobbertin et al. and o-polynomials that give rise to the Subiaco and Adelaide classes of hyperovals is found. This allows to expand the class of bent functions that corresponds to Subiaco hyperovals, in the case when $m\equiv 2 (\bmod 4)$.

Motivation & Objective

• To establish a structural connection between binomial Niho bent functions and o-polynomials that generate Subiaco and Adelaide hyperovals.
• To extend the known class of bent functions corresponding to Subiaco hyperovals when $ m \equiv 2 \pmod{4} $.
• To demonstrate that the existence of certain Niho bent functions is equivalent to the existence of specific $ q $-clans and o-polynomials.
• To unify the representation of bent functions and hyperoval constructions through trace functions and bivariate forms over $ \mathbb{F}_{2^m} $.

Proposed method

• The authors use the bivariate representation of bent functions over $ \mathbb{F}_{2^m} \times \mathbb{F}_{2^m} $, expressing them as trace functions of polynomials in two variables.
• They analyze the univariate and bivariate forms of Niho bent functions, particularly focusing on the case $ f(t) = \mathrm{Tr}_1^m(a t^{2^m+1}) + \mathrm{Tr}_1^n(b t^{(2^m-1)/6 + 1}) $, with $ b^{2^m+1} = a $.
• The paper derives the corresponding o-polynomial $ G(z) $ from the bent function's structure, showing it matches the form of o-polynomials from Subiaco and Adelaide hyperovals.
• It employs trace identities and algebraic manipulations over $ \mathbb{F}_{2^n} $, particularly using $ \mathrm{Tr}_m^n $ and $ \mathrm{Tr}_1^n $, to relate the function's exponents to geometric invariants.
• The authors verify that the derived $ G(z) $ satisfies the o-polynomial condition $ G(0) = 0 $, $ G(1) = 1 $, and $ G(x) + G(y) + G(x+y) $ is a square for all $ x \neq y $, $ x,y \in \mathbb{F}_{2^m}^* $.
• For $ m \equiv 0 \pmod{4} $, they show that the o-polynomial form matches the known $ q $-clan construction, confirming equivalence to the Adelaide family.

Experimental results

Research questions

• RQ1 How are binomial Niho bent functions related to o-polynomials of Subiaco and Adelaide hyperovals?
• RQ2 Can the class of bent functions associated with Subiaco hyperovals be extended beyond the known cases?
• RQ3 What is the precise algebraic condition under which a Niho bent function corresponds to an o-polynomial of a hyperoval?
• RQ4 Does the existence of a specific Niho bent function imply the existence of a corresponding $ q $-clan or o-polynomial?
• RQ5 What role does the trace function play in linking the univariate and bivariate representations of these bent functions to geometric objects?

Key findings

• The paper proves that the set of Niho bent functions with $ b \in \mathbb{F}_{2^n}^* $ corresponds exactly to the o-polynomials from the Subiaco and Adelaide hyperoval families.
• For $ m \equiv 2 \pmod{4} $, the class of bent functions linked to Subiaco hyperovals is expanded beyond previously known constructions.
• The authors derive an explicit expression for the o-polynomial $ G(z) $ in terms of trace functions and rational expressions involving $ w $, $ u $, and $ z $, showing its equivalence to known forms.
• When $ m \equiv 0 \pmod{4} $, the o-polynomial $ G(z) $ is shown to match the form derived from the $ q $-clan associated with the Adelaide family.
• The paper confirms that the bent function $ f(t) = \mathrm{Tr}_1^m(t^{2^m+1}) + \mathrm{Tr}_1^n(t^{(2^m-1)/6 + 1}) $, with $ a = b = 1 $, generates an o-polynomial matching the Adelaide hyperoval structure.
• The transformation from the univariate bent function to the bivariate form via $ (u,1) $ as a basis of $ \mathbb{F}_{2^n} $ over $ \mathbb{F}_{2^m} $ yields a consistent and verifiable o-polynomial expression.