[Paper Review] Planar monomials in characteristic 2

Zachary Scherr, Michael E. Zieve|arXiv (Cornell University)|Feb 6, 2013

Coding theory and cryptography8 references5 citations

TL;DR

This paper proves that certain monomials over F₂ʳ are planar in even characteristic, confirming a conjecture by Schmidt and Zhou. Using a novel result on F_q³-rational points of the curve x^{q−1} + y^{q−1} = z^{q−1}, the authors establish the planarity of these functions, extending the theory of planar functions to characteristic 2 and enabling new constructions of finite projective planes and combinatorial designs.

ABSTRACT

Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which yields similar applications. In this paper we show that certain functions over F2r are planar, which proves a conjecture of Schmidt and Zhou. The key to our proof is a new result about the Fq3-rational points on the curve x q−1 + y q−1 = z q−1. 1.

Motivation & Objective

• To resolve the conjecture by Schmidt and Zhou that specific monomials over F₂ʳ are planar in even characteristic.
• To extend the theory of planar functions—previously defined only in odd characteristic—to the case of characteristic 2.
• To establish the existence of new planar functions that yield finite projective planes and other combinatorial objects in even characteristic.
• To develop a new arithmetic result on F_q³-rational points of the curve x^{q−1} + y^{q−1} = z^{q−1} as a key technical tool.
• To provide a constructive characterization of planar monomials in F₂ʳ using algebraic geometry over finite fields.

Proposed method

• The authors analyze the curve defined by x^{q−1} + y^{q−1} = z^{q−1} over finite fields of characteristic 2.
• They derive a new result on the number of F_q³-rational points on this curve, which is central to proving planarity.
• Using this point-counting result, they verify that specific monomials f(x) = x^d over F₂ʳ satisfy the planarity condition in even characteristic.
• The proof relies on algebraic geometry techniques, particularly the study of rational points on algebraic curves over finite fields.
• The authors apply a criterion for planarity in characteristic 2, introduced by Zhou, to the monomials under consideration.
• They combine field-theoretic arguments with properties of trace and norm functions to analyze the differential uniformity of the monomials.

Experimental results

Research questions

• RQ1 Are there monomials over F₂ʳ that satisfy the planarity condition in even characteristic, as conjectured by Schmidt and Zhou?
• RQ2 What is the structure of F_q³-rational points on the curve x^{q−1} + y^{q−1} = z^{q−1} in characteristic 2?
• RQ3 Can the number of rational points on this curve be used to establish the planarity of specific monomials?
• RQ4 How can the theory of planar functions be extended from odd to even characteristic?
• RQ5 What new finite projective planes or combinatorial designs can be constructed from planar monomials in characteristic 2?

Key findings

• The paper confirms that certain monomials f(x) = x^d over F₂ʳ are planar in even characteristic, resolving a long-standing conjecture by Schmidt and Zhou.
• A new result is established on the number of F_q³-rational points on the curve x^{q−1} + y^{q−1} = z^{q−1}, which is crucial for the proof.
• The planarity of these monomials implies the existence of new finite projective planes and other combinatorial configurations in characteristic 2.
• The method provides a constructive framework for identifying planar functions in even characteristic using algebraic geometry.
• The result extends the applicability of planar functions beyond odd characteristic, enriching the class of functions useful in design theory and cryptography.
• The proof demonstrates that the curve x^{q−1} + y^{q−1} = z^{q−1} has a controlled number of rational points over F_q³, which directly supports the planarity criterion.