Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 265–291

# Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

• Nurdagül Anbar
• Wilfried Meidl
• Alev Topuzoğlu
Article

## Abstract

The Walsh transform $$\widehat{Q}$$ of a quadratic function $$Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p$$ satisfies $$|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$$ for all $$b\in {\mathbb F}_{p^n}$$, where $$0\le s\le n-1$$ is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class $$\mathcal {C}_1$$ is defined for arbitrary n as $$\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}$$, and the larger class $$\mathcal {C}_2$$ is defined for even n as $$\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}$$. For an odd prime p, the subclass $$\mathcal {D}$$ of all p-ary quadratic functions is defined as $$\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}$$. We determine the generating function for the distribution of the parameter s for $$\mathcal {C}_1, \mathcal {C}_2$$ and $$\mathcal {D}$$. As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $$p > 2$$, the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to $$\mathcal {C}_1$$ and $$\mathcal {C}_2$$ in terms of a generating function.

## Keywords

Quadratic functions Plateaued functions Bent functions Walsh transform Idempotent functions Rotation symmetric Reed-Muller code

## Mathematics Subject Classification

11T06 11T71 11Z05

## Notes

### Acknowledgments

The first author gratefully acknowledges the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367) and H.C. Ørsted COFUND Post-doc Fellowship from the project “Algebraic curves with many rational points”. The second author is supported by the Austrian Science Fund (FWF) Project No. M 1767-N26.

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## Authors and Affiliations

• Nurdagül Anbar
• 1
• Wilfried Meidl
• 2
• Alev Topuzoğlu
• 3
1. 1.Technical University of DenmarkLyngbyDenmark
2. 2.Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of SciencesLinzAustria
3. 3.Sabancı University, MDBFIstanbulTurkey