Cryptology ePrint Archive: Report 2020/705

On the minimal value set size of APN functions

Ingo Czerwinski

Abstract: We give a lower bound for the size of the value set of almost perfect nonlinear (APN) functions \(F\colon \mathbb{F}_2^n \to \mathbb{F}_2^n\) in explicit form and proof it with methods of linear programming. It coincides with the bound given in [CHP17]. For \(n\) even it is \(\frac{ 2^n + 2 }{3}\) and sharp as the simple example \(F(x) = x^3\) shows. The sharp lower bound for \(n\) odd has to lie between \(\frac{ 2^n + 1 }{3}\) and \(2^{n-1}\). Sharp bounds for the cases \(n = 3\) and \(n = 5\) are explicitly given.

Category / Keywords: foundations / Boolean functions, Cryptographic S-boxes, Almost perfect nonlinear (APN), Value set size

Date: received 12 Jun 2020, last revised 5 May 2021

Contact author: ingo at czerwinski eu

Available format(s): PDF | BibTeX Citation

Version: 20210505:214423 (All versions of this report)

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