Cryptology ePrint Archive: Report 2016/887

A generalisation of Dillon's APN permutation with the best known differential and nonlinear properties for all fields of size $2^{4k+2}$

Anne Canteaut and Sébastien Duval and Léo Perrin

Abstract: The existence of Almost Perfect Nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et al. in~2009. However it is still unknown whether this example can be generalised to any even number of inputs. In a recent work, Perrin et al. described an infinite family of permutations, named butterflies, operating on (4k+2) variables and with differential uniformity at most 4, which contains the Dillon APN permutation. In this paper, we generalise this family, and we completely solve the two open problems raised by Perrin et al. Indeed we prove that all functions in this larger family have the best known nonlinearity. We also show that this family does not contain any APN permutation besides the Dillon permutation, implying that all other functions have differential uniformity exactly four.

Category / Keywords: Boolean function, Sbox, APN, differential uniformity, nonlinearity

Original Publication (in the same form): IEEE Transactions on Information Theory

Date: received 9 Sep 2016, last revised 28 Feb 2017

Contact author: Anne Canteaut at inria fr

Available format(s): PDF | BibTeX Citation

Note: Minor modifications compared to the previous submission. This version is the same as the paper to appear in the IEEE Transactions on Information Theory.

Version: 20170228:155426 (All versions of this report)

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