On the exponents of APN power functions and Sidon sets, SUM-free sets, and Dickson Polynomials

On the exponents of APN power functions and Sidon sets, SUM-free sets, and Dickson Polynomials

Carlet, Claude (University of Bergen and Bjerknes Centre for Climate Research; University of Paris 8)
Picek, S. (TU Delft Cyber Security; University of Paris 8)

2023

We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents d ∈ Z/(2ⁿ − 1)Z, which are such that F (x) = x^d is an APN function over F₂n (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents d. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to n = 48, providing the number of exponents satisfying all the conditions for a function to be APN. We also show a new connection between APN exponents and Dickson polynomials: F (x) = x^d is APN if and only if the reciprocal polynomial of the Dickson polynomial of index d is an injective function from {y ∈ F^∗2n; trn(y) = 0} to F₂n \ {1}. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.

Almost Perfect Nonlinear Functions
Dickson polynomial
Power functions
Sidon
Sum-free

http://resolver.tudelft.nl/uuid:548f510a-fdd1-45a1-9b8f-b4c2249c39c4

https://doi.org/10.3934/amc.2021064

2023-01-01

1930-5346

Advances in Mathematics of Communications, 17 (6), 1507-1525

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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© 2023 Claude Carlet, S. Picek