On Subspaces of Kloosterman Zeros and Permutations of the Form $$L_1(x^{-1})+L_2(x)$$

Göloğlu, Faruk; Kölsch, Lukas; Kyureghyan, Gohar M.; Perrin, Léo

doi:10.1007/978-3-030-68869-1_12

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2021

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Cited by 2 publications

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“…Instead, we are going to use the following proposition which connects both of the problems we mentioned to a special type of permutation polynomial. The proof of the proposition can be found in [11], co-written by the author of this paper, that deals with the same questions as this paper but only achieved partial results.…”

Section: Introductionmentioning

confidence: 83%

On CCZ-equivalence of the inverse function

Kölsch¹

2020

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The inverse function x → x −1 on F 2 n is one of the most studied functions in cryptography due to its widespread use as an S-box in block ciphers like AES. In this paper, we show that, if n ≥ 5, every function that is CCZ-equivalent to the inverse function is already EA-equivalent to it. This confirms a conjecture by Budaghyan, Calderini and Villa. We also prove that every permutation that is CCZ-equivalent to the inverse function is already affine equivalent to it. The majority of the paper is devoted to proving that there are no permutation polynomials of the form L 1 (x −1 ) + L 2 (x) over F 2 n if n ≥ 5, where L 1 , L 2 are nonzero linear functions. In the proof, we combine Kloosterman sums, quadratic forms and tools from additive combinatorics.

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Section: Introductionmentioning

confidence: 83%