# Multidimensional WalshTransform andaCharacterization ofBentFunctions

01 Jan 2007-

About: The article was published on 2007-01-01 and is currently open access. It has received None citations till now.

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08 Apr 1991TL;DR: In this article, it was shown that for a perfect nonlinear S-box, the number of input variables is at least twice the size of output variables, and two different construction methods were given.

Abstract: A perfect nonlinear S-box is a substitution transformation with evenly distributed directional derivatives. Since the method of differential cryptanalysis presented by E. Biham and A. Shamir makes use of nonbalanced directional derivatives, the perfect nonlinear S-boxes are immune to this attack. The main result is that for a perfect nonlinear S-box the number of input variables is at least twice the number of output variables. Also two different construction methods are given. The first one is based on the Maiorana-McFarland construction of bent functions and is easy and efficient to implement. The second method generalizes Dillon's construction of difference sets.

369 citations

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TL;DR: A perfect nonlinear S-box is a substitution transformation with evenly distributed directional derivatives and the number of input variables is at least twice thenumber of output variables.

Abstract: A perfect nonlinear S-box is a substitution transformation with evenly distributed directional derivatives. Since the method of differential cryptanalysis presented by E. Biham and A. Shamir makes use of nonbalanced directional derivatives, the perfect nonlinear S-boxes are immune to this attack. The main result is that for a perfect nonlinear S-box the number of input variables is at least twice the number of output variables. Also two different construction methods are given. The first one is based on the Maiorana-McFarland construction of bent functions and is easy and efficient to implement. The second method generalizes Dillon's construction of difference sets.

349 citations

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TL;DR: It is proved that a large class of quadratic functions cannot be APN, and some results of Nyberg (1994) are generalized and a conjecture on the upper bound of nonlinearity of APN functions is strengthened.

Abstract: We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic 2. We provide new characterizations of APN functions and of APN permutations by means of their component functions. We generalize some results of Nyberg (1994) and strengthen a conjecture on the upper bound of nonlinearity of APN functions. We also focus on the case of quadratic functions. We contribute to the current works on APN quadratic functions by proving that a large class of quadratic functions cannot be APN

124 citations