On “bent” functions

Binary sequences of length n = 2m- 1 whose autocorrelation function is either 1 or -1/n have been known for a long time, and are called pseudo-random (or PN) sequences, or maximal-length shift-register sequences. Two-dimensional arrays of area n = 2lm- 1 with the same property have rcently been found by several authors. This paper gives a simple description of such sequences and arrays and their many nice properties.

Pseudo-random sequences and arrays

Generalized bent functions and their properties

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.

/pdf/perfect-nonlinear-s-boxes-3j3wdnvq27.pdf

Robert L McFarland

Papers

A family of difference sets in non-cyclic groups