Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles. (English) Zbl 1136.94006

LeBel, Alain; Horadam, K. J.

J. Comb. Des. 16, No. 3, 173-181 (2008).

Summary: Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by K. Nyberg [Lect. Notes Comput. Sci. 547, 378–386 (1991; Zbl 0766.94012)] on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg’s result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non-cyclic abelian groups and use it to find all the orthogonal cocycles over \(\mathbb{Z}_{2}^{t}\), \(2 \leq t \leq 4\). We conjecture that any orthogonal cocycle over \(\mathbb{Z}_{2}^{t}, t \geq 2\), must be multiplicative.

MSC:

94A60	Cryptography
94A55	Shift register sequences and sequences over finite alphabets in information and communication theory
20J06	Cohomology of groups

Keywords:

perfect nonlinear function; balanced function; orthogonal cocycle; relative difference set; generalized Hadamard matrix; exponential sum

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