# zbMATH — the first resource for mathematics

Characterization of linear structures. (English) Zbl 0963.94021
Let $$F$$ be a function defined from $$F_2^m$$ to $$F_2^n$$ and $$(\alpha,a)$$ be an element of $$F_2^m\times F_2^n$$, with $$\alpha\neq 0$$. Then $$(\alpha,a)$$ is defined to be a linear structure of $$F$$ if $$F(x)= F(x+\alpha)+ a$$ for all $$x\in F_2^m$$.
In this paper the existence of linear structures for $$F$$ is characterized using the Fourier transform of the function. In particular the case of Boolean functions, i.e. where $$n=1$$ is carefully studied. For cryptographic applications Boolean functions without linear structures are desirable. Two constructions of resilient (i.e., correlation-immune and balanced) Boolean functions which have no linear structure are presented.

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: