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Vectorial Boolean functions for cryptography. (English) Zbl 1209.94036
Crama, Yves (ed.) et al., Boolean models and methods in mathematics, computer science, and engineering. Cambridge: Cambridge University Press (ISBN 978-0-521-84752-0/hbk). Encyclopedia of Mathematics and its Applications 134, 398-469 (2010).
From the introduction: This chapter deals with multi-output Boolean functions viewed from a cryptographic viewpoint, that is, functions from the vector space $$\mathbb F_2^n$$, of all binary vectors of length $$n$$, to the vector space $$\mathbb F_2^m$$, for some positive integers $$n$$ and $$m$$, where $$\mathbb F_2$$ is the finite field with two elements. Obviously, these functions include the (single-output) Boolean functions that correspond to the case $$m=1$$. The, present chapter follows the chapter “Cryptography and Error-Correcting Codes” [the author, in: Crama, Yves (ed.) et al., Boolean models and methods in mathematics, computer science, and engineering. Cambridge: Cambridge University Press. Encyclopedia of Mathematics and its Applications 134, 257–397 (2010; Zbl 1209.94035)] (dedicated to Boolean functions), to which we refer for all the definitions and properties that will be needed in the present chapter. As in this previous chapter, additions of bits performed in characteristic 0 (that is, in $$\mathbb Z$$, i.e., not modulo 2) will be denoted by +, and additions modulo 2 (in $$\mathbb F_2$$) will be denoted by $$\oplus$$. The multiple sums will be denoted by $$\Sigma_i$$ when they are calculated in characteristic 0 and by $$\bigoplus_i$$, when they are calculated modulo 2. These two different notations are necessary because some representations of (vectorial) Boolean functions live in characteristic 2 and some representations of the same functions live in characteristic 0. However, the additions of elements of the finite field $$\mathbb F_{2^n}$$ will be denoted by +, as is usual in mathematics, despite the fact they are performed in characteristic 2. So, for simplicity (because $$\mathbb F_2^n$$ will often be identified with $$\mathbb F_{2^n}$$) and because there will be no ambiguity, we shall also denote by + the addition of vectors of $$\mathbb F_2^n$$ when $$n>1$$.
For the entire collection see [Zbl 1196.06001].

##### MSC:
 94A60 Cryptography