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On a family of perfect nonlinear binomials. (English) Zbl 1198.94098

Preneel, Bart (ed.) et al., Boolean functions in cryptology and information security. Selected papers based on the presentations at the NATO-Russia Advanced Study Institute on Boolean functions in cryptology and information security, Zvenigorod, Russia, September 8–18, 2007. Amsterdam: IOS Press (ISBN 978-1-58603-878-6/hbk). NATO Science for Peace and Security Series D: Information and Communication Security 18, 126-138 (2008).
Summary: A mapping \(f: \text{GF}(p^n)\to \text{GF}(p^n)\) is called differentially \(k\)-uniform if \(k\) is the maximum number of solutions \(x\in\text{GF}(p^n)\) of \(f(x+a)-f(x)=b\), where \(a,b\in\text{GF}(p^n)\) and \(a\neq 0\). A 1-uniform mapping is called perfect nonlinear (PN). In this paper we discuss some problems related to the equivalence of perfect nonlinear functions, and describe a class of perfect nonlinear binomials \(ux^{p^k+1}+x^2\) in \(\text{GF}(p^{2k})\). These are the first PN binomials known to us which are composed with inequivalent monomials. We show that this family of binomials is equivalent to the monomial \(x^2\). We survey some of the close connections between perfect nonlinear functions and finite affine planes, in particular those which are important for equivalence proofs.
For the entire collection see [Zbl 1149.68014].

MSC:

94A60 Cryptography
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