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Abstract
The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and \(q\)-ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. We show that the hyperovals of \(PG_{2}(2^m)\) from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of \(2^r\)-ary (\(r\) being a divisor of \(m\)) simplex linear codes (whose duals are the perfect \(2^r\)-ary Hamming codes) over an extension field \({\mathbb F}_{2^{r}}\) of \({\mathbb F}_{2^{}}\). The following diagram gives an indication of the main topics and interconnections arising in this paper.
A polynomial \(f\in {\mathbb F}_{2^{n}} [x]\) is a permutation polynomial of \({\mathbb F}_{2^{n}}\) if the function \(f:c\mapsto f(c)\) from \({\mathbb F}_{2^{n}}\) into itself induces a permutation. Alternatively, the equation \(f(x)=a\) has a unique solution for each \(a\in {\mathbb F}_{2^{n}}\).
A subset \(S\) of \(PG_n(q)\) is called \(\rho \)-saturating when every point \(P\) in \(PG_n(q)\) can be written as a linear combination of at most \(\rho +1\) points of \(S\).
The sphere-covering bound is given by the inequality: \(\#\mathcal {C}\times V_{q}(n, R)\ge q^n\) where \(V_{q}(n, R)\) denotes the size of the Hamming ball of radius \(R\).
