Abstract
Depending on the parity of n and the regularity of a bent function f from {{mathbb F}_{p}^{n}} to {mathbb F}_{p}, f can be affine on a subspace of dimension at most n/2, (n − 1)/2 or n/2 − 1. We point out that many p-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2-normal, i.e. affine on a n/2-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not n/2-normal. We develop an algorithm for testing normality for functions from {{mathbb F}_{p}^{n}} to {mathbb F}_{p}. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.
Highlights
Let p be a prime, and let f be a function from an n-dimensional vector space Vn over Fp to Fp
In this article we investigate normality for p-ary bent functions, which describes a feature of the corresponding relative difference set
By [2, Lemma 25] this guarantees the existence of non-weakly-normal Boolean bent functions in dimension n ≥ 10
Summary
Let p be a prime, and let f be a function from an n-dimensional vector space Vn over Fp to Fp. Non-weakly-normal bent functions in dimension 10 (and 12) were presented in. By [2, Lemma 25] this guarantees the existence of non-weakly-normal Boolean bent functions in (even) dimension n ≥ 10. Our algorithm is not a straightforward generalization of the algorithm in [2], which was used to find non-weakly-normal Boolean bent functions in dimension 14 [2], and 10 and 12 [14]. Applying this algorithm we find the first examples of p-ary bent functions (in small dimensions) which do not possess k-normality with maximal possible k. Generalizing Lemma 25 of [2] we can obtain bent functions with this property in every larger dimension of the same parity
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