Affiliation:
1. NP “GST” , Moscow , Russia
Abstract
Abstract
The nonlinearity of vectorial functions and of their restrictions to manifolds are defined as the Hamming distance to the set of affine mappings and of their restrictions to the manifold, respectively. Relations between the parameters of the nonlinearity of a vectorial function and their analogues for its coordinate functions and its restrictions to manifolds are established. An analogue of the Parseval identity for such parameters of vectorial functions is proved, which implies the upper bound (qk
− 1)q
n−k
− q
n/2−k
for the nonlinearity of a mapping over a q-element field of n variables with k coordinates. Attainability conditions for this estimate are found, and a class of Boolean vectorial functions with high value of nonlinearity is constructed. Estimates characterizing the distribution of the nonlinearity of a vectorial function and of its restrictions to manifolds are obtained.
Reference14 articles.
1. Ambrosimov A. S., “Properties of bent functions of q-valued logic over finite fields”, Discrete Math. Appl., 4:4 (1994), 341–350.
2. Ambrosimov A. S., “Approximation of k-ary functions by functions from the given system”, Fundam. Prikl. Matem., 3:3 (1997), 653–674 (in Russian).
3. Gorshkov S. P., Dvinyaninov A. V., “Lower and upper bounds for the affinity order of transformations of Boolean vector spaces”, Prikladnaya Diskretnaya Matematika, 2013, №2(20), 653–674 (in Russian).
4. Zubkov A. M., Serov A. A., “Bounds for the number of Boolean functions that have affine approximations of given accuracy”, Discrete Math. Appl., 20:5–6 (2010), 467–486.
5. Ryabov V. G., “On the degree of restrictions of q-valued logic functions to linear manifolds”, Prikladnaya Diskretnaya Matematika, 2019, №45, 653–674 (in Russian).