Analiza ukrivljenih Boolovih funkcij glede na nove metrike / Analysis of bent Boolean functions in terms of new metrics

Glavni cilj projekta je revizija ideje metrike, ki se uporablja za določanje razdalje med Boolovimi funkcijami, s posebnim poudarkom na doseganju lastnosti ukrivljenosti. Običajno se z dobro znano Hammingovo metriko definira razdalja med dvema Boolovima funkcijama, ki je opredeljena kot kardinalnost nabora vhodov, na katerih se dani dve funkciji razlikujeta. Na podlagi te metrike določimo, da je Boolova funkcija ukrivljena, če stoji na največji možni razdalji od prostora linearnih funkcij. Tako je lastnost ukrivljenosti Boolovih funkcij neposredno povezana z definicijo osnovne metrike. Tema raziskovalnega predloga je analiza učinka različnih metrik na lastnost ukrivljenosti Boolovih funkcij. Obe raziskovalni skupini, slovenska in bosanska, sestavljajo starejši raziskovalci z dokazanimi znanstvenimi dosežki na več sorodnih področjih matematike. Če bodo imeli člani obeh ekip priložnost, da se redno osebno srečujejo, smo trdno prepričani, da bo združevanje njihovega znanja in izkušenj privedlo do pomembnih novih rezultatov na tem. The main goal of the project is to revise the notion of a metric used for defining the distance between Boolean functions, with the special focus on achieving the bentness property. Traditionally, the well-known Hamming metric is used to define the distance between two Boolean functions, which is defined as the cardinality of the set of inputs on which given two functions differ. Based on this metric, we say that a Boolean function is bent if it stands at maximal distance to the space of linear functions. Hence, the bentness property of Boolean functions is directly related to the definition of the underlying metric. The topic of this research proposal is the analysis of the effect of different metrics on the bentness property of Boolean functions. Both research teams, the Slovenian and the Bosnian one, contain senior researchers with a proven record of scientific achievements in several related areas of mathematics. Therefore, if the members of the two teams will have a chance to meet regularly in person, we strongly believe that combining their knowledge and experience will lead to strong new results in the field.