Bent functions: results and applications. A survey

Abstract

Приводится краткий обзор основных результатов в области бент-функций. Рассматриваются их теоретические и практические приложения. Ключевые слова: булева функция, АНФ, преобразование Уолша Адамара, нелинейность, бент-функция. 1 Исследование выполнено при финансовой поддержке интеграционного проекта СО РАН № 35 Древовидный каталог математических интернет-ресурсов mathtree.ru , РФФИ (проекты 07-01-00248, 08-01-00671, 09-01-00528-а) и Фонда содействия отечественной науке.

Generalized bent Boolean functions and strongly regular Cayley graphs

“…Let f ∈ GB q n , where 2 k−1 < q ≤ 2 k , then we can represent f uniquely as f (x) = a 0 (x) + 2a 1 (x) + · · · + 2 k−1 a k−1 (x) for some Boolean functions a i , 0 ≤ i ≤ k − 1 (this representation comes from the binary representation of the elements in the image set Z 2 k ). For results on classical bent functions and related topics, the reader can consult [5,8,13,18].…”

Quasi-random multilinear polynomials

“…Thus each of (−1) ξ and (−1) η is a perfectly flat multilinear Littlewood function; as we have seen, this is possible only due to their high degree. Bent functions have been extensively studied particularly in view of applications in cryptography [51,6] and quantum computation [7].…”

On minimal distance between q-ary bent functions

“…III. BENT FUNCTIONS A function f : Z n q → Z q is called a q-ary bent function iff |W f (y)| = 1 for each y ∈ Z n q or ξ f · ξ f = I, where I is equal to 1 everywhere (see [3], [10]). By using (2) we can obtain that the definition of bent function is equivalent to the equation…”

“…It is well known that Q 0 is a bent function from Maiorana-McFarland class (see [10]). The following proposition is proved, for example, in [1] (p.274, Lemma 9.4.1)).…”