A New Representation of Boolean Functions

Abstract

Abstract. We study a representation of Boolean functions (and more generally of integer-valued / complex-valued functions), not used until now in coding and cryptography, which yields more information than the currently known representations, on the combinatorial, spectral and cryptographic properties of the functions.

p-Ary and q-ary versions of certain results about bent functions and resilient functions

“…for all ð1; y; 1ÞauAU: Theorem 1.1 is a combination of Proposition 1 of [4] and a result of [3]. A binary function f AFðF n 2 ; F 2 Þ is called t-resilient if fðcÞ ¼ 0 for all cAF n 2 with jcjpt:…”

“…Theorem 1.2 is not explicitly stated in the literature. However, it follows from the results of[3] and is thus considered known. A binary function f AFðF n 2 ; F 2 Þ is called bent iffðcÞ ¼ 72 n=2 for all cAF n 2 : (n must be even.)…”

Highly nonlinear mappings

“…See for example, [10,[12][13][14][15][17][18][19][20][37][38][39][40]67,71]. Non-Boolean functions have also important applications in cryptography [8,9,64], sequences [57,68] and coding theory [43,69], but they have been less studied.…”

Primitivity of PRESENT and other lightweight ciphers

“…Then there exist U, W < (F 2 ) m such that dim(U ) = dim(W ) = m − 1 and U γ = W . As the non-linearity of a vectorial Boolean function is affine invariant (see for instance [12]), without loss of generality, we may assume U = W = ⟨e 1 , . .…”