О четверичных и двоичных бент-функциях.pdf In this paper direct links between Boolean bent functions (Rothaus, [1], 1976), generalizedBoolean bent functions (Schmidt, [2], 2006) and quaternary bent functions (Kumar,Scholtz, Welch, [3], 1985) are explored. We also study Gray images of bent functions andnotions of generalized nonlinearity for Boolean functions.Let n, q be integers, q ^ 2. We consider the following mappings:1) f : Zn ^ Z 2 - B o o lean function in n variables. Its sign function is F :== (-1)f . The Walsh Hadamard transform (WHT) of f is F (x) := ^ yeZn (-1)f(y)+x-y == yoxn Fy(-1)x'y. Here x.y is a usual inner product of vectors. A Boolean function f issaid to be bent, iff |F(x)| = 2n/2 for all x Е Zn. It is near bent iff F(x) Е {0, ± 2 (n+1)/2}.Note that Boolean bent (resp. near bent) functions exist only if the number of variables,n, is even (resp. odd).2) f : Zn ^ Z q - generalized Bo o lean function in n variables. Its sign function isF := wf , with w a primitive complex root of unity of order q, i. e. w = e2nj/q. When q = 4,we write w = i. Its WHT is given as F (x) := ^ yeZn wf(y)(-1)x'y = ^ yeZn Fy(-1)x'y.As above, a generalized Boolean function f is bent, iff |i^(x)| = 2n/2 for all x Е Zn. Incomparison to the previous case it not follows that n should be even if f is bent. Suchfunctions for q = 4 were studied in [2]. Here we consider q = 4 only.3) f : Z J ^ Z q - q-ary function in n variables. Its sign function is given by F :=as in the previous case. Its WHT is defined by F(x) := . yeZn (y)+x-y = . yeZn Fyu x'y.Note that the matrix of this transform is no longer a Sylvester type Hadamard matrix asin the previous case, but a generalized (complex) Hadamard matrix. A q-ary function fis called bent, iff |F(x)| = qn/2 for all x . Z J. Notice that again it not follows from thedefinition that q-ary bent functions do not exist if n is odd. Kumar, Scholtz and Welch [3]have studied q-ary bent functions in 1985. They proved that such functions exist for anyeven n and q = 2(mod4). Later Ambrosimov described all quadratic q-ary bent functionsover an arbitrary finite field and Agievich proposed an approach to describe regular q-arybent functions in terms of bent rectangles. If q = 4 we call f a quaternary function.Here we study such functions only.Let f : Z ^ ^ Z 4 be any generalized Boolean function. Represent it as f (x,y) == а ^ , y) + 2b(x, y), for any x, y . ZJ, where а, b : Z ^ ^ Z 2 are Boolean functions.T h e o rem 1. The following statements are equivalent:(i) the generalized Boolean function f is bent in 2n variables;(ii) the Boolean functions of 2n variables b and а + b are both bent.Define a quaternary function g : ZJ ^ Z 4 as g(x + 2y) = f (x, y).We say that two Boolean functions c and d in 2n variables are bent correlated (withrespect to dividing variables into two halves) if for any x ,y . ZJ, the conditions hold1) C72(x,y) + C72(x + y,y) + D 2(x,y) + D 2(x + y,y) = 4n+i;2) C7(x, y) = D (x + y, y) = ± 2 n C7(x + y, y) = D (x, y) = ± 2 n.It is easy to construct examples of such functions.T h e o rem 2. The following statements are equivalent:(i) the quaternary function g is bent in n variables;(ii) the Boolean functions b and а + b are bent correlated.Now let f be a generalized Boolean function from ZJ to Z 4. The Gray map + b(z).Using results from [2] we proveP r op o s ition 3. If f is bent then

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