On some properties of known isometric mappings of the set of bent functions

We prove that there doesn't exist an isometry on the set of all Boolean functions in 2k variables which acts on the set of bent functions by assigning the dual bent functions. We state the affine equivalence of a bent function and its dual bent function in the case of small number of variables. Keywords: Boolean function, bent function, isometry, dual bent function. Miloserdov A. V. PERMUTATION BINOMIALS OVER FINITE FIELDS. CONDITIONS OF EXISTENCE. Let 1 ^ j < i ^ 2ⁿ - 1, 1 ^ k ^ 2ⁿ - 1, a is a primitive element of the field F₂n. It is proved that: 1) if a function f : F₂n ^ F₂n of the form f (y) = a^kyⁱ + y^j is one-to-one function, then gcd(i - j, 2ⁿ - 1) doesn't divide gcd(k, 2ⁿ - 1); 2) if 2ⁿ - 1 is prime, then one-to-one function f : F₂n ^ F₂n of the form f (x) = a^kxⁱ + x^j doesn't exist; 3) if n is a composite number, then there is one-to-one function f : F₂n ^ F₂n n of the form f (x) = a^kxⁱ+x^j; 4) if 2ⁿ - 1 has a divisor d < --- - 1, then there is one-to- ^{2 lo}g2 ⁽ⁿ⁾ one function f : F₂n ^ F₂n of the form f (y) = ayⁱ + y^j for some a G F2n, 0 < j < i < 2ⁿ - 1.

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