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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