On the inverses of Kasami and Bracken–Leander exponents

We explicitly determine the binary representation of the inverse of all Kasami exponents $$K_r=2^{2r}-2^r+1$$ K r = 2 2 r - 2 r + 1 modulo $$2^n-1$$ 2 n - 1 for all possible values of n and r. This includes as an important special case the APN Kasami exponents with $$\gcd (r,n)=1$$ gcd ( r , n ) = 1 . As a corollary, we determine the algebraic degree of the inverses of the Kasami functions. In particular, we show that the inverse of an APN Kasami function on $${\mathbb {F}}_{2^n}$$ F 2 n always has algebraic degree $$\frac{n+1}{2}$$ n + 1 2 if $$n\equiv 0 \pmod 3$$ n ≡ 0 ( mod 3 ) . For $$n\not \equiv 0 \pmod 3$$ n ≢ 0 ( mod 3 ) we prove that the algebraic degree is bounded from below by $$\frac{n}{3}$$ n 3 . We consider Kasami exponents whose inverses are quadratic exponents or Kasami exponents. We also determine the binary representation of the inverse of the Bracken–Leander exponent $$BL_r=2^{2r}+2^r+1$$ B L r = 2 2 r + 2 r + 1 modulo $$2^n-1$$ 2 n - 1 where $$n=4r$$ n = 4 r and r odd. We show that the algebraic degree of the inverse of the Bracken–Leander function is $$\frac{n+2}{2}$$ n + 2 2 .