Abstract
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
.