We explicitly determine the binary representation of the inverse of all Kasami exponents Kr = 2 − 2 + 1 modulo 2 − 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. 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 F2n always has algebraic degree n+1 2 if n ≡ 0 (mod 3). For n 6≡ 0 (mod 3) we prove that the algebraic degree is bounded from below by 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 BLr = 2 2r + 2 + 1 modulo 2−1 where n = 4r and r odd. We show that the algebraic degree of the inverse of the Bracken-Leander function is n+2 2 .

中文翻译：

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关于 Kasami 和 Bracken-Leander 指数的逆

对于 n 和 r 的所有可能值，我们明确地确定了所有 Kasami 指数 Kr = 2 − 2 + 1 modulo 2 − 1 的逆的二进制表示。这包括一个重要的特殊情况，即具有 gcd(r, n) = 1 的 APN Kasami 指数。作为推论，我们确定 Kasami 函数的逆的代数次数。特别地，我们证明了 F2n 上的 APN Kasami 函数的逆函数总是具有代数次数 n+1 2 如果 n ≡ 0 (mod 3)。对于 n 6≡ 0 (mod 3)，我们证明代数次数的下界为 3 。我们考虑其逆为二次指数或 Kasami 指数的 Kasami 指数。我们还确定了 Bracken-Leander 指数 BLr = 2 2r + 2 + 1 模 2−1 的倒数的二进制表示，其中 n = 4r 和 r 奇数。