关于Kasami和Bracken-Leander指数的逆
On the inverses of Kasami and Bracken-Leander exponents
摘要：
我们明确地确定的二进制表示的逆的所有Kasami指数$K_r=2^{2r}-2^r+1$模$2^n-1$的$n$和$r$的所有可能的值。这包括APN Kasami指数与$\gcd(r,n)=1$的重要特殊情况。作为推论，我们确定了Kasami函数的逆的代数次数。特别地，我们证明了$\mathbb{F}_{2^n}$上的APN Kasami函数的逆总是具有代数次数$\frac{n+1}{2}$，如果$n\equiv 0 \pmod 3$。对于$n\not\equiv 0 \pmod 3$，我们证明了代数次数由$\frac{n}{3}$从下有界。我们考虑Kasami指数的逆是二次指数或Kasami指数。我们还确定了Bracken-Leander指数$BL_r=2^{2r}+2^r+1$模$2^n-1$的逆的二进制表示，其中$n=4r$和$r$是奇数。我们证明了Bracken-Leander函数的逆的代数次数是$\frac{n+2}{2}$。
英文摘要：
We explicitly determine the binary representation of the inverse of all Kasami exponents $K_r=2^{2r}-2^r+1$ modulo $2^n-1$ for all possible values of $n$ and $r$. This includes as an important special case the APN Kasami exponents with $\gcd(r,n)=1$. 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 $\mathbb{F}_{2^n}$ always has algebraic degree $\frac{n+1}{2}$ if $n\equiv 0 \pmod 3$. For $n\not\equiv 0 \pmod 3$ we prove that the algebraic degree is bounded from below by $\frac{n}{3}$. 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 $BL_r=2^{2r}+2^r+1$ modulo $2^n-1$ where $n=4r$ and $r$ odd. We show that the algebraic degree of the inverse of the Bracken-Leander function is $\frac{n+2}{2}$.
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