论文标题
在Kasami和Bracken-Leander指数上
On the inverses of Kasami and Bracken-Leander exponents
论文作者
论文摘要
我们明确确定所有kasami指数倒数的二进制表示，$ k_r = 2^{2r} -2^r+1 $ modulo $ 2^n-1 $，所有可能的$ n $和$ r $的值。这包括作为重要的特殊情况，即具有$ \ gcd（r，n）= 1 $的APN KASAMI指数。作为推论，我们确定了卡萨米函数的倒置的代数程度。特别是，我们表明，$ \ mathbb {f} _ {2^n} $ apn kasami函数的倒数始终具有代数$ \ frac {n+1} {2} {2} $，如果$ n \ equiv 0 \ equiv 0 \ pmod 3 $。对于$ n \ not \ equiv 0 \ pmod 3 $，我们证明了代数学位以$ \ frac {n} {3} $从下面界定。我们认为其逆向二次指数或kasami指数的kasami指数。我们还确定了Bracken-Leander指数$ bl_r = 2^{2r}+2^r+1 $ modulo $ 2^n-1 $其中$ n = 4r $和$ r $奇数的二进制表示。我们表明，Bracken-Leander函数倒数的代数度为$ \ frac {n+2} {2} $。
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 $\frac{n+2}{2}$.
