有限域上的不可约根与r-次根的求法
Irreducibility and r-th root finding over finite fields
摘要：
构造有限域上的$r$次非剩余是一个基本的计算问题.一个相关的问题是在给定的特征为$p$的有限域$\mathbb{F}_q$上构造一个次数为$r^e$的不可约多项式（其中$r$是素数）（等价地，构造更大的域$\mathbb{F}_{q^{r^e}}$）。这两个问题都有著名的随机算法，但去随机化是一个开放的问题。我们给出了这两个问题及其变体之间的一些新的联系。1897年，Stickelberger证明了如果一个多项式有奇数个偶数次因子，那么它的判别式是域上的二次非剩余。我们给出了Stickelberger引理的一个推广，我们从一个有$d$的多项式$f$中构造了$r$ -th非剩余，使得$r|d$和$r\nmid\,$ #（$f(x)$的不可约因子）.我们的定理有以下有趣的结果：（1）如果$m$是一个$r$ -幂且$f$已知，我们可以在确定性的多（deg（$f$），$m\log q$）-时间中构造$\mathbb{F}_{q^m}$;（2）如果$r$是常数且$r|\gcd(m,p-1)$已知，我们可以在确定性的多（$m\log p$）-时间中找到$\mathbb{F}_{p^m}$的$r$ -次根.我们还讨论了一个比广义黎曼假设弱得多的猜想，得到了$r$ -次根的确定性多时间算法。
英文摘要：
Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of characteristic $p$ (equivalently, constructing the bigger field $\mathbb{F}_{q^{r^e}}$). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its discriminant is a quadratic nonresidue in the field. We give an extension of Stickelberger's Lemma; we construct $r$-th nonresidues from a polynomial $f$ for which there is a $d$, such that, $r|d$ and $r\nmid\,$#(irreducible factor of $f(x)$ of degree $d$). Our theorem has the following interesting consequences: (1) we can construct $\mathbb{F}_{q^m}$ in deterministic poly(deg($f$),$m\log q$)-time if $m$ is an $r$-power and $f$ is known; (2) we can find $r$-th roots in $\mathbb{F}_{p^m}$ in deterministic poly($m\log p$)-time if $r$ is constant and $r|\gcd(m,p-1)$. We also discuss a conjecture significantly weaker than the Generalized Riemann hypothesis to get a deterministic poly-time algorithm for $r$-th root finding.
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