arXiv math/03111202003-11-07math.NT

Kummer和Artin-Schreier扩张下的有界位数和离散对数问题

On the Bounded Sum-of-digits Discrete Logarithm Problem in Kummer and Artin-Schreier Extensions

Qi Cheng

备注：: 10 pages

摘要：

本文研究有限域$\F_{q^n}$上的离散对数问题，其中$n|q-1$ .该域称为库默尔域或$\F_q$的库默尔扩展。这对改进AKS素性证明算法具有重要作用。我们知道，我们可以有效地构造一个元素$g$与阶大于$2^n$的字段。令$S_q(\bullet)$是从整数到其$q$ -元展开式中的数字之和的函数。我们提出了一个算法，给定$g^e$（$ 0\leq e < q^n $）发现$e$在随机多项式时间，如果$S_q (e) < n$。然后，我们证明了这个问题是可解的随机多项式时间的大部分指数$e$与$S_q (e) < 1.32 n $。后一结果的主要工具是Guruswami-Sudan列表解码算法。在此基础上，证明了在$\F_{q^{q-1}}$域上，关于$g$的有界位数和离散对数可以在随机时间$O(f(w) \log^4 (q^{q-1}))$内计算，其中$f$是次指数函数，$w$是指数的$q$元位数和上的界.因此，该问题是固定参数易处理的。这些结果可推广到Artin-Schreier扩张$\F_{p^p}$，其中$p$是素数.由于每个有限域都有一个合理次的扩张是Kummer域，我们的结果揭示了离散对数问题的一个意想不到的性质，即任何有限域上的有界位数和离散对数问题在某些低次扩张中成为多项式时间可解的.

英文摘要：

In this paper, we study the discrete logarithm problem in the finite fields $\F_{q^n}$ where $n|q-1$. The field is called a Kummer field or a Kummer extension of $\F_q$. It plays an important role in improving the AKS primality proving algorithm. It is known that we can efficiently construct an element $g$ with order greater than $2^n$ in the fields. Let $S_q(\bullet)$ be the function from integers to the sum of digits in their $q$-ary expansions. We present an algorithm that given $g^e$ ($ 0\leq e < q^n $) finds $e$ in random polynomial time, provided that $S_q (e) < n$. We then show that the problem is solvable in random polynomial time for most of the exponent $e$ with $S_q (e) < 1.32 n $. The main tool for the latter result is the Guruswami-Sudan list decoding algorithm. Built on these results, we prove that in the field $\F_{q^{q-1}}$, the bounded sum-of-digits discrete logarithm with respect to $g$ can be computed in random time $O(f(w) \log^4 (q^{q-1}))$, where $f$ is a subexponential function and $w$ is the bound on the $q$-ary sum-of-digits of the exponent. Hence the problem is fixed parameter tractable. These results are shown to be extendible to Artin-Schreier extension $\F_{p^p}$ where $p$ is a prime. Since every finite field has an extension of reasonable degree which is a Kummer field, our result reveals an unexpected property of the discrete logarithm problem, namely, the bounded sum-of-digits discrete logarithm problem in any given finite field becomes polynomial time solvable in certain low degree extensions.

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