Constructing high order elements through subspace polynomials

Constructing high order elements through subspace polynomials
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DOI：
10.1137/1.9781611973099.115
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发表时间：
2012-01
期刊：
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影响因子：
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通讯作者：
Qi Cheng;Shuhong Gao;D. Wan
Qi Cheng;Shuhong Gao;D. Wan
中科院分区：
其他
文献类型：
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作者：
Qi Cheng;Shuhong Gao;D. Wan

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每个有限域都有许多乘法发生器。然而，在多项式时间内找到一个是一个重要的开放问题。事实上，即使是寻找高阶元素也没有得到令人满意的解决。本文给出了在确定时间（qc）O(1)下，对任意正整数c和素数幂q，在有限域[方程]中求exp（Ω（√qc））阶元的算法。我们还证明了在这些域中对于某些基的离散对数问题存在exp（Ω（√qc））多个弱密钥。
Every finite field has many multiplicative generators. However, finding one in polynomial time is an important open problem. In fact, even finding elements of high order has not been solved satisfactorily. In this paper, we present an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω(√qc)) in the finite field [EQUATION] in deterministic time (qc)O(1). We also show that there are exp(Ω(√qc)) many weak keys for the discrete logarithm problems in those fields with respect to certain bases.