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arXiv 2011.07483cs.CR
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离散对数密码体制中的弱密钥

Removable Weak Keys for Discrete Logarithm Based Cryptography

Michael John Jacobson,, Prabhat Kushwaha

摘要:

本文提出了一种新的弱密码私钥,它可以存在于素数阶$p$群中的任何基于离散对数的公钥密码系统中,其中$p-1$具有小因子.与基于\textit{numerical size}的弱私钥(例如较小的私钥或位于区间中的私钥)不同,\textit{always}将存在于任何DLP密码系统中,我们的弱私钥类型纯粹是由于$p$的参数选择而产生的,因此,可以通过适当的$p$值来删除。利用隐式群表示的理论,我们提出的算法,可以确定是否一个密钥是弱的,如果是这样,恢复私钥从相应的公钥。我们分析了文献中提出的几个椭圆曲线和各种标准,给出了可以用相对少量的计算来破解的密钥的数量。我们的研究结果表明,许多这些曲线,包括一些标准,有相当数量的这种弱私钥。我们还使用我们的方法来表明,没有14个突出的Certicom挑战问题的情况下,在我们的意义上是弱的,达到一定的弱点。

英文摘要:

We describe a novel type of weak cryptographic private key that can exist in any discrete logarithm based public-key cryptosystem set in a group of prime order $p$ where $p-1$ has small divisors. Unlike the weak private keys based on \textit{numerical size} (such as smaller private keys, or private keys lying in an interval) that will \textit{always} exist in any DLP cryptosystems, our type of weak private keys occurs purely due to parameter choice of $p$, and hence, can be removed with appropriate value of $p$. Using the theory of implicit group representations, we present algorithms that can determine whether a key is weak, and if so, recover the private key from the corresponding public key. We analyze several elliptic curves proposed in the literature and in various standards, giving counts of the number of keys that can be broken with relatively small amounts of computation. Our results show that many of these curves, including some from standards, have a considerable number of such weak private keys. We also use our methods to show that none of the 14 outstanding Certicom Challenge problem instances are weak in our sense, up to a certain weakness bound.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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