Ring-LWE的可证弱实例
Provably weak instances of Ring-LWE
摘要：
环和多项式错误学习问题（Ring-LWE和Poly-LWE）已被提出作为硬问题，以形成密码系统的基础，并已提出各种安全性约简到硬格问题。到目前为止，这些问题已经说明了一般（数）环，但只有仔细审查分圆数环。在本文中，我们的状态和审查的Ring-LWE问题的一般数环，并证明可证明弱的情况下，Ring-LWE。我们构造了一个显式的数域族，我们有一个有效的攻击。我们在理论和实践中演示了攻击，提供了攻击的代码和运行时间。攻击在q中以时间线性运行，其中q是模数。我们的攻击是基于对Poly-LWE的攻击，该攻击在[Hellenträger-Hallgren-Lauter]中提出。我们扩展的EHL攻击适用于更大的一类数域，并显示它如何适用于攻击环LWE的一个大类的字段。某些Ring-LWE实例可以转换为Poly-LWE实例，而不会过多地扭曲错误，从而提供Ring-LWE问题的第一个弱实例。我们还提供了其他的例子，领域容易受到我们的攻击的聚-LWE，包括电源的$2$分圆领域，提出了使用最小多项式的$ζ_{2^n} \pm 1$。
英文摘要：
The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of Ring-LWE. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus. Our attack is based on the attack on Poly-LWE which was presented in [Eisenträger-Hallgren-Lauter]. We extend the EHL-attack to apply to a larger class of number fields, and show how it applies to attack Ring-LWE for a heuristically large class of fields. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the first weak instances of the Ring-LWE problem. We also provide additional examples of fields which are vulnerable to our attacks on Poly-LWE, including power-of-$2$ cyclotomic fields, presented using the minimal polynomial of $ζ_{2^n} \pm 1$.
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