关于随机有理素数上的理想最短向量问题
On the ideal shortest vector problem over random rational primes
摘要：
数域中的任何理想都可以分解为素理想的乘积。本文研究了环$ \Z[x]/(x^{2^n} + 1) $上的素理想最短向量问题（SVP），这是理想格密码系统设计中的一个常用问题。我们发现，大多数的合理素数谎言下的素理想承认一个多项式时间算法的SVP。虽然理想格的最短向量问题是Ring-LWE密码系统安全性的基础，但本文的工作并没有破坏Ring-LWE，因为安全性的降低是从最坏情况的理想SVP到平均情况的Ring-LWE，而且是单向的。
英文摘要：
Any ideal in a number field can be factored into a product of prime ideals. In this paper we study the prime ideal shortest vector problem (SVP) in the ring $ \Z[x]/(x^{2^n} + 1) $, a popular choice in the design of ideal lattice based cryptosystems. We show that a majority of rational primes lie under prime ideals admitting a polynomial time algorithm for SVP. Although the shortest vector problem of ideal lattices underpins the security of Ring-LWE cryptosystem, this work does not break Ring-LWE, since the security reduction is from the worst case ideal SVP to the average case Ring-LWE, and it is one-way.
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