标题： 一个快速乘法算法以及最大实子域的$2^r p^s$-次分圆域的 RLWE-PLWE 等价性 显示英文标题

标题： A Fast Multiplication Algorithm and RLWE-PLWE Equivalence for the Maximal Real Subfield of the $2^r p^s$-th Cyclotomic Field

摘要： 本文证明了在导数为$n = 2^r p^s$的分圆域的最大实子域中 RLWE-PLWE 等价性，其中$p$是一个奇素数，而$r \geq 0$和$s \geq 1$是整数。 特别地，我们证明了作为线性变换的规范嵌入的条件数被一个关于$n$的多项式所界。 此外，我们描述了这些实子域的整数环中的快速乘法算法。 该乘法算法使用快速离散余弦变换（DCT），其计算复杂度为$\mathcal{O}(n \log n)$。 RLWE-PLWE 等价性的证明和快速乘法算法都是 Ahola 等人先前结果的推广，他们在那篇论文中对单个素数$p = 3$证明了相同的结论。 显示英文摘要

摘要： This paper proves the RLWE-PLWE equivalence for the maximal real subfields of the cyclotomic fields with conductor $n = 2^r p^s$, where $p$ is an odd prime, and $r \geq 0$ and $s \geq 1$ are integers. In particular, we show that the canonical embedding as a linear transform has a condition number bounded above by a polynomial in $n$. In addition, we describe a fast multiplication algorithm in the ring of integers of these real subfields. The multiplication algorithm uses the fast Discrete Cosine Transform (DCT) and has computational complexity $\mathcal{O}(n \log n)$. Both the proof of the RLWE-PLWE equivalence and the fast multiplication algorithm are generalizations of previous results by Ahola et al., where the same claims are proved for a single prime $p = 3$.