关于小素数生成乘法群

中文摘要 AI 辅助

受Regev在其改进的量子因式分解算法中提出的问题启发，我们研究当每个素数只能以指数0或1使用时，需要多少个小素数才能生成群$({\mathbb Z}/q{\mathbb Z})^\times$。我们证明，对于每个固定的$\varepsilon>0$和$A>0$，存在一个绝对常数$C_*$和一个至多包含$(\log Q)^{1+\varepsilon}$个素数的集合，这些素数均不超过$(\log Q)^{C_*(A+1)}$，使得对于所有除了$O(Q(\log Q)^{-A})$个（隐含常数仅依赖于$\varepsilon$和$A$）满足$q\leq Q$的整数$q$，$({\mathbb Z}/q{\mathbb Z})^\times$中的每个元素都是这些素数模$q$的某个子集的乘积。素数个数中的指数$1+\varepsilon$在指数中任意$\varepsilon$的意义下是最优的。

英文摘要

Motivated by a question of Regev arising from his improved quantum factoring algorithm, we study how many small primes are needed to generate the group $({\mathbb Z}/q{\mathbb Z})^\times$ when each prime may be used with exponent only $0$ or $1$. We prove that, for every fixed $\varepsilon>0$ and $A>0$, there is an absolute constant $C_*$ and a set of at most $(\log Q)^{1+\varepsilon}$ primes, all at most $(\log Q)^{C_*(A+1)}$, such that for all but $O(Q(\log Q)^{-A})$ (with the implied constant depending only on $\varepsilon$ and $A$) integers $q\leq Q$, every element of $({\mathbb Z}/q{\mathbb Z})^\times$ is a product of a subset of these primes modulo $q$. The exponent $1+\varepsilon$ in the number of primes is best possible up to the arbitrary $\varepsilon$ in the exponent.

发表机构

• University of Bristol（布里斯托大学）
• University of New South Wales（新南威尔士大学）
• University of Cambridge（剑桥大学）

机构由 AI 辅助整理，请以论文原文为准。