利用仿射随机过程生成（Z/pZ）^d中的随机向量
Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process
摘要：
本文考虑一类随机过程X_{n+1} =TX_n+B_n（mod p），其中B_n和X_n是（Z/pZ）^d上的随机变量，T是固定的d × d整数矩阵，且在复数上可逆.对于B_n的一个特殊分布，本文改进了Asci的结果，证明了如果T没有长度为1的复特征值，则对于与det（T）互素的整数p，阶（log p）^2步足以使X_n接近均匀分布，其中X_0是零向量。本文还证明了，如果T有一个复特征值是单位单位根，则X_n需要p^b阶步才能接近一致，其中b是一个可能依赖于T的值，X_0是零向量。
英文摘要：
This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.
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。
