利用$p$ -adic遍历变换生成伪随机数
Pseudorandom number generation by $p$-adic ergodic transformations
摘要：
本文研究反相依伪随机发生器，后者是在工作时状态转移函数（和输出函数）动态变化的发生器：对于这样的发生器，状态递归序列满足同余式$x_{i+1}\equiv f_i(x_i)\pmod{2^n}$，而其输出序列的形式为$z_{i}=F_i(u_i)$ .本文介绍的技术和结构，使一个组成的发电机输出均匀分布的序列的最大周期长度和高线性和2-adic跨度。相应的流chipher是可证明的强大对已知明文攻击（一个似是而非的猜想）。状态转移函数和输出函数都可能是依赖于密钥的，所以密码分析师唯一可用的信息是这些函数属于某个（指数级大的）类。这些函数是标准机器指令（如加法、乘法、按位逻辑运算等）的组合。组合应满足相当宽松的条件，因此相应的生成器是足够灵活的，可以很容易地实现为计算机程序。
英文摘要：
The paper study counter-dependent pseudorandom generators; the latter are generators such that their state transition function (and output function) is being modified dynamically while working: For such a generator the recurrence sequence of states satisfies a congruence $x_{i+1}\equiv f_i(x_i)\pmod{2^n}$, while its output sequence is of the form $z_{i}=F_i(u_i)$. The paper introduces techniques and constructions that enable one to compose generators that output uniformly distributed sequences of a maximum period length and with high linear and 2-adic spans. The corresponding stream chipher is provably strong against a known plaintext attack (up to a plausible conjecture). Both state transition function and output function could be key-dependent, so the only information available to a cryptanalyst is that these functions belong to some (exponentially large) class. These functions are compositions of standard machine instructions (such as addition, multiplication, bitwise logical operations, etc.) The compositions should satisfy rather loose conditions; so the corresponding generators are flexible enough and could be easily implemented as computer programs.
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。
