有限状态马尔可夫链随机比特的有效生成
Efficient Generation of Random Bits from Finite State Markov Chains
摘要：
从不相关的随机源（未知概率分布）生成随机数的问题可以追溯到冯·诺依曼1951年的工作。Elias（1972）推广了von Neumann的方案，并展示了如何在无偏随机比特生成中实现最佳效率。因此，一个自然的问题是，如果源是相关的呢？埃利亚斯和萨缪尔森都提出了在相关源（未知概率分布）的情况下生成无偏随机比特的方法，特别是，他们考虑了有限马尔可夫链。然而，他们提出的方法不是有效的，或有实施困难。Blum（1986）设计了一种在期望的线性时间内从二次有限马尔可夫链有效地生成随机比特的算法，然而，他的漂亮的方法在信息效率上仍然远未达到最优。在本文中，我们推广Blum的算法到任意程度的有限马尔可夫链，并结合Elias的方法，有效地产生无偏位。其结果是，我们提供了第一个已知的算法，从任意有限马尔可夫链产生无偏的随机位，在预期的线性时间内运行，并达到信息理论上的效率上限。
英文摘要：
The problem of random number generation from an uncorrelated random source (of unknown probability distribution) dates back to von Neumann's 1951 work. Elias (1972) generalized von Neumann's scheme and showed how to achieve optimal efficiency in unbiased random bits generation. Hence, a natural question is what if the sources are correlated? Both Elias and Samuelson proposed methods for generating unbiased random bits in the case of correlated sources (of unknown probability distribution), specifically, they considered finite Markov chains. However, their proposed methods are not efficient or have implementation difficulties. Blum (1986) devised an algorithm for efficiently generating random bits from degree-2 finite Markov chains in expected linear time, however, his beautiful method is still far from optimality on information-efficiency. In this paper, we generalize Blum's algorithm to arbitrary degree finite Markov chains and combine it with Elias's method for efficient generation of unbiased bits. As a result, we provide the first known algorithm that generates unbiased random bits from an arbitrary finite Markov chain, operates in expected linear time and achieves the information-theoretic upper bound on efficiency.
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