A Universal Scheme for Transforming Binary Algorithms to Generate Random Bits from Loaded Dice

In this paper, we present a universal scheme for transforming an arbitrary algorithm for biased 2-face coins to generate random bits from the general source of an m-sided die, hence enabling the application of existing algorithms to general sources. In addition, we study approaches of efficiently generating a prescribed number of random bits from an arbitrary biased coin. This contrasts with most existing works, which typically assume that the number of coin tosses is fixed, and they generate a variable number of random bits.Comment: 2 columns, 10 page

Efficiently Generating Random Bits from Finite State Markov Chains

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

Efficiently Extracting Randomness from Imperfect Stochastic Processes

We study the problem of extracting a prescribed number of random bits by reading the smallest possible number of symbols from non-ideal stochastic processes. The related interval algorithm proposed by Han and Hoshi has asymptotically optimal performance; however, it assumes that the distribution of the input stochastic process is known. The motivation for our work is the fact that, in practice, sources of randomness have inherent correlations and are affected by measurement's noise. Namely, it is hard to obtain an accurate estimation of the distribution. This challenge was addressed by the concepts of seeded and seedless extractors that can handle general random sources with unknown distributions. However, known seeded and seedless extractors provide extraction efficiencies that are substantially smaller than Shannon's entropy limit. Our main contribution is the design of extractors that have a variable input-length and a fixed output length, are efficient in the consumption of symbols from the source, are capable of generating random bits from general stochastic processes and approach the information theoretic upper bound on efficiency.Comment: 2 columns, 16 page

Linear Transformations for Randomness Extraction

Information-efficient approaches for extracting randomness from imperfect sources have been extensively studied, but simpler and faster ones are required in the high-speed applications of random number generation. In this paper, we focus on linear constructions, namely, applying linear transformation for randomness extraction. We show that linear transformations based on sparse random matrices are asymptotically optimal to extract randomness from independent sources and bit-fixing sources, and they are efficient (may not be optimal) to extract randomness from hidden Markov sources. Further study demonstrates the flexibility of such constructions on source models as well as their excellent information-preserving capabilities. Since linear transformations based on sparse random matrices are computationally fast and can be easy to implement using hardware like FPGAs, they are very attractive in the high-speed applications. In addition, we explore explicit constructions of transformation matrices. We show that the generator matrices of primitive BCH codes are good choices, but linear transformations based on such matrices require more computational time due to their high densities.Comment: 2 columns, 14 page

Synthesis of Stochastic Flow Networks

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network, and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. Stochastic flow networks can be easily implemented by DNA-based chemical reactions, with promising applications in molecular computing and stochastic computing. In this paper, we address a fundamental synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability transformation dates back to von Neumann's 1951 work and was followed, among others, by Knuth and Yao in 1976. Most existing works have been focusing on the "simulation" of target distributions. In this paper, we design optimal-sized stochastic flow networks for "synthesizing" target distributions. It shows that when each splitter has two outgoing edges and is unbiased, an arbitrary rational probability \frac{a}{b} with a\leq b\leq 2^n can be realized by a stochastic flow network of size n that is optimal. Compared to the other stochastic systems, feedback (cycles in networks) strongly improves the expressibility of stochastic flow networks.Comment: 2 columns, 15 page