Distributed Computing with Imperfect Randomness

Abstract

Randomness is a critical resource in many computational scenarios, enabling solutions where deterministic ones are elusive or even provably impossible. However, randomized solutions typically assume access to a source of unbiased, independent coins. Physical sources of randomness, on the other hand, are rarely unbiased and independent although they do seem to exhibit somewhat imperfect randomness. This gap in modeling questions the relevance of current randomized solutions to computational tasks. Indeed, there has been substantial investigation of this issue in complexity theory in the context of the applications to efficient algorithms and cryptography.In this paper, we seek to determine whether imperfect randomness, modeled appropriately, is "good enough" for distributed algorithms. Namely, can we do with imperfect randomness all that we can do with perfect randomness, and with comparable efficiency?We examine two natural models of imperfect randomness in a distributed setting -namely, where each player has an (independent) block source (resp. weak source). For the first model (that of block sources), we construct a general-purpose compiler that converts distributed algorithms that assume perfect randomness, to ones that work even with imperfect coins. The compiled protocol has the same fault-tolerance and round-complexity as the original protocol. For the second model (that of weak sources), we construct Byzantine Agreement protocols in a variety of scenarios (synchronous or asynchronous networks, with or without private channels). Our protocols have comparable fault-tolerance and round complexity as the best known Byzantine Agreement protocols that expect perfect randomness.

Fast asynchronous Byzantine agreement and leader election with full information

“…Also, protocols which rely on cryptographic primitives may require sources of perfect randomness [13], whereas in the full information model, lower entropy sources of randomness have been shown to suffice. [17].…”

New independent source extractors with exponential improvement

“…Historically, Goldwasser, Sudan, and Vaikuntanathan [11] were the first to consider this problem. They showed that it is possible to run distributed computing applications (e.g., Byzantine agreement) with imperfect randomness.…”

Network Extractor Protocols

“…Instead, the proper tool for distributed computing is a network extractor protocol, which was introduced (but left unnamed) by Dodis and Oliveira [10] in the cryptographic context and by Goldwasser et al [16] in the context of Byzantine agreement. The idea is that each processor starts with a weak source, and these weak sources are independent.…”

“…For protocols using weak sources, the only results are due to Goldwasser et al [16]. They require all weak sources to have min-entropy rate at least 1/2.…”

“…Goldwasser, Sudan, and Vaikuntanathan [16] addressed the natural question of whether weak sources can be used in distributed computing. They showed how to solve Byzantine agreement even if each processor has a weak source, and these weak sources are independent.…”