[论文解读] Applications of Derandomization Theory in Coding

Mahdi Cheraghchi|arXiv (Cornell University)|Jan 1, 2010

Cryptography and Data Security参考文献 156被引用 11

一句话总结

本文应用去随机化工具——特别是随机性提取器和凝聚器——来解决编码理论与组合群组检测中的基本问题。它为窃听信道、在噪声查询下的鲁棒群组检测，以及达到容量的纠错编码，构建了显式、信息论最优的协议，表明伪随机性技术能够实现高效、显式的构造，在标准计算假设下达到理论极限。

ABSTRACT

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. Towards this goal, numerous fundamental notions have been developed to provide a unified framework for approaching various derandomization problems and to improve our general understanding of the power of randomness in computation. Two important classes of such tools are pseudorandom generators and randomness extractors. Pseudorandom generators transform a short, purely random, sequence into a much longer sequence that looks random, while extractors transform a weak source of randomness into a perfectly random one (or one with much better qualities, in which case the transformation is called a randomness condenser). In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions. We utilize randomness extractors to construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Next, we use randomness condensers and extractors to design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. Finally, we consider the problem of explicit construction of error-correcting codes on the Gilbert-Varshamov bound and extend the original idea of Nisan and Wigderson to obtain a small ensemble of codes, mostly achieving the bound, under suitable computational hardness assumptions.

研究动机与目标

• 通过将伪随机性工具应用于通信与组合问题，弥合去随机化理论与编码理论之间的鸿沟，构建高效、显式的解决方案。
• 通过使用随机性提取器设计信息论最优、显式的通信协议，解决窃听信道问题。
• 通过使用随机性凝聚器，在不可靠查询结果和阈值模型下，设计鲁棒的群组检测方案。
• 构建能够达到一大类信道信息论容量的纠错码集合。
• 在Nisan-Wigderson风格的困难性假设下，提供满足Gilbert-Varshamov界的小型显式码构造。

提出的方法

• 利用随机性提取器将弱随机源转换为均匀随机字符串，以在窃听信道中实现安全通信。
• 利用随机性凝聚器处理群组检测中的噪声或不可靠查询结果，确保在噪声下仍能正确识别缺陷项目。
• 应用提取器与凝聚器生成码集合，使其达到无记忆信道的容量，从而实现显式达到容量的纠错码。
• 采用受Nisan和Wigderson启发的困难性构造，减小码集合的规模，同时保持性能接近Gilbert-Varshamov界。
• 设计显式、确定性的构造，避免依赖随机选择，确保实际可实现性。
• 结合伪随机性工具与编码理论模型，在查询复杂度、抗错能力与码率之间实现最优权衡。

实验结果

研究问题

• RQ1 能否使用随机性提取器构建显式、信息论最优的窃听信道通信协议？
• RQ2 如何应用随机性凝聚器设计在噪声或不可靠查询响应下的鲁棒群组检测方案？
• RQ3 能否使用提取器与凝聚器构建显式纠错码集合，使其达到一大类通信信道的容量？
• RQ4 在计算困难性假设下，能否以小型显式码集合实现Gilbert-Varshamov界？
• RQ5 在噪声或基于阈值的查询模型下，识别大小为n的群体中d个缺陷项目所需的最少查询次数是多少？

主要发现

• 本文利用随机性提取器构建了显式、信息论最优的窃听信道通信协议，实现了完美保密且速率损失最小。
• 首次提出显式群组检测方案，对高度不可靠的查询结果具有鲁棒性，利用随机性凝聚器确保在噪声下的正确性。
• 通过提取器与凝聚器构造码集合，实现了对一大类无记忆信道的容量达到，从而获得容量达到的纠错码。
• 在标准计算困难性假设下，构建了一个小型显式码集合，使其对大部分码实现Gilbert-Varshamov界。
• 构造过程显式且确定，避免了概率方法，在标准与阈值模型下均实现了最优或近似最优的查询复杂度。
• 结果表明，去随机化工具可系统性地应用于实现编码与组合搜索问题中的最优性能，统一了理论极限与实际构造。