Invertible Extractors and Wiretap Protocols

TL;DR

This paper introduces invertible extractors to construct efficient wiretap protocols, achieving optimal trade-offs between rate and resilience, and applies these to complex communication scenarios including active intruders and network coding.

Contribution

It develops invertible extractors for various sources and uses them to create wiretap protocols with optimal rate-resilience trade-offs, also providing new explicit extractors for affine sources.

Findings

Constructed invertible extractors for symbol-fixing, affine, and general sources.

Designed wiretap protocols with asymptotically optimal rate and resilience.

Produced new explicit extractors for affine sources over large fields.

Abstract

A wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message, while knowledge of the entire encoding reveals the message using the decoder. In this paper we study the notion of efficiently invertible extractors and show that a wiretap protocol can be constructed from such an extractor. We will then construct invertible extractors for symbol-fixing, affine, and general sources and apply them to create wiretap protocols with asymptotically optimal trade-offs between their rate (ratio of the length of the message versus its encoding) and resilience (ratio of the observed positions of the encoding and the length of the encoding). We will then apply our results to create wiretap protocols for challenging communication problems, such as active intruders who change portions of the encoding, network coding, and intruders observing arbitrary boolean functions of the encoding. As a by-product of our constructions we obtain new explicit extractors for a restricted family of affine sources over large fields (that in particular generalizes the notion of symbol-fixing sources) which is of independent interest. These extractors are able to extract the entire source entropy with zero error. Keywords: Wiretap Channel, Extractors, Network Coding, Active Intrusion, Exposure Resilient Cryptography.