Hiding Symbols and Functions: New Metrics and Constructions for Information-Theoretic Security

TL;DR

This paper introduces new information-theoretic metrics and constructions, such as symbol secrecy and list-source codes, to analyze and enhance the security of symmetric-key encryption schemes beyond perfect secrecy.

Contribution

It proposes a novel metric called symbol secrecy, develops explicit list-source code constructions, and connects these to rate-distortion theory for security analysis.

Findings

Maximum symbol secrecy achieved with MDS codes

Bounds on eavesdropper's success probability in function reconstruction

Extension of security analysis to functional encryption scenarios

Abstract

We present information-theoretic definitions and results for analyzing symmetric-key encryption schemes beyond the perfect secrecy regime, i.e. when perfect secrecy is not attained. We adopt two lines of analysis, one based on lossless source coding, and another akin to rate-distortion theory. We start by presenting a new information-theoretic metric for security, called symbol secrecy, and derive associated fundamental bounds. We then introduce list-source codes (LSCs), which are a general framework for mapping a key length (entropy) to a list size that an eavesdropper has to resolve in order to recover a secret message. We provide explicit constructions of LSCs, and demonstrate that, when the source is uniformly distributed, the highest level of symbol secrecy for a fixed key length can be achieved through a construction based on minimum-distance separable (MDS) codes. Using an analysis related to rate-distortion theory, we then show how symbol secrecy can be used to determine the probability that an eavesdropper correctly reconstructs functions of the original plaintext. We illustrate how these bounds can be applied to characterize security properties of symmetric-key encryption schemes, and, in particular, extend security claims based on symbol secrecy to a functional setting.