On Bounding Problems of Quantitative Information Flow

TL;DR

This paper studies the computational hardness of precisely bounding the quantitative information flow in programs, showing it is highly complex and not a k-safety property, with implications for security analysis techniques.

Contribution

It proves that the bounding problem for quantitative information flow is not a k-safety property and establishes its PP-hardness for loop-free boolean programs, highlighting computational challenges.

Findings

Bounding problem is not a k-safety property.

PP-hardness for loop-free boolean programs.

Complexity gap with non-interference.

Abstract

Researchers have proposed formal definitions of quantitative information flow based on information theoretic notions such as the Shannon entropy, the min entropy, the guessing entropy, belief, and channel capacity. This paper investigates the hardness of precisely checking the quantitative information flow of a program according to such definitions. More precisely, we study the "bounding problem" of quantitative information flow, defined as follows: Given a program M and a positive real number q, decide if the quantitative information flow of M is less than or equal to q. We prove that the bounding problem is not a k-safety property for any k (even when q is fixed, for the Shannon-entropy-based definition with the uniform distribution), and therefore is not amenable to the self-composition technique that has been successfully applied to checking non-interference. We also prove complexity theoretic hardness results for the case when the program is restricted to loop-free boolean programs. Specifically, we show that the problem is PP-hard for all definitions, showing a gap with non-interference which is coNP-complete for the same class of programs. The paper also compares the results with the recently proved results on the comparison problems of quantitative information flow.