# Quantifying information flow in interactive systems

## TL;DR

This paper analyzes how information flows in deterministic interactive systems modeled as finite-state transducers, showing that the flow is either logarithmic or linear and providing algorithms to distinguish these cases efficiently.

## Contribution

It extends the theory of information leakage to interactive systems and links the flow rate to the growth rate of antichains in regular languages, with efficient algorithms for deterministic cases.

## Key findings

- Information flow in deterministic systems is either logarithmic or linear.
- A polynomial-time algorithm can distinguish the two flow types and compute the flow rate.
- Flow rate corresponds to the growth rate of antichains in a regular language.

## Abstract

We consider the problem of quantifying information flow in interactive systems, modelled as finite-state transducers in the style of Goguen and Meseguer. Our main result is that if the system is deterministic then the information flow is either logarithmic or linear, and there is a polynomial-time algorithm to distinguish the two cases and compute the rate of logarithmic flow. To achieve this we first extend the theory of information leakage through channels to the case of interactive systems, and establish a number of results which greatly simplify computation. We then show that for deterministic systems the information flow corresponds to the growth rate of antichains inside a certain regular language, a property called the width of the language. In a companion work we have shown that there is a dichotomy between polynomial and exponential antichain growth, and a polynomial time algorithm to distinguish the two cases and to compute the order of polynomial growth. We observe that these two cases correspond to logarithmic and linear information flow respectively. Finally, we formulate several attractive open problems, covering the cases of probabilistic systems, systems with more than two users and nondeterministic systems where the nondeterminism is assumed to be innocent rather than demonic.