# On the Commitment Capacity of Unfair Noisy Channels

## TL;DR

This paper investigates the maximum rate at which unfair noisy channels can be used for cryptographic commitments, providing a single-letter formula and extending prior results to more realistic adversarial scenarios.

## Contribution

It derives a single-letter characterization of the commitment capacity for unfair noisy channels, generalizing previous work on fair channels.

## Key findings

- Commitment capacity reduces to binary symmetric channel capacity in the fair case.
- Provides a single-letter formula for the commitment capacity of unfair noisy channels.
- Extends cryptographic resource analysis to channels with adversarial control.

## Abstract

Noisy channels are a valuable resource from a cryptographic point of view. They can be used for exchanging secret-keys as well as realizing other cryptographic primitives such as commitment and oblivious transfer. To be really useful, noisy channels have to be consider in the scenario where a cheating party has some degree of control over the channel characteristics. Damg\r{a}rd et al. (EUROCRYPT 1999) proposed a more realistic model where such level of control is permitted to an adversary, the so called unfair noisy channels, and proved that they can be used to obtain commitment and oblivious transfer protocols. Given that noisy channels are a precious resource for cryptographic purposes, one important question is determining the optimal rate in which they can be used. The commitment capacity has already been determined for the cases of discrete memoryless channels and Gaussian channels. In this work we address the problem of determining the commitment capacity of unfair noisy channels. We compute a single-letter characterization of the commitment capacity of unfair noisy channels. In the case where an adversary has no control over the channel (the fair case) our capacity reduces to the well-known capacity of a discrete memoryless binary symmetric channel.