# Entropy Estimation of Physically Unclonable Functions via Chow Parameters

## TL;DR

This paper estimates the entropy of physically unclonable functions (PUFs) using Chow parameters, extending previous methods to larger systems and providing insights into their security properties.

## Contribution

It introduces a novel approach leveraging Chow parameters to estimate PUF entropy for larger systems, up to n=10, improving upon prior estimations.

## Key findings

- Shannon entropy of PUFs is close to the maximum entropy.
- Entropy increases quadratically with the number of delay elements.
- The method extends entropy estimation to larger PUFs beyond previous limits.

## Abstract

A physically unclonable function (PUF) is an electronic circuit that produces an intrinsic identifier in response to a challenge. These identifiers depend on uncontrollable variations of the manufacturing process, which make them hard to predict or to replicate. Various security protocols leverage on such intrinsic randomness for authentification, cryptographic key generation, anti-counterfeiting, etc. Evaluating the entropy of PUFs (for all possible challenges) allows one to assess the security properties of such protocols. In this paper, we estimate the probability distribution of certain kinds of PUFs composed of $n$ delay elements. This is used to evaluate relevant R\'enyi entropies and determine how they increase with $n$. Such a problem was known to have extremely high complexity (in the order of $2^{2^n}$) and previous entropy estimations were carried out up to $n=7$. Making the link with the theory of Boolean threshold functions, we leverage on the representation by Chow parameters to estimate probability distributions up to $n=10$. The resulting Shannon entropy of the PUF is close to the max-entropy, which is asymptotically quadratic in $n$.