On the Response Entropy of APUFs

• Main Authors: Dumoulin, Vincent, Rao, Wenjing, Devroye, Natasha
• Format: Journal Article
• Language: English
• Published: 28-06-2024
• Subjects: Computer Science - Cryptography and Security
• DOI: 10.48550/arxiv.2406.19975

A Physically Unclonable Function (PUF) is a hardware security primitive used
for authentication and key generation. It takes an input bit-vector challenge
and produces a single-bit response, resulting in a challenge-response pair
(CRP). The truth table of all challenge-response pairs of each manufactured PUF
should look different due to inherent manufacturing randomness, forming a
digital fingerprint. A PUF's entropy (the entropy of all the responses, taken
over the manufacturing randomness and uniformly selected challenges) has been
studied before and is a challenging problem. Here we explore a related notion
-- the response entropy, which is the entropy of an arbitrary response given
knowledge of one (and two) other responses. This allows us to explore how
knowledge of some CRP(s) impacts the ability to guess another response. The
Arbiter PUF (APUF) is a well-known PUF architecture based on accumulated delay
differences between two paths. In this paper, we obtain in closed form the
probability mass function of any arbitrary response given knowledge of one or
two other arbitrary CRPs for the APUF architecture. This allows us to obtain
the conditional response entropy and then to define and obtain the size of the
entropy bins (challenge sets with the same conditional response entropy) given
knowledge of one or two CRPs. All of these results depend on the probability
that two different challenge vectors yield the same response, termed the
response similarity of those challenges. We obtain an explicit closed form
expression for this. This probability depends on the statistical correlations
induced by the PUF architecture together with the specific known and
to-be-guessed challenges. As a by-product, we also obtain the optimal
(minimizing probability of error) predictor of an unknown challenge given
access to one (or two) challenges and the associated predictability.