  During the COVID-19 (SARS-CoV-2) epidemic, Contact Tracing emerged as an
essential tool for managing the epidemic. App-based solutions have emerged for
Contact Tracing, including a protocol designed by Apple and Google (influenced
by an open-source protocol known as DP3T). This protocol contains two
well-documented de-anonymisation attacks. Firstly that when someone is marked
as having tested positive and their keys are made public, they can be tracked
over a large geographic area for 24 hours at a time. Secondly, whilst the app
requires a minimum exposure duration to register a contact, there is no
cryptographic guarantee for this property. This means an adversary can scan
Bluetooth networks and retrospectively find who is infected. We propose a novel
"cross hashing" approach to cryptographically guarantee minimum exposure
durations. We further mitigate the 24-hour data exposure of infected
individuals and reduce computational time for identifying if a user has been
exposed using $k$-Anonymous buckets of hashes and Private Set Intersection. We
empirically demonstrate that this modified protocol can offer like-for-like
efficacy to the existing protocol.
