Symmetric-Key Corruption Detection: When XOR-MACs Meet Combinatorial Group Testing

We consider two combinatorial problems. The first we call ``search with wildcards'': given an unknown $n$-bit string $x$, and the ability to check whether any subset of the bits of $x$ is equal to a provided query string, the goal is to output $x$. We give a nearly optimal $O(\sqrt{n} \log n)$ quantum query algorithm for search with wildcards, beating the classical lower bound of $\Omega(n)$ queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most $k$ special items out of a set of $n$ items, given the ability to make queries of the form ``does the set $S$ contain any special items?''\ for any subset $S$ of the $n$ items. We give a simple quantum algorithm which uses $O(k)$ queries to solve this problem, as compared with the classical lower bound of $\Omega(k \log(n/k))$ queries.

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A Non-Adaptive Combinatorial Group Testing Strategy to Facilitate Healthcare Worker Screening During the Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) Outbreak

10.1101/2020.07.21.20157677 ◽

2020 ◽

Author(s):

John Henry McDermott ◽

Duncan Stoddard ◽

Peter Woolf ◽

Jamie M Ellingford ◽

David Gokhale ◽

...

Keyword(s):

Web Application ◽

Group Testing ◽

Return Of Results ◽

Testing Strategy ◽

Healthcare Facilities ◽

Staff Members ◽

High Prevalence ◽

Combinatorial Group Testing ◽

Low Prevalence ◽

Combinatorial Group

Background: Regular SARS-CoV-2 testing of healthcare workers (HCWs) has been proposed to prevent healthcare facilities becoming persistent reservoirs of infectivity. Using monoplex testing, widespread screening would be prohibitively expensive, and throughput may not meet demand. We propose a non-adaptive combinatorial (NAC) group-testing strategy to increase throughput and facilitate rapid turnaround via a single round of testing. Methods: NAC matrices were constructed for sample sizes of 700, 350 and 250 with replicates of 2, 4 and 5, respectively. Matrix performance was tested by simulation under different SARS-CoV-2 prevalence scenarios of 0.1-10%, with each simulation ran for 10,000 iterations. Outcomes included the proportions of re-tests required and the proportion of true negatives identified. NAC matrices were compared to Dorfman Sequential (DS) approaches. A web application (www.samplepooling.com) was designed to decode results. Findings: NAC matrices performed well at low prevalence levels with an average number of 585 tests saved per assay in the n=700 matrix at a 1% prevalence. As prevalence increased, matrix performance deteriorated with n=250 most tolerant. In simulations of low to medium (0.1%-3%) prevalence levels all NAC matrices were superior, as measured by fewer repeated tests required, to the DS approaches. At very high prevalence levels (10%) the DS matrix was marginally superior, however both group testing approaches performed poorly at high prevalence levels. Interpretation: This testing strategy maximises the proportion of samples resolved after a single round of testing, allowing prompt return of results to staff members. Using the methodology described here, laboratories can adapt their testing scheme based on required throughput and the current population prevalence, facilitating a data-driven testing strategy.

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