Abstract
A number of databases around the world currently host a wealth of genomic data that is invaluable to researchers conducting a variety of genomic studies. However, patients who volunteer their genomic data run the risk of privacy invasion. In this work, we give a cryptographic solution to this problem: to maintain patient privacy, we propose encrypting all genomic data in the database. To allow meaningful computation on the encrypted data, we propose using a homomorphic encryption scheme.
Specifically, we take basic genomic algorithms which are commonly used in genetic association studies and show how they can be made to work on encrypted genotype and phenotype data. In particular, we consider the Pearson Goodness-of-Fit test, the \(D'\) and \(r^2\)-measures of linkage disequilibrium, the Estimation Maximization (EM) algorithm for haplotyping, and the Cochran-Armitage Test for Trend. We also provide performance numbers for running these algorithms on encrypted data.
Adriana López-Alt—Research conducted while visiting Microsoft Research.
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Notes
- 1.
The running time is linear in the population size for a fixed parameter set. For larger population sizes, parameters need to be increased and performance degrades, but not by a large factor (see Table 1 for a comparison of the running times for two typical parameter sets).
- 2.
1 degree of freedom = 3 genotypes \(-\) 2 alleles.
- 3.
Common choices for the set of weights \({{\varvec{w}}} = (w_0, w_1, w_2)\) are: \({{\varvec{w}}} = (0,1,2)\) for the additive (co-dominant) model, \({{\varvec{w}}} = (0,1,1)\) for the dominant model (\(A\) is dominant over \(a\)), and \({{\varvec{w}}} = (0,0,1)\) for the recessive model (\(A\) is recessive to allele \(a\)).
- 4.
For a bi-allelic gene with alleles \(A\) and \(a\), the value 0 corresponds to the genotype \(AA\), the value 1 corresponds to the genotype \(Aa\) and the value 2 corresponds to the genotype \(aa\).
- 5.
An arithmetic circuit over \(\mathbb {F}_t\) has addition and multiplication gates modulo \(t\).
- 6.
- 7.
The only modification we make to the scheme of López-Alt and Naehrig is removing a step called “relinearization” or “key switching”, needed to make decryption independent of the function that was homomorphically evaluated. In our implementation, decryption depends on the number of homomorphic multiplications that were performed. We make this change for efficiency reasons, as relinearization is very costly.
- 8.
Informally, a function has degree \(D\) if it can be represented as a (possibly multivariate) polynomial of degree \(D\). See Sect. 4.4 for more details.
- 9.
Recall from Sect. 3 that we cannot perform homomorphic divisions.
- 10.
Admittedly, the size of the parameters needed does depend on the magnitude of the genotype and phenotype counts, which can be as large as the size of the population sample. This is because the size of the message encrypted at any given time (i.e. the size of the counts and all the intermediate values in the computation) cannot grow too large relative to the modulus \(q\). Therefore, larger population sizes (and therefore larger counts) require a larger modulus \(q\), which in turn requires a larger dimension \(n\) for security. However, for a fixed parameter set, it is possible to compute an upper bound on the size of the population sample and the homomorphic computations detailed in this work do work correctly for any population sample with size smaller than the given bound.
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Acknowledgments
We thank Tancrède Lepoint for suggesting the encoding in Sect. 4.1.
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Lauter, K., López-Alt, A., Naehrig, M. (2015). Private Computation on Encrypted Genomic Data. In: Aranha, D., Menezes, A. (eds) Progress in Cryptology - LATINCRYPT 2014. LATINCRYPT 2014. Lecture Notes in Computer Science(), vol 8895. Springer, Cham. https://doi.org/10.1007/978-3-319-16295-9_1
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