Louis J. M. Aslett

Pedro M. Esperança, Louis J. M. Aslett, Chris C. Holmes

Information that is stored in an encrypted format is, by definition, usually not amenable to statistical analysis or machine learning methods. In this paper we present detailed analysis of coordinate and accelerated gradient descent algorithms which are capable of fitting least squares and penalised ridge regression models, using data encrypted under a fully homomorphic encryption scheme. Gradient descent is shown to dominate in terms of encrypted computational speed, and theoretical results are proven to give parameter bounds which ensure correctness of decryption. The characteristics of encrypted computation are empirically shown to favour a non-standard acceleration technique. This demonstrates the possibility of approximating conventional statistical regression methods using encrypted data without compromising privacy.

Louis J. M. Aslett

The precise design of a system may be considered a trade secret which should be protected, whilst at the same time component manufacturers are sometimes reluctant to release full test data (perhaps only providing mean time to failure data). In this situation it seems impractical to both produce an accurate reliability assessment and satisfy all parties' privacy requirements. However, we present recent developments in cryptography which, when combined with the recently developed survival signature in reliability theory, allows almost total privacy to be maintained in a cryptographically strong manner in precisely this setting. Thus, the system designer does not have to reveal their trade secret design and the component manufacturer can retain component test data in-house.

Louis J. M. Aslett, Pedro M. Esperança, Chris C. Holmes

We present two new statistical machine learning methods designed to learn on fully homomorphic encrypted (FHE) data. The introduction of FHE schemes following Gentry (2009) opens up the prospect of privacy preserving statistical machine learning analysis and modelling of encrypted data without compromising security constraints. We propose tailored algorithms for applying extremely random forests, involving a new cryptographic stochastic fraction estimator, and naïve Bayes, involving a semi-parametric model for the class decision boundary, and show how they can be used to learn and predict from encrypted data. We demonstrate that these techniques perform competitively on a variety of classification data sets and provide detailed information about the computational practicalities of these and other FHE methods.

Louis J. M. Aslett, Pedro M. Esperança, Chris C. Holmes

Recent advances in cryptography promise to enable secure statistical computation on encrypted data, whereby a limited set of operations can be carried out without the need to first decrypt. We review these homomorphic encryption schemes in a manner accessible to statisticians and machine learners, focusing on pertinent limitations inherent in the current state of the art. These limitations restrict the kind of statistics and machine learning algorithms which can be implemented and we review those which have been successfully applied in the literature. Finally, we document a high performance R package implementing a recent homomorphic scheme in a general framework.