Most homomorphic encryption (FHE) schemes exploit a technique called single-instruction multiple-data (SIMD) to process several messages in parallel. However, they base their security in somehow strong assumptions, such as the hardness of approximate lattice problems with superpolynomial approximation factor. On the other extreme of the spectrum, there are lightweight FHE schemes that have much faster bootstrapping but no SIMD capabilities. On the positive side, the security of these schemes is based on lattice problems with (low-degree) polynomial approximation factor only, which is a much weaker security assumption. Aiming the best of those two options, Micciancio and Sorrell (ICALP'18) proposed a new amortized bootstrapping that can process many messages at once, yielding sublinear time complexity per message, and allowing one to construct FHE based on lattice problems with polynomial approximation factor.

Machine learning as a service scenario typically requires the client to trust the server and provide sensitive data in plaintext. However, with the recent improvements in fully homomorphic encryption (FHE) schemes, many such applications can be designed in a privacy-preserving way. In this work, we focus on such a problem, private decision tree evaluation (PDTE) --- where a server has a decision tree classification model, and a client wants to use the model to classify her private data without revealing the data or the classification result to the server. We present an efficient non-interactive design of PDTE, that we call SortingHat, based on FHE techniques. As part of our design, we solve multiple cryptographic problems related to FHE: (1) we propose a fast homomorphic comparison function where one input can be in plaintext format; (2) we design an efficient binary decision tree evaluation technique in the FHE setting, which we call homomorphic traversal, and apply it together with our homomorphic comparison to evaluate private decision tree classifiers, obtaining running times orders of magnitude faster than the state of the art; (3) we improve both the communication cost and the time complexity of transciphering, by applying our homomorphic comparison to the FiLIP stream cipher. Through a prototype implementation, we demonstrate that our improved transciphering solution runs around 400 times faster than previous works. We finally present a choice in terms of PDTE design: we present a version of SortingHat without transciphering that achieves significant improvement in terms of computation cost compared to prior works, and another version t-SortingHat with transciphering that has a communication cost about 20 thousand times smaller but comparable running time.

We propose homomorphic algorithms for privacy-preserving applications where we are given an encrypted dataset and we want to compute the number of elements that share a common property. We consider a two party scenario between a client and a server, where the storage and computation is outsourced to the server. We present two new efficient methods to solve this problem by homomorphically evaluating a selection function encoding the desired property, and counting the number of elements which evaluates to the same value. Our first method programs the homomorphic computation in the style of the the functional bootstrapping of TFHE and can be instantiated with essentially any homomorphic encryption scheme that operates on polynomials, like FV or BGV. Our second method relies on new homomorphic operations and ciphertext formats, and it is more suitable for applications where the number of possible inputs is much larger than the number of possible values for the property. We illustrate the feasibility of our methods by presenting a publicly available proof-ofconcept implementation in C++ and using it to evaluate a heatmap function over encrypted geographic points.