One of the tasks in the 2017 iDASH secure genome analysis competition was to enable training of logistic regression models over encrypted genomic data. More precisely, given a list of approximately 1500 patient records, each with 18 binary features containing information on specific mutations, the idea was for the data holder to encrypt the records using homomorphic encryption, and send them to an untrusted cloud for storage. The cloud could then homomorphically apply a training algorithm on the encrypted data to obtain an encrypted logistic regression model, which can be sent to the data holder for decryption. In this way, the data holder could successfully outsource the training process without revealing either her sensitive data, or the trained model, to the cloud. Our solution to this problem has several novelties: we use a multi-bit plaintext space in fully homomorphic encryption together with fixed point number encoding; we combine bootstrapping in fully homomorphic encryption with a scaling operation in fixed point arithmetic; we use a minimax polynomial approximation to the sigmoid function and the 1-bit gradient descent method to reduce the plaintext growth in the training process. Our algorithm for training over encrypted data takes 0.4–3.2 hours per iteration of gradient descent. We demonstrate the feasibility but high computational cost of training over encrypted data. On the other hand, our method can guarantee the highest level of data privacy in critical applications.

In this paper, we present dynamic systems over encrypted data that compute the next state and the output using homomorphic properties of the cryptosystem, which has equivalent performance to the linear dynamic controllers over real-valued signals. Assuming that the input as well as the output of the plant is encrypted and transmitted to the system, the state matrix of the system is designed to consist of integers. This allows the proposed dynamic system to operate for infinite time horizon, without decryption or reset of the state. For implementation in practice, the use of cryptosystems based on Learning With Errors problem is considered, which allows both multiplication and addition over encrypted data. The effect of injecting errors during encryption is in turn controlled under closed-loop stability.

Homomorphic encryption (HE), which enables computation on ciphertexts without any leakage, rise as a most promising solution for privacy-preserving data processing including secure machine learning and secure out-sourcing computation. Despite the extensive applicability of HE, the current constructions are sometimes considered as impractical due to its inefficiency. In this work, we propose improvements on the linear transformation in bootstrapping, a technique allowing the infinite number of operation for HE, and homomorphic discrete Fourier transformation (DFT) using batch homomorphic encryption. We observe that the multiplication of a sparse diagonal matrix and a ciphertext of a vector can be done within O(1) homomorphic computations. This observation induces the faster algorithm for linear transformation in bootstrapping and homomorphic DFT. To achieve this, we use Cooley-Tukey matrix factorization and construct a new recursive factorization of the linear transformation in bootstrapping. Our method with radix r only requires O(r logr n) constant vector multiplication and O( √r logr n) rotations by consuming O(logr n) depth when the input vector size is n. The previous method used in HEAAN library, a library that implements a homomorphic encryption for approximate computation, requires O(n) and O(√ n) respectively. To show the performance improvement, we implement our method on top of HEAAN library. Our implementation, along with further few techniques, of these algorithms show the significant improvements compared to the previous algorithm. New homomorphic DFT with length 214 only takes about 8 seconds which is about 150 times faster result than the previous method. Furthermore, the bootstrapping takes about 2 minutes for C32768 plaintext space with 8 bit precision, which takes 26 hours with same bit precision using the previous method.