Machine learning algorithms based on deep neural networks have achieved remarkable results and are being extensively used in different domains. However, the machine learning algorithms requires access to raw data which is often privacy sensitive. To address this issue, we develop new techniques to provide solutions for running deep neural networks over encrypted data. In this paper, we develop new techniques to adopt deep neural networks within the practical limitation of current homomorphic encryption schemes. More specifically, we focus on classification of the well-known convolutional neural networks (CNN). First, we design methods for approximation of the activation functions commonly used in CNNs (i.e. ReLU, Sigmoid, and Tanh) with low degree polynomials which is essential for efficient homomorphic encryption schemes. Then, we train convolutional neural networks with the approximation polynomials instead of original activation functions and analyze the performance of the models. Finally, we implement convolutional neural networks over encrypted data and measure performance of the models. Our experimental results validate the soundness of our approach with several convolutional neural networks with varying number of layers and structures. When applied to the MNIST optical character recognition tasks, our approach achieves 99.52% accuracy which significantly outperforms the state-of-the-art solutions and is very close to the accuracy of the best non-private version, 99.77%. Also, it can make close to 164000 predictions per hour. We also applied our approach to CIFAR-10, which is much more complex compared to MNIST, and were able to achieve 91.5% accuracy with approximation polynomials used as activation functions. These results show that CryptoDL provides efficient, accurate and scalable privacy-preserving predictions. *

Machine learning algorithms based on deep Neural Networks (NN) have achieved remarkable results and are being extensively used in different domains. On the other hand, with increasing growth of cloud services, several Machine Learning as a Service (MLaaS) are offered where training and deploying machine learning models are performed on cloud providers’ infrastructure. However, machine learning algorithms require access to the raw data which is often privacy sensitive and can create potential security and privacy risks. To address this issue, we present CryptoDL, a framework that develops new techniques to provide solutions for applying deep neural network algorithms to encrypted data. In this paper, we provide the theoretical foundation for implementing deep neural network algorithms in encrypted domain and develop techniques to adopt neural networks within practical limitations of current homomorphic encryption schemes. We show that it is feasible and practical to train neural networks using encrypted data and to make encrypted predictions, and also return the predictions in an encrypted form. We demonstrate applicability of the proposed CryptoDL using a large number of datasets and evaluate its performance. The empirical results show that it provides accurate privacy-preserving training and classification.

Abstract. We make use of a result of Hurwitz and Reznick [8] [19], and a consequence of this result due to Fidalgo and Kovacec [5], to determine a new sufficient condition for a polynomial f ∈ R[X 1 , . . . , X n ] of even degree to be a sum of squares. This result generalizes a result of Lasserre in [10] and a result of Fidalgo and Kovacec in [5], and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound f gp for f , and we explain how fgp can be computed using geometric programming. The lower bound f gp is generally not as good as the lower bound f sos introduced by Lasserre [11] and Parrilo and Sturmfels [15], which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of f gp is much faster. The computation is simplest when the highest degree term of f has the form. . , n. The lower bounds for f established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points.

The multivariate moment problem is investigated in the general context of the polynomial algebra R[

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