300 research outputs found
The Dirac and Klein-Gordon equations and Greybody Radiation for the Regular Hayward Black Hole
We investigate the Dirac and Klein-Gordon equations, as well as greybody
radiation, for the Hayward black hole (BH) spacetime. We first consider the
Dirac equation using a null tetrad in the Newman- Penrose (NP) formalism. The
equations are then separated into angular and radial parts. A pair of
one-dimensional Schr\"odinger like wave equations with effective potentials is
obtained from the radial part. In order to examine the behavior of potentials,
they are plotted with respect to radial distances. Additionally, the
Klein-Gordon equation is considered in the Hayward BH spacetime. At the end, we
compute greybody factors for bosons and fermions and our results are shown
graphically and discussed
Demystifying Bootstrapping in Fully Homomorphic Encryption
Bootstrapping is a term used very often in the context of Fully Homomorphic Encryption (FHE). Anyone who is familiar with FHE knows that bootstrapping is the most sophisticated and compute-intensive component of an FHE scheme. However, very few non-FHE-experts understand what the bootstrapping operation really is and that there are various bootstrapping methods, each with its own tradeoffs. The goal of this paper is to provide a high-level introduction to common bootstrapping methods and evaluate their performance using the existing implementations in OpenFHE and HElib open-source libraries.
Our performance evaluation suggests that the bootstrapping in the Cheon-Kim-Kim-Song (CKKS) scheme provides highest throughput and efficiently achieves large precision for vectors of real numbers, which are often used in machine learning applications. The Ducas-Micciancio (DM) and Chillotti-Gama-Georgieva-Izabachene (CGGI) schemes achieve the smallest latency (typically for small integers or small-precision fixed-point numbers) and provide a general capability for evaluating arbitrary functions (programmable bootstrapping) via lookup tables. The Brakerski-Gentry-Vaikuntanathan (BGV) and Brakerski/Fan-Vercauteren (BFV) schemes provide higher bootstrapping throughput than DM/CGGI for vectors of small integers or finite-field elements but do not support programmable bootstrapping.
The target audience is anyone interested in FHE. We intend to keep this paper up-to-date to include new bootstrapping results as they become available
Application-Aware Approximate Homomorphic Encryption: Configuring FHE for Practical Use
Fully Homomorphic Encryption (FHE) is a powerful tool for performing privacy-preserving analytics over encrypted data. A promising method for FHE over real and complex numbers is approximate homomorphic encryption, instantiated with the Cheon-Kim-Kim-Song (CKKS) scheme. The CKKS scheme enables efficient evaluation for many privacy-preserving machine learning applications. Despite its high efficiency, there is currently a lot of confusion on how to securely instantiate CKKS for a given application, especially after secret-key recovery attacks were proposed by Li and Micciancio (EUROCRYPT\u2721) for the setting, i.e., where decryption results are shared with other parties. On the one hand, the generic definition of is application-agnostic and often requires impractically large parameters. On the other hand, practical CKKS implementations target specific applications and use tighter parameters. A good illustration are the recent secret-key recovery attacks against a CKKS implementation in the OpenFHE library by Guo et al. (USENIX Security\u2724). We show that these attacks misuse the library by employing different (incompatible) circuits during parameter estimation and run-time computation, yet they do not violate the generic (application-agnostic) definition.
To address this confusion, we introduce the notion of application-aware homomorphic encryption and devise related security definitions, which correspond more closely to how homomorphic encryption schemes are implemented and used in practice. We then formulate the guidelines for implementing the application-aware homomorphic encryption model to achieve security for practical applications of CKKS. We also show that our application-aware model can be used for secure, efficient instantiation of exact homomorphic encryption schemes
Multi-GPU design and performance evaluation of homomorphic encryption on GPU clusters
We present a multi-GPU design, implementation and performance evaluation of the Halevi-Polyakov-Shoup (HPS) variant of the Fan-Vercauteren (FV) levelled Fully Homomorphic Encryption (FHE) scheme. Our design follows a data parallelism approach and uses partitioning methods to distribute the workload in FV primitives evenly across available GPUs. The design is put to address space and runtime requirements of FHE computations. It is also suitable for distributed-memory architectures, and includes efficient GPU-to-GPU data exchange protocols. Moreover, it is user-friendly as user intervention is not required for task decomposition, scheduling or load balancing. We implement and evaluate the performance of our design on two homogeneous and heterogeneous NVIDIA GPU clusters: K80, and a customized P100. We also provide a comparison with a recent shared-memory-based multi-core CPU implementation using two homomorphic circuits as workloads: vector addition and multiplication. Moreover, we use our multi-GPU Levelled-FHE to implement the inference circuit of two Convolutional Neural Networks (CNNs) to perform homomorphically image classification on encrypted images from the MNIST and CIFAR - 10 datasets. Our implementation provides 1 to 3 orders of magnitude speedup compared with the CPU implementation on vector operations. In terms of scalability, our design shows reasonable scalability curves when the GPUs are fully connected.This work is supported by A*STAR under its RIE2020 Advanced Manufacturing and Engineering (AME) Programmtic Programme (Award A19E3b0099).Peer ReviewedPostprint (author's final draft
Towards the AlexNet Moment for Homomorphic Encryption: HCNN, theFirst Homomorphic CNN on Encrypted Data with GPUs
Deep Learning as a Service (DLaaS) stands as a promising solution for
cloud-based inference applications. In this setting, the cloud has a
pre-learned model whereas the user has samples on which she wants to run the
model. The biggest concern with DLaaS is user privacy if the input samples are
sensitive data. We provide here an efficient privacy-preserving system by
employing high-end technologies such as Fully Homomorphic Encryption (FHE),
Convolutional Neural Networks (CNNs) and Graphics Processing Units (GPUs). FHE,
with its widely-known feature of computing on encrypted data, empowers a wide
range of privacy-concerned applications. This comes at high cost as it requires
enormous computing power. In this paper, we show how to accelerate the
performance of running CNNs on encrypted data with GPUs. We evaluated two CNNs
to classify homomorphically the MNIST and CIFAR-10 datasets. Our solution
achieved a sufficient security level (> 80 bit) and reasonable classification
accuracy (99%) and (77.55%) for MNIST and CIFAR-10, respectively. In terms of
latency, we could classify an image in 5.16 seconds and 304.43 seconds for
MNIST and CIFAR-10, respectively. Our system can also classify a batch of
images (> 8,000) without extra overhead
Implementation and Performance Evaluation of RNS Variants of the BFV Homomorphic Encryption Scheme
Homomorphic encryption is an emerging form of encryption that provides the ability to compute on encrypted data without ever decrypting them. Potential applications include aggregating sensitive encrypted data on a cloud environment and computing on the data in the cloud without compromising data privacy. There have been several recent advances resulting in new homomorphic encryption schemes and optimized variants. We implement and evaluate the performance of two optimized variants, namely Bajard-Eynard-Hasan-Zucca (BEHZ) and Halevi-Polyakov-Shoup (HPS), of the most promising homomorphic encryption scheme in CPU and GPU. The most interesting (and also unexpected) result of our performance evaluation is that the HPS variant in practice scales significantly better (typically by 15%-30%) with increase in multiplicative depth of the computation circuit than BEHZ, implying that the HPS variant will always outperform BEHZ for most practical applications. For the multiplicative depth of 98, our fastest GPU implementation performs homomorphic multiplication in 51 ms for 128-bit security settings, which is faster by two orders of magnitude than prior results and already practical for cloud environments supporting GPU computations. Large multiplicative depths supported by our implementations are required for applications involving deep neural networks, logistic regression learning, and other important machine learning problems
TREBUCHET: Fully Homomorphic Encryption Accelerator for Deep Computation
Secure computation is of critical importance to not only the DoD, but across
financial institutions, healthcare, and anywhere personally identifiable
information (PII) is accessed. Traditional security techniques require data to
be decrypted before performing any computation. When processed on untrusted
systems the decrypted data is vulnerable to attacks to extract the sensitive
information. To address these vulnerabilities Fully Homomorphic Encryption
(FHE) keeps the data encrypted during computation and secures the results, even
in these untrusted environments. However, FHE requires a significant amount of
computation to perform equivalent unencrypted operations. To be useful, FHE
must significantly close the computation gap (within 10x) to make encrypted
processing practical. To accomplish this ambitious goal the TREBUCHET project
is leading research and development in FHE processing hardware to accelerate
deep computations on encrypted data, as part of the DARPA MTO Data Privacy for
Virtual Environments (DPRIVE) program. We accelerate the major secure
standardized FHE schemes (BGV, BFV, CKKS, FHEW, etc.) at >=128-bit security
while integrating with the open-source PALISADE and OpenFHE libraries currently
used in the DoD and in industry. We utilize a novel tile-based chip design with
highly parallel ALUs optimized for vectorized 128b modulo arithmetic. The
TREBUCHET coprocessor design provides a highly modular, flexible, and
extensible FHE accelerator for easy reconfiguration, deployment, integration
and application on other hardware form factors, such as System-on-Chip or
alternate chip areas.Comment: 6 pages, 5figures, 2 table
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