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Privacy and security have increasingly become a concern for computing se Modern deep learning applications urge to push the model inference takin Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

The next step in the machine learning revolution would be Deep Learning as a Service DLaaS which seeks to take advantage of the benefits that cloud computing brings.

Cloud servers are excellent machine learning platforms, offering cheap data storage, near-zero deployment cost and high computational services.

However, it is not all-powerful and there are important questions that need to be resolved before DLaaS can become widespread.

One of the main questions is that cloud platforms do not guarantee data privacy. In the DLaaS setting, one uploads their data into the cloud, runs the model on it and gets the results back from the cloud.

At every step along the way, there are numerous opportunities for hackers and other malicious actors to compromise the data.

Privacy-preserving machine learning was considered previously by Graepel et al. Following them, Dowlin et al.

Since then, others. We follow the framework put forward in CryptoNets. Although the framework is available, there are still challenges to realizing performant HCNNs.

Informally, it works as follows. Encryption masks the input data, called a plaintext, by a random error sampled from some distribution, resulting in a ciphertext that reveals nothing about what it encrypts.

Decryption uses the secret key to filter out the noise and retrieve the plaintext as long as the noise is within some threshold.

Note that during computation, the noise in ciphertexts grows, but in a controlled manner. At some point, it grows to a point where no further computation can be done without resulting in decryption failure.

Bootstrapping can be used to refresh a ciphertext with large noise into one with less noise that can be used for computation.

By doing this indefinitely, theoretically, any function can be computed. However, this approach is still impractical and bootstrapping is not used in most cases.

Instead, the class of functions that can be evaluated is restricted to depth L arithmetic circuits, yielding a levelled FHE scheme to avoid bootstrapping.

For performance, L should be minimized which means that we have to carefully design HCNNs with this in mind.

Furthermore, the model of computation in FHE, arithmetic circuits with addition HAdd and multiplication HMult gates, is not compatible with non-polynomial functions such as sigmoid, R e L U and m a x.

This means that we should use polynomial approximations to the activation functions where possible and consider if pooling layers are useful in practice.

Besides that, we have to encode decimals in a form that is compatible with FHE plaintext data, which are usually integers.

These can have high precision which mean that they will require integers of large bit-size to represent them in the commonly used scalar encoding.

The main drawback of this encoding is that we cannot re-scale encoded data mid-computation; therefore, successive homomorphic operations will cause data size to increase rapidly.

Managing this scaling expansion is a necessary step towards scaling HCNNs to larger datasets and deeper neural networks.

Our Contributions. We provide a rich set of optimization techniques to enable easy designs of HCNN and reduce the overall computational overhead. These include low-precision training, optimized choice of HE scheme and parameters, and a GPU-accelerated implementation.

Related Work. Dowlin et al. They proposed using polynomial approximations of the most widespread R e L U activation function and using pooling layers only during the training phase to reduce the circuit depth of their neural network.

This makes it very difficult to scale to deeper networks since intermediate layers in those networks will quickly reach several hundred bits with their settings.

Following them, Bourse et al. Each neuron computes a weighted sum of its inputs and the activation function is the sign function,. Some of the main limitations of pure FHE-based is the need to approximate non-polynomial activation functions and high computation time.

They take commonly used protocols in deep learning and transform them into oblivious protocols. With MPC, they could evaluate neural networks without changing the training phase, preserving accuracy since there is no approximation needed for activation functions.

However, MPC comes with its own set of drawbacks. In this setting, each computation requires communication between the data owner and model owner, thus resulting in high bandwidth usage.

In a similar vein, Juvekar et al. Instead of applying levelled FHE, they alternate between an additive homomorphic encryption scheme for convolution-type layers and garbled circuits for activation and pooling layers.

This way, communication complexity is reduced compared to MiniONN but unfortunately is still significant. Organization of the Paper.

In this section, we review a set of notions that are required to understand the paper. Next, we introduce neural networks and how to tweak them to become compatible with FHE computation model.

First proposed by Rivest et al. FHE would support operations on ciphertexts that translate to functions on the encrypted messages within.

The blueprint of this construction remains the only method to design FHE schemes. The modernized blueprint is a simple two-step process. First, a somewhat homomorphic encryption scheme that can evaluate its decryption function is designed.

Then, we perform bootstrapping, which decrypts a ciphertext using an encrypted copy of the secret key. Note that the decryption function here is evaluated homomorphically, i.

As bootstrapping imposes high computation costs, we adopt a levelled FHE scheme instead, which can evaluate functions up to a pre-determined multiplicative depth without bootstrapping.

KeyGen is the algorithm that generates the keys used in an FHE scheme given the parameters chosen. Encrypt and Decrypt are the encyption and decryption algorithms respectively.

The differentiation between FHE and standard public-key encryption schemes is the operations on ciphertexts; which we call HAdd and HMult.

HAdd outputs a ciphertext that decrypts to the sum of the two input encrypted messages while HMult outputs one that decrypts to the product of the two encrypted inputs.

Tensor compute. Scale and Relinearize output. Correctness of the Scheme. We characterize when decryption will succeed in the following theorem.

To see why HAdd works, part of the decryption requires computing. This equation remains correct modulo q as long as the errors are small, i.

For HMult , the procedure is more complicated but observe that. Computation Model with Fully Homomorphic Encryption.

The set of functions that can be evaluated with FHE are arithmetic circuits over the plaintext ring R t. However, this is not an easy plaintext space to work with; elements in R t are polynomials of degree up to several thousand.

Therefore, the computation model generally used with homomorphic encryption is arithmetic circuits with modulo t gates.

For efficiency, the circuits evaluated using the HAdd and HMult algorithms should be levelled. This means that the gates of the circuits can be organized into layers, with inputs in the first layer and output at the last, and the outputs of one layer are inputs to gates in the next layer.

In particular, the most important property of arithmetic circuits for HE is its depth. The depth of a circuit is the maximum number of multiplication gates along any path of the circuit from the input to output layers.

A levelled FHE scheme with input level L can evaluate circuits of at most depth L which affects the choice of parameter q due to noise in ciphertexts.

In particular, the HMult operation on ciphertext is the main limiting factor to homomorphic evaluations.

Successive calls to HMult have outputs that steadily grow. A neural network, by which we mean artificial feed-forward neural networks , can be seen as a circuit made up of levels called layers.

Each layer is made up of a set of nodes, with the first being the inputs to the network. Nodes in the layers beyond the first take the outputs from a subset of nodes in the previous layer and output the evaluation of some function over them.

The values of the nodes in the last layer are the outputs of the neural network. In the literature, many different layers are used but these can generally be grouped into three categories.

Activation Layers: Each node in this layer takes the output, z , of a single node of the previous layer and outputs f z for some function z.

Pooling Layers: Each node in this layer takes the outputs, z , of some subset of nodes from the previous layer and outputs f z for some function f.

Although commonly used in practice, some have questioned the utility of pooling layers. Springenberg et al. To adapt neural networks operations over encrypted data, we do not use pooling and focus on the following layers:.

Fully Connected Layer: Similar to the convolution layer, each node outputs a weighted-sum, but over the entire previous layer rather than a subset of it.

Homomorphic encryption HE enables computation directly on encrypted data. This is ideal to handle the challenges that machine learning face when it comes to questions of data privacy.

Although HE promises a lot, there are several obstacles, ranging from the choice of plaintext space to translating neural network operations, that prevent straightforward translation of standard techniques for traditional CNNs to HCNNs.

The first problem is the choice of plaintext space for HCNN computation. Weights and inputs of a neural network are usually decimals, which are represented in floating-point.

Unfortunately, these cannot be directly encoded and processed in most HE libraries and thus require some adjustments.

Note that we can classify the entire MNIST testing dataset at once as the number of slots is more than 10, Encoding into the Plaintext Space.

We adopt the scalar encoding , which approximates these decimals with integers. Then, numbers encoded with the same scaling factor can be combined with one another using integer addition or multiplication.

Although straightforward to use, there are some downsides to this encoding. The scale factor cannot be adjusted mid-computation and mixing numbers with different scaling factors is not straightforward.

This means that as homomorphic operations are done on encoded data, the scaling factor in the outputs increases without a means to control it.

Therefore, the plaintext modulus t has to be large enough to accommodate the maximum number that is expected to result from homomorphic computations.

Thus, we require a way to handle large plaintext moduli of possibly several hundred bits. Arithmetic modulo t is replaced by component-wise addition and multiplication modulo the prime t i for the i -th entry.

We can recover the output of any computation as long as it is less than t because the inverse map C R T will return a modulo t result.

The actual output f m is obtained by applying the CRT map to v , i. Computation in HE schemes are generally limited to addition and multiplication operations over ciphertexts.

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Unfortunately, these cannot be directly encoded and processed in most HE libraries and thus require some adjustments.

Note that we can classify the entire MNIST testing dataset at once as the number of slots is more than 10, Encoding into the Plaintext Space.

We adopt the scalar encoding , which approximates these decimals with integers. Then, numbers encoded with the same scaling factor can be combined with one another using integer addition or multiplication.

Although straightforward to use, there are some downsides to this encoding. The scale factor cannot be adjusted mid-computation and mixing numbers with different scaling factors is not straightforward.

This means that as homomorphic operations are done on encoded data, the scaling factor in the outputs increases without a means to control it.

Therefore, the plaintext modulus t has to be large enough to accommodate the maximum number that is expected to result from homomorphic computations.

Thus, we require a way to handle large plaintext moduli of possibly several hundred bits. Arithmetic modulo t is replaced by component-wise addition and multiplication modulo the prime t i for the i -th entry.

We can recover the output of any computation as long as it is less than t because the inverse map C R T will return a modulo t result.

The actual output f m is obtained by applying the CRT map to v , i. Computation in HE schemes are generally limited to addition and multiplication operations over ciphertexts.

As a result, it is easy to compute polynomial functions with HE schemes. As with all HE schemes, encryption injects a bit of noise into the data and each operation on ciphertexts increases the noise within it.

As long as the noise does not exceed some threshold, decryption is possible. Otherwise, the decrypted results are essentially meaningless.

Approximating Non-Polynomial Activations. For CNNs, a major stumbling block for translation to the homomorphic domain is the activation functions.

These are usually not polynomials, and therefore unsuitable for evaluation with HE schemes. The effectiveness of the R e L U function in convolutional neural networks means that it is almost indispensable.

Therefore, it should be approximated by some polynomial function to try to retain as much accuracy as possible. The choice of approximating polynomial depends on the desired performance of the HCNN.

However, with the use of scalar encoding, there is another effect to consider. Namely, the scaling factor on the output will be dependent on the depth of the approximation, i.

Handling Pooling Layers. Still, that is not the only choice that is available. For a simpler CNN, we chose to remove the pooling layers used in CryptoNets during training and apply the same network for both training and inference, with the latter over encrypted data.

Convolution-Type Layers. Lastly, we have the convolutional-type layers. Since these are weighted sums, they are straightforward to compute over encrypted data; the weights can be multiplied to encrypted inputs with HMult and the results summed with HAdd.

Nevertheless, we still have to take care of the scaling factor of outputs from this layer. But, there is actually the potential for numbers to increase in bit-size from the additions done in weighted sums.

In practice, this bound is usually not achieved since the summands are almost never all positive. Implementation is comprised of two parts: 1 training on unencrypted data, and 2 classifying encrypted data.

This part is quite straightforward and can be simply verified by classifying the unencrypted test dataset. For neural networks design, one of the major constraints posed by homomorphic encryption is the limitation of numerical precision of layer-wise weight variables.

Training networks with lower precision weights would significantly prevent the precision explosion in ciphertext as network depth increases, and thus speed up inference rate in encrypted domain.

To this end, we propose to train low-precision networks from scratch, without incurring any loss in accuracy compared to networks trained in floating point precision.

The second part is more involved since it requires running the network with the pre-learned model on encrypted data. First, we need to fix HE parameters to accommodate for both the network multiplicative depth and precision.

We optimized the scaling factor in all aspects of the HCNN. Inputs were normalized to [ 0 , 1 ] , scaled by 4 and then rounded to their nearest integers.

With the low-precision network trained from scratch, we convert the weights of the convolution-type layers to short 4 -bit integers, using a small scaling factor of 15 ; no bias was used in the convolutions.

NTL is used to facilitate the treatment of the scaled inputs and accommodate for precision expansion of the intermediate values during the computation.

We found that the largest precision needed is less than 2 This is low enough to fit in a single word on bit platforms without overflow.

The next step is to implement the network using a HE library. The purpose of implementing the network in SEAL is to facilitate a more unified comparison under the same system parameters.

Before delving into the details of our implementation, we introduce an approach that is commonly followed to choose the FHE parameters.

Similar to other cryptographic schemes, one needs to select FHE parameters to bound the known attacks computationally infeasible.

A widely acceptable estimate for. In this work, we used a levelled BFV scheme that can be configured to support a known multiplicative depth L.

L can be controlled by three parameters: Q , t and noise growth. Q and t are problem dependent whereas noise growth is scheme dependent.

As mentioned in the previous section, we found that t should be at least a bit integer to accommodate the precision expansion in HCNN evaluation.

For our HCNN, five multiplication operations are required: 2 ciphertext by ciphertext in the square layer and 3 ciphertext by plaintext in convolution and fully connected layers operations.

It is known that the latter has less effect on noise growth. This means that L needs not be set to 5. Next, we try to estimate n to ensure a certain security level.

The above discussion suggests that the design space of HCNN is not limited depending on the choice of the plaintext coefficient modulus t.

We identify a set of possible designs that fit different requirements. The designs vary in the number of factors in t i. Note that, in the 1-CRT channel, we set t as a bit prime number, whereas in the 2-CRT channels, we use 2 bit prime numbers whose product is a bit number.

Support for arbitrary scaling factors per layer is included for flexibility and allows us to easily define neural network layers for HCNN inference.

Now, we briefly describe how our library realizes these layers. For convolution-type layers, they are typically expressed with matrix operations but only require scalar additions and multiplications.

For the other two, activation and pooling, some modifications had to be done for compatibility with HE schemes. These are non-polynomial functions and thus cannot be directly evaluated over HE encrypted data.

The BFV scheme is considered among the most promising HE schemes due to its simple structure and low overhead primitives compared to other schemes.

Moreover, it is a scale-invariant scheme where the ciphertext coefficient modulus is fixed throughout the entire computation.

This contrasts to other scale-variant schemes that keep a chain of moduli and switch between them during computation.

We use our GPU-based BFV implementation as an underlying HE engine to perform the core HE primitives: key generation, encryption, decryption and homomorphic operations such as addition and multiplication.

Polynomial Arithmetic Unit PAU : performs basic polynomial arithmetic such as addition and multiplication.

We note that further task parallelism can be extracted from HCNN by decomposing the computation into smaller independent parts that can run in parallel.

In this scenario, the computation is completely separable requiring communication only at the beginning and end of computation for CRT calculations.

Nevertheless, our implementation executes the channels sequentially on a single GPU. In this section, we describe our experiments to evaluate the performance of HCNN using the aforementioned mentioned designs.

We start by describing the hardware configuration. Next, we present the results together with discussion and remarks on the performance. Timing results can be reduced into half if the network is run simultaneously on two GPUs.

This also applies for SEAL as well. We include the timing of all the aforementioned parameter sets. In particular, the speedup factors achieved are The amortized time represents the per-image inference time.

Note that in parameter sets 3,4 and 5 we can classify the entire testing dataset of MNIST in a single network evaluation. The results also show the importance of low-precision training which reduced the required precision to represent the network output.

This allows running a single instance of the network without plaintext decomposition 1-CRT channel. We remark that CryptoNets used higher precision training and required plaintext modulus of higher precision 2 Therefore, they had to run the network twice using 2-CRT channels.

We also note that our timing results shown here for SEAL are much higher than those reported in CryptoNets seconds at bit security.

Lastly, we compare our best results with the currently available solutions in the literature. As we can see, our solution outperforms both solutions in total and amortized time.

Note that E2DM classifies 64 images in a single evaluation. This means that to classify the entire dataset, one would need more than 1 hour. The main motivation of this work was to show that privacy-preserving deep learning with FHE is dramatically accelerated with GPUs and offers a way towards efficient DLaaS.

Our implementation included a set of techniques such as low-precision training, unified training and testing network, optimized FHE parameters and a very efficient GPU implementation to achieve high performance.

In terms of performance, our best results show that we could classify the entire testing dataset in This packing scheme is ideal for applications that require the inference of large batches of images which can be processed in parallel in a single HCNN evaluation.

Other application may have different requirements such as classifying 1 or small number of images. For this particular case, other packing methods that pack more pixels of the same image in the ciphertext can be used.

As future work, we will investigate other packing methods that can fit a wide-range of applications. Moreover, we will target more challenging problems with larger datasets and deeper networks.

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