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Last week, the team at Julia Computing published a research based on cutting edge cryptographic techniques. The research involved cryptography techniques to practically perform computation on data without ever decrypting it.

For example, the user would send encrypted data (e.g. images) to the cloud API, which would run the machine learning model and then return the encrypted answer. Nowhere is the user data decrypted and in particular the cloud provider does not have access to either the original image nor is it able to decrypt the prediction it computed. The team made this possible by building a machine learning service for handwriting recognition of encrypted images (from the MNIST dataset).

The ability to compute on encrypted data is generally referred to as “secure computation” and is a fairly large area of research, with many different cryptographic approaches and techniques for a plethora of different application scenarios. For their research, Julia team focused on using a technique known as “homomorphic encryption”.

What is homomorphic encryption

Homomorphic encryption is a form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on the plaintext.

This technique can be used for privacy-preserving outsourced storage and computation. It allows data to be encrypted and out-sourced to commercial cloud environments for processing, all while encrypted. In highly regulated industries, such as health care, homomorphic encryption can be used to enable new services by removing privacy barriers inhibiting data sharing.

In this research, the Julia Computing team used a homomorphic encryption system which involves the following operations:

pub_key, eval_key, priv_key = keygen()

encrypted = encrypt(pub_key, plaintext)

decrypted = decrypt(priv_key, encrypted)

encrypted′ = eval(eval_key, f, encrypted)

So the first three are fairly straightforward and are familiar to anyone who has used asymmetric cryptography before. The last one is important as it evaluates some function f on the encryption and returns another encrypted value corresponding to the result of evaluating f on the encrypted value. It is this property that gives homomorphic computation its name.

Further the Julia Computing team talks about CKKS (Cheon-Kim-Kim-Song), a homomorphic encryption scheme that allowed homomorphic evaluation on the following primitive operations:

Element-wise addition of length n vectors of complex numbers

Element-wise multiplication of length n complex vectors

Rotation (in the circshift sense) of elements in the vector

Complex conjugation of vector elements

But they also mentioned that computations using CKKS were noisy, and hence they tested to perform these operations in Julia.

Which convolutional neural network did the Julia Computing team use

As a starting point the Julia Computing team used the convolutional neural network example given in the Flux model zoo. They kept training the loop, prepared the data and tweaked the ML model slightly.

It is essentially the same model as the one used in the paper “Secure Outsourced Matrix Computation and Application to Neural Networks”, which uses the same (CKKS) cryptographic scheme. This paper also encrypts the model, which the Julia team neglected for simplicity and they involved bias vectors after every layer (which Flux does by default). This resulted in a higher test set accuracy of the model used by Julia team which was (98.6% vs 98.1%).

An unusual feature in this model are the x.^2 activation functions. More common choices here would have been tanh or relu or something more advanced. While those functions (relu in particular) are cheap to evaluate on plaintext values, they would however, be quite expensive to evaluate on encrypted values. Also, the team would have ended up evaluating a polynomial approximation had they adopted these common choices. Fortunately x.^2 worked fine for their purpose.

How was the homomorphic operation carried out

The team performed homomorphic operation on Convolutions and Matrix Multiply assuming a batch size of 64. They precomputed each convolution window of 7×7 extraction from the original images which gave them 64 7×7 matrices per input image. Then they collected the same position in each window into one vector and got a 64-element vector for each image, (i.e. a total of 49 64×64 matrices), and encrypted these matrices. In this way the convolution became a scalar multiplication of the whole matrix with the appropriate mask element, and by summing all 49 elements later, the team got the result of the convolution.

Then the team moved to Matrix Multiply by rotating elements in the vector to effect a re-ordering of the multiplication indices. They considered a row-major ordering of matrix elements in the vector. Then shifted the vector by a multiple of the row-size, and got the effect of rotating the columns, which is a sufficient primitive for implementing matrix multiply. The team was able to get everything together and it worked. You can take a look at the official blog post to know the step by step implementation process with codes.

Further they also executed the whole encryption process in Julia as it allows powerful abstractions and they could encapsulate the whole convolution extraction process as a custom array type.

The Julia Computing team states, “Achieving the dream of automatically executing arbitrary computations securely is a tall order for any system, but Julia’s metaprogramming capabilities and friendly syntax make it well suited as a development platform.”

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