User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, addbmm() should return a new tensor and addbmm_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, clone() should return a new tensor that's an exact copy. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, le() should return a new tensor and le_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

NOTE: This ticket requires minimal work on ISSUE 10 to have been already started. Specifically, all work on this ticket should be over the generic Tensor type, the specs of which are described within that issue.

Description: Homomorphic Encryption has a fundamental limitation in that after a certain number of additions and multiplications, noise builds up in the cyphertext and endangers the decrypt-ability of the cyphertext. In order to make our tensors easy to use, we need to provide tracking around noise growth by tracking the number of additions and multiplications involved in creating each particular value in any given tensor. This tracking needs to either automatically bootstrap (remove noise) or flag a warning (error) if the noise is about to render the cyphertext undecryptable. Furthermore, as each encryption technique has slightly different needs in this area, our tensor object needs to implement this functionality in a way that is agnostic to the encryption technique.

Acceptance Criteria:

1. For the Paillier, FV, (and YASHE if it's up in time) algorithms, be able to predict when a number is about to become undecryptable with > 99.99999% recall given arbitrary parameterization of the encryption algorithm. The "how" is not descriptive (be creative!) as long as its efficient. Unit Tests should demonstrate this level of accuracy.
2. Implement the logic of (1) in such a way that the user is not aware of it (abstracted behind the operations)
3. Implement the logic of (1) in such a way that it is agnostic of the HE algorithm (including changing the required functions in the encryption algorithm's abstract interface.
4. Implement 1 of 3 settings as options for when a number is about to become undecryptable:
   a. the number is locally bootstrapped (make this the default if available)
   b. a call is made to an outside server (which you should also build in Syft) where the number is decrypted and then re-encrypted with the same key (make this the first backup after the default)
   c. an error is thrown (as a last resort)
5. All functionality should work given arbitrary encryption parameters (Increasing key sizes can increase the max noise tolerated regardless of the algorithm being used)
6. Complete benchmarking test suite for all abstract tensor operations so that their speed can be automatically reported by running the script with a given encryption algorithm. In this way, we should be able to see the tradeoff between speed and stability (bigger keys == slower but larger multiplicative depth). This should allow us to find the optimal tradeoff between the speed of any operation (which gets faster if you bootstrap more often) and the frequency of bootstrapping (which takes time).

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, fmod() should return a new tensor and fmod_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, chunk() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, cauchy_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, half() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, cumprod() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, eq() should return a new tensor and eq_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, gt() should return a new tensor and gt_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, dot() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

Description: All of our Homomorphic Encryption algorithms operate on Integers under the hood. However, we want to be able to perform math on numbers of varying precision (most popularly, Floats). Thus, we need someone to put together a generic Fixed Point Precision wrapper that can provide this functionality around encryption algorithms that are Integers such that the following types are supported:

• Byte (8-bit float)

• Half (16-bit float)

• Float (32-bit float)

• Double (64-bit float)

• Arbitrary (n-bit float)

• Char (8-bit int)

• Short (16-bit int)

• Int (32-bit int)

• Long (64-bit int)

• Arbitrary (n-bit int)

Acceptance Criteria:

• All types above are supported assuming 32-bit integer as input.
• All basic operations are supported (+-*/ and inline versions).
• Exponents can be decrypted
• Unit Tests show that it works with all types above given Integers underneath
• Unit Tests show that it works in the encrypted domain (using Paillier is fine)

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, exp() should return a new tensor and exp_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, fill_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, floor() should return a new tensor and floor_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, mean() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, cumsum() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

Description: We are currently wrapping a Python implementation of the Paillier cryptosystem. While it is a great implementation, we want to be able to integrate one into our project a bit more closely. Furthermore, we would like to leverage lower level libraries for C, BLAS, and/or GPU integration when available. As such, this project endeavors to rebuild Paillier from scratch in a way that is more performant than the current (while also leaning on (ISSUE 12) for fixed point precision instead of having it deeply coupled to the encryption algorithm (as in python-paillier).

Acceptance Criteria:

1. The new implementation supports the major homomorphic operations (with unit tests demonstrating)
   a. Public / Private Key Generation, Serialization, and Deserialization
   b. Encryption and Decryption
   c. Addition Between Encrypted Numbers
   d. Multiplication of an Encrypted Number by a Decrypted Scalar
   e. Leverages vector "packing" for increased performance (thus, the base data structure is a vector, not a scalar)
2. The new implementation by default calls a highly performant runtime at least as fast as THIS
3. The new implementation can support BLAS if available (and if BLAS is faster than (2))
4. The new implementation can support CUDA and/or OpenCL GPU if available on a machine.
5. This implementation is constructed in a new library called PyPaillier (OpenMined/PyPaillier) and integrated into Syft proper.
6. Benchmarking suite documents the performance across a range of key sizes and backends.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, ge() should return a new tensor and ge_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, bernoulli() should return a new tensor and bernoulli_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, ceil() should return a new tensor and ceil_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing the batch matrix multiply + add of a Tensor of arbitrary type. addcdiv() should return a new tensor and addcdiv_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, histc() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, masked_select() should return a new tensor and masked_scatter_(), masked_fill_() should perform their operations inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, log(), log1p(), and log_normal() should return a new tensor and log_(), log1p_(), and log_normal_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing each operation on a Tensor of arbitrary type. addmm_() should return a new tensor and addmm_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation of addmm() and addmm_() on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, bmm() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, gather() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

In this ticket, we want to create a new basic type called a "Tensor". A Tensor is a nested list of numbers with an arbitrary number of dimensions. With one dimension, it's a list of numbers known as a "vector". With two dimensions, it's a list of lists of numbers known as a "matrix". In this ticket, we want to build our implementation of Tensor with inspiration from PyTorch's Tensor object (Tensor Docs).

So, in this ticket, you should build a basic tensor class. You should be able to pass in as an argument an arbitrary shape (number of rows, columns, etc.) when you create it. Furthermore, we want this basic class to support elementwise addition, subtraction, multiplication, and division.

Acceptance Criteria:

• init supports initializing a tensor with an arbitrary number of dimensions and arbitrary scalar type (test with float and int)
• elementwise operators for +-/* and their inline operators should be overridden
• elementwise operations should check to make sure both tensors are of identical dimension
• elementwise operators should also support (and automatically detect!) the input of a single number, which should be applied to the entire tensor (aka... multiply every number by "5").

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, geometric_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, addmv() should return a new tensor and addmv_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, frac() should return a new tensor and frac_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, expand_as() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, index() and index_select() should return a new tensor and index_add_(), index_copy_(), and index_fill_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, inverse should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, diag() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, baddbmm() should return a new tensor and baddbmm_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

Description: In the Proof-of-Concept Syft implementation, the basic type was a Float object contained within the Paillier.py class. This class contains overridden functions for various mathematical operators, so that the class can then be used inside of numpy arrays, as in our basic linear network linear.py. However, this approach, while simple, has several issues:

• Homomorphic Encryption uses a technique called "packing" which can make efficient use when entire vectors of information are encrypted all at once. This was impossible when forcing all encrypted values to be simple scalars, resulting in a negative impact on performance.
• Any function was callable upon the numpy arrays holding encrypted values, even those which were almost guaranteed to cause overflow within the encrypted cyphertexts, corrupting their values. While this was convenient, we need to implement the basic Tensor type ourselves so that we can have better warnings/automation for when bootstrapping should occur.
• There was not a clear path to fast GPU support, which also requires vectorized operations.

In this work, we want to remedy these issues by building a generic tensor type, allowing us to handle encrypted values in the optimal way under the hood without requiring advanced knowledge by the user. The API for this tensor type will take direct inspiration from the unencrypted tensor classes done in the PyTorch framework, as it is a long term desire to eventually integrate with PyTorch if it becomes feasible to do so.

Acceptance Criteria:

1. The generic tensor type must support the following operations inline with PyTorch:
   a. abs(), abs_(), add(value), add_(value), addbmm(), addbmm_(), addcdiv(), addcdiv_(), addcmul(), addcmul_(), addmm(), admm_(), addmv(), addmv_(), addr(), addr_(), baddbmm(), baddbmm_(), bernoulli(), bernoulli_(), bmm(), cauchy_(), ceil(), ceil_(), char(), chunk(), clamp(), clamp_(), clone(), contiguous(), copy_(), cpu(), cuda() - returns "not yet supported", cross(), cumprod(), cumsum(), diag(), dim(), dist(), div(), div_(), dot(), double(), eq(), eq_(), equal(), exp(), exp_(), expand(), expand_as(), exponential(), fill_(), float(), floor(), floor_(), fmod(), fmod_(), frac(), frac_(), gather(), ge(), ge_(), geometric_(), gt(), gt_(), half(), histc(), index(), index_add_(), index_copy_(), index_fill_(), index_select(), int(), inverse(), is_contiguous(), is_cuda(), is_signed(), le(), le_(), lerp(), lerp_(), log(), log1p(), log1p_(), log_(), log_normal_(), long(), lt(), lt_(), masked_scatter_(), masked_fill_(), masked_select(), matmul(), max(), mean(), median(), min(), mm(), mode(), mul(), mul_(), multinomial(), mv(), narrow(), ndimension(), ne(), ne_(), neg(), neg_(), nelement(), new(), nonzero(), norm(), normal_(), numel(), numpy(), permute(), pow(), pow_(), prod(), random_(), reciprocal(), reciprocal_(), remainder(), remainder_(), renorm(), renodm_(), repeat(), resize_(), resize_as_(), round(), round_(), rsqrt(), rsqrt_(), scatter_(), select(), set_(), short(), sigmoid(), sigmoid_(), sign(), sign_(), size(), split(), sqrt(), sqrt_(), squeeze(), squeeze_(), stride(), sub(), sub_(), sum(), t(), t_(), tolist(), topk(), trace(), transpose(), transpose_(), type(), unfold(), uniform_(), unsqueeze(), unsqueeze_(), view(), view_as(), zero_()
   b. For all functions above that pull from a distribution, optionally use a lookup table (abstract this?)
   c. For all functions above that use a function, optionally use an interpolation (abstract this?)

2. The generic tensor type must be able to support an arbitrary number of dimensions for all operations.

3. The generic tensor type must overload +-*/ and apply the appropriate operator (based on the dimensions of the tensors involved.

4. The generic tensor type must prefer to use vector storage (under the hood) when encrypted to maximize the usefulness of packing in homomorphic encryption

5. Any operations that are unsupported in the encrypted domain should include error messages

6. str and repr should follow the conventions set in PyTorch.

7. Additionally, there should be "encrypt(publickey)" and "decrypt(prikey)" functions which accept any key of the abstract type public or private key.

8. All functionality within this base class should be of abstract type regardless of encryption type (specialized functionality for any encryption scheme should live within that scheme).

9. Before this ticket is closed, this functionality should be shown to work with at least one homomorphic encryption strategy

Ticket Creator: @iamtrask

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing the elementwise absolute value of a Tensor of arbitrary type. abs() should return a new tensor and abs_ should perform the operation inline. For a great reference on how

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation of abs() and abs_() on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, addcmul() should return a new tensor and addcmul_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, exponential(). For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, lerp() should return a new tensor and lerp_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, diag() should return a list of the dimensions of the tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, lt() should return a new tensor and lt_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, equal() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, clamp() should return a new tensor and clamp_() should perform the operation inline. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, matmul() should return a new tensor. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.

Background: It has been alleged that vanilla Learning with Errors can be extremely performant for additive homomorphic encryption. In this work, we seek to verify that this is indeed true and achieve similar performance gains.

Acceptance Criteria:

• the LWE approach in the above paper is constructed and shown to work for homomorphic addition and scalar multiplication (with unit tests)
• key switching interfaces are added
• the homomorphic encryption algorithm is in a new repository called PyFastLWE

User Story A: As a Data Scientist using Syft's Base Tensor type, we want to implement a default method for computing operations on a Tensor of arbitrary type. For this ticket to be complete, max() should return functionality based on what is passed into the method. For a reference on the operation this performs check out PyTorch's documentation.

Acceptance Criteria:

• If the Base Tensor type's attribute "encrypted" is set to True, it should return a NotImplemented error.
• a unit test demonstrating the correct operation on the Base Tensor type implemented over int and float Tensors.
• inline documentation in the python code. For inspiration on inline documentation, please check out PyTorch's documentation for this operator.