Add Linear Algebra DSL about homomorphic-implementors-toolkit HOT 2 CLOSED

Currently, encryptRowVec() takes the row vector to encrypt (as columns) _and* the width of the encoding unit (so we know how many times to replicate it). This seems unnecessary: we know the encoding unit from context, so we can derive the number of columns.

from homomorphic-implementors-toolkit.

The full paper describes a formalized encoding system for linear algebra objects as CKKS plaintexts with respect to an encoding unit. It also describes basic linear algebra operations using this encoding. A single linear algebra object is encoded as one or more plaintexts, depending on the encoding and the size of the linear algebra objects. This induces extra work for linear algebra operations. For example, if we encode a matrix as several plaintexts, then the linear algebra operation of “multiply the matrix by a constant c” amounts to multiplying each plaintext by the constant c.
Putting this level of detail into a homomorphic algorithm unnecessarily complicates the algorithm and makes it harder to reason about. As a concrete example of this problem, the Python compiler/code generator only supports objects encoded as a single ciphertext due to the complexity of dealing with coordinating the individual plaintexts that make up a linear algebra object.
To make writing homomorphic linear algebra operations as easy as possible, we will create a higher-level linear algebra (LA) API on top of the existing CKKSEvaluator API. The LA API will directly call the CKKSEvaluator API. As a result:

• Existing ciphertext objects will become simpler because they will no longer contain linear algebra meta-data
• New types for CKKSMatrix and CKKSVector will be introduced. They will hold a vector of ciphertexts, but users can treat them as a single object.

from homomorphic-implementors-toolkit.