lol-0.4.0.0: README

Most of the functionality in Lol is exported by two modules:

* 'Crypto.Lol' exports the primary *interfaces* of Lol

* 'Crypto.Lol.Types' exports concrete types that would be used by most
  instantiations including tensors, base rings, and cryptographic

For a brief introduction to relevant mathematical notation, see 'Crypto.Lol'.

Overview of key modules, roughly from highest- to lowest-level:

Cyclotomic layer:
* 'Crypto.Lol.Cyclotomic.Cyc', which defines an interface for using cyclotomic
  fields, rings \( R \), and quotient rings \( R_q=R/qR \); as well as many
  other commonly used operations, e.g., converting
  between rings, decoding and decomposing elements, modulus
  reduction/rounding, etc. 'Cyc' is a safe wrapper around the
  'UCyc' type, which exposes some representation-dependent operations.
  'UCyc' (and hence 'Cyc') is implemented using a generic 'Tensor'
  (described below).


Tensor layer:
* 'Crypto.Lol.Cyclotomic.Tensor', which defines a class that encapsulates all
  the necessary linear transformations for operating on representations of
  \( R \)- and \( R_q \)-elements, e.g., the CRT transform, converting between
  the powerful and decoding bases, generating error terms, etc.

* 'Crypto.Lol.Cyclotomic.Tensor.RepaTensor', which gives an
  implementation of the 'Tensor' class based on the "repa"
  package, a highly optimized and parallelizable array library.

* 'Crypto.Lol.Cyclotomic.Tensor.CTensor', which gives an
  implementation of the 'Tensor' class using a C++ backend via Haskell's FFI.


Base ring layer:
* 'Crypto.Lol.Types.FiniteField', which gives an unoptimized implementation of
  finite field arithmetic. To use this module, you will need an instance of
  'IrreduciblePoly'. These instances provide irreducible polynomials
  for various degrees and base fields.  One (orphan) instance is provided for
  characteristic 2 fields of size up to 2^128 in
  'Crypto.Lol.Types.IrreducibleChar2', and is exported by 'Crypto.Lol.Types'.
  If you need to use an unsupported finite field,  define your own
  instance of 'IrreduciblePoly' and do **not** import 'IrreducibleChar2'.

* 'Crypto.Lol.Types.ZqBasic', which is a basic implementation of
  \( \Z_q=\Z/q\Z \) arithmetic.

* 'Crypto.Lol.Factored', which contains type-level support code for
  factored integers. It also supports "reifying" 'Int's at runtime as static
  types and "reflecting" those types as integers back to the code..
  'Factored' types are mainly used to represent cyclotomic indices.