These appear to be unused for extension fields, so we're free to change the mapping without breaking anything.
As the TODO says, the mapping that's currently implemented doesn't seem natural or useful. It seems more natural to treat the `BigUint` as a base field element, potentially in a non-canonical form.
It seems redundant in most contexts, e.g. `use plonky2::field::extension_field::Extendable;`. One could import `extension_field`, but it's not that common in Rust, and `field::extension` is now about as short.
* Halo2 style lookup arguments in System Zero
It's a really nice and simple protocol, particularly for the verifier since the constraints are trivial (aside from the underlying batched permutation checks, which we already support). See the [Halo2 book](https://zcash.github.io/halo2/design/proving-system/lookup.html) and this [talk](https://www.youtube.com/watch?v=YlTt12s7vGE&t=5237s) by @daira.
Previously we generated the whole trace in row-wise form, but it's much more efficient to generate these "permuted" columns column-wise. So I changed our STARK framework to accept the trace in column-wise form. STARK impls now have the flexibility to do some generation row-wise and some column-wise (without extra costs; there's a single transpose as before).
* sorting
* fixes
* PR feedback
* into_iter
* timing
* Initial implementation of quintic extensions.
* Update to/from_biguint() methods.
* Draft of fast multiplication on quintic extensions over 64-bit base.
* cargo fmt
* Typo.
* Document functions (a bit).
* Refactor reduction step.
* Change multiplication call so that LLVM generates better assembly.
* Use one main accumulator instead of two minor ones; faster reduce.
* Use one main accumulator in square too; clean up redundant code.
* Call faster routines from Mul and Square impls.
* Fix reduction function.
* Fix square calculation.
* Slightly faster reduction.
* Clean up names and types.
* cargo fmt
* Move extension field mul/sqr specialisations to their own file.
* Rename functions to have unique prefix.
* Add faster quadratic multiplication/squaring.
* Faster quartic multiplication and squaring.
* cargo fmt
* clippy
* Alternative reduce160 function.
* Typo.
* Remove alternative reduction function.
* Remove delayed reduction implementation of squaring.
* Enforce assumptions about extension generators.
* Make the accumulation variable a u32 instead of u64.
* Add test to trigger carry branch in reduce160.
* cargo fmt
* Some documentation.
* Clippy; improved comments.
* cargo fmt
* Remove redundant Square specialisations.
* Fix reduce*() visibility.
* Faster reduce160 from Jakub.
* Change mul-by-const functions to operate on 160 bits instead of 128.
* Move code for extensions of GoldilocksField to its own file.
* Batch multiple perm args into one Z and compute Z columnwise
It's slightly complex because we batch `constraint_degree - 1` permutation arguments into a single `Z` polynomial. This is a slight generalization of the [technique](https://zcash.github.io/halo2/design/proving-system/lookup.html) described in the Halo2 book.
Without this batching, we would simply have `num_challenges` random challenges (betas and gammas). With this batching, however, we need to use different randomness for each permutation argument within the same batch. Hence we end up generating `batch_size * num_challenges` challenges for all permutation arguments.
* Feedback + updates for recursion code
* Initial implementation of quintic extensions.
* Update to/from_biguint() methods.
* cargo fmt
* Fix call to test suite.
* Small optimisation in try_inverse().
* Replace multiplicative group generator and document requirement.
* Column definitions for addition, range checks & lookups
I implemented addition (unsigned for now) as an example of how the arithmetic unit can interact with the 16-bit range check unit.
Range checks and lookups aren't implemented yet.
* Missing constraints
* Tweaks to get tests passing
* Reorg registers into files
* Minor
* trim_to_len helper function
Seems a little nicer IMO to only remove a certain number of zeros, vs removing all trailing zeros then re-adding some.
* PR feedback
* Rename `PrimeField` -> `Field64`
And add TODOs for moving around various methods which aren't well-defined in their current traits, or would be well-defined in a supertrait.
* fix test
* TODOs as per PR feedback
