Finish documenting / testing Zeromorph (#142)

* fix: remove a TODO using RefCast

* fix: remove unsightly clone

* doc: comment intermediate ZM steps

* refactor: Refactor `open` function in zeromorph provider

Extracted the computation of `q_hat` in the `open` function into a new function `batched_lifted_degree_quotient` for more modular code.

* test: test batched_lifted_degree_quotient

Addition of a new test `test_batched_lifted_degree_quotient` to validate batched degree quotient lifting for polynomials.

* refactor: Refine computations in non_hiding_zeromorph.rs

- Overhauled the 'eval_and_quotient_scalars' function with a revised return type with pair of vectors rather than scalar and a vector,
- correspondingly split scalars `degree_check_q_scalars` and `zmpoly_q_scalars` within the `open` method of `non_hiding_zeromorph.rs`.
- Adjusted the 'verify' method to accommodate the modified 'eval_and_quotient_scalars' function output changes.

* test: test partially evaluated quotient \zeta_x

- Introduced a new test case `test_partially_evaluated_quotient_zeta` to validate `zeta_x` construction.
- correct an off-by-one error in the y_k terms

* test: test partially evaluated quotient

- Created a new function `phi` for evaluating using an inefficient formula in `non_hiding_zeromorph.rs`
- Implemented a new test function `test_partially_evaluated_quotient_z` for validation of `Z_x` computation method in `non_hiding_zeromorph.rs`

* fix: refine q_hat comment

Cargo.toml

@@ -43,7 +43,8 @@ tracing-texray = "0.2.0"
 tracing-subscriber = { version = "0.3.17", features = ["env-filter"] }
 cfg-if = "1.0.0"
 once_cell = "1.18.0"
-rand = "0.8.4"
+rand = "0.8.5"
+ref-cast = "1.0.20"
 
 [target.'cfg(any(target_arch = "x86_64", target_arch = "aarch64"))'.dependencies]
 pasta-msm = { git="https://github.com/lurk-lab/pasta-msm", branch="dev", version = "0.1.4" }
@@ -60,7 +61,6 @@ pprof = { version = "0.11" }
 sha2 = "0.10.7"
 tracing-test = "0.2.4"
 proptest = "1.2.0"
-rand = "0.8.5"
 
 [[bench]]
 name = "recursive-snark"