* Update FoldingSchemes trait, fit Nova+CycleFold

- update lib.rs's `FoldingScheme` trait interface
- fit Nova+CycleFold into the `FoldingScheme` trait
- refactor `src/nova/*`

* Add `examples` dir, with Nova's `FoldingScheme` example

* polishing

* expose poseidon_test_config outside tests

* Add KZG commitment scheme adapted to vector commitment

Add KZG commitment scheme adapted to vector commitment
Also move the `src/pedersen.rs` into `src/commitment/pedersen.rs` where
it will coexist with `kzg.rs` and the trait defined in
`src/commitment/mod.rs`.

* Adapt Pedersen into the new CommitmentProver trait

* add CommitmentProver (Pedersen&KZG) homomorphic property test

* polishing

* Use divide_with_q_and_r, rename skip_first_zero_coeffs

Co-authored-by: han0110 <tinghan0110@gmail.com>

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Co-authored-by: han0110 <tinghan0110@gmail.com>

* feat: init sumcheck.rs

* chore: rename

* feat: update lib and add trait for transcript with vec storing challenges

* bugfix: mut self ref of transcript

* feat: tentative sum-check using poseidon

* refactor: remove extension trait and use initial trait

* refactor: stop using extension trait, use initial Transcript trait

* feat: generic over CurveGroup sum-check verifier and algorithm

* feat: implement generic sum-check veriy

* bugfix: cargo clippy --fix

* chore: cargo fmt

* feat: (unstable) sum-check implementation

* feat: start benches

* chore: run clippy

* chore: run cargo fmt

* feat: add sum-check tests + benches

* chore: clippy + fmt

* chore: remove unstable sumcheck

* chore: delete duplicated sum-check code

* chore: remove deleted sum-check code from lib.rs imports

* feat: remove non generic traits, implement sum-check with generic trait and add test

* chore: remove non-generic struct

* chore: remove non generic verifier

* feat: make nifms generic over transcript and update to use poseidon transcript

* chore: cargo fmt

* chore: remove tmp benches

* chore: update cargo.toml

* refactor: remove Generic suffix

* feat: prover state generic over CurveGroup

* chore: disable clippy type complexity warning

* refactor: remove Transcript type and espresso transcript dependency

* refactor: SumCheckProver generic over CurveGroup

* feat: init sumcheck.rs

* chore: rename

* feat: update lib and add trait for transcript with vec storing challenges

* bugfix: mut self ref of transcript

* feat: tentative sum-check using poseidon

* refactor: stop using extension trait, use initial Transcript trait

* feat: generic over CurveGroup sum-check verifier and algorithm

* feat: implement generic sum-check veriy

* bugfix: cargo clippy --fix

* chore: cargo fmt

* feat: (unstable) sum-check implementation

* feat: start benches

* chore: run clippy

* chore: run cargo fmt

* feat: add sum-check tests + benches

* chore: clippy + fmt

* chore: remove unstable sumcheck

* chore: delete duplicated sum-check code

* chore: remove deleted sum-check code from lib.rs imports

* feat: remove non generic traits, implement sum-check with generic trait and add test

* chore: remove non-generic struct

* chore: remove non generic verifier

* feat: make nifms generic over transcript and update to use poseidon transcript

* chore: cargo fmt

* chore: remove tmp benches

* chore: update cargo.toml

* refactor: remove Generic suffix

* feat: prover state generic over CurveGroup

* chore: disable clippy type complexity warning

* refactor: remove Transcript type and espresso transcript dependency

* refactor: SumCheckProver generic over CurveGroup

* feat: adds `compute_lagrange_poly`, returning a `DensePolynomial` to extract coeffs from

* chore: add assert on interpolated poly degree vs initial poly degree

* refactor: use `compute_lagrange_poly` in `SumCheckVerifier` instead of `interpolate_uni_poly`

* refactor: have `TranscriptVar` be generic over `CurveGroup` for consistency

* refactor: change back to being generic over field

* feat: start to use `PoseidonTranscriptVar` struct

* chore: add line to eof for `Cargo.toml`

* chore: naming consistency with espresso/sum_check folder

* bugfix: add error handling on sum-check prove and verify

* chore: clippy fix

* chore: add line at eof

* feat: switch to using coeffs instead of evals in sum-check

* bugfix: tmp remove sanity check in nimfs

* refactor: update sanity check

* refactor: update verifier evaluation form + add comment

* chore: run clippy

* fix: correct merge artifacts

* feat: verify circuit passing

* refactor: change naming to use the `Gadget` suffix, update `verify_sumcheck` to not have `&self` as first argument, update test

* feat: testing on polynomials with various number of variables

* refactor: update method name

* fix: avoid rust-analyzer from complaining

* fix: fix clippy complains

* chore: cargo clippy

* chore: udpate arg name for `SumCheckVerifierGadget`

* refactor: remove unnecessary cloning in sumcheck circuit

* refactor: impl `get_poly_vars_from_sumcheck_proof` for `IOPProof`

* chore: group imports

* refactor: update `P(0) + P(1)` and name it `eval`

* chore: clippy + fmt

* refactor: move `compute_lagrange_poly` to `utils`

* fix: wrong import

* chore: cargo fmt

* refactor: absorb num vars and max degree within sumcheck circuit

* refactor: update name to `compute_lagrange_interpolated_poly` and add comment

* feat: create `IOPProofVar`, which implements the `AllocVar` trait

* fix: clippy allow `type_complexity` on return type of `SumCheckVerifierGadget`

* refactor: use `VPAuxInfo` instead of virtual type, remove `_` prefix in params name

* feat: add tests on computed & returned values from `verify`

* refactor: use `new_witness` instead of `new_variable`

Co-authored-by: arnaucube <root@arnaucube.com>

* chore: clippy

* fix: remove `unwrap()` within `verify()`

Co-authored-by: arnaucube <root@arnaucube.com>

* chore: add comment on `unwrap()`

* refactor: move `compute_lagrange_interpolated_poly` tests to `lagrange_poly.rs`

---------

Co-authored-by: arnaucube <root@arnaucube.com>

* feat: init sumcheck.rs

* chore: rename

* feat: update lib and add trait for transcript with vec storing challenges

* bugfix: mut self ref of transcript

* feat: tentative sum-check using poseidon

* refactor: remove extension trait and use initial trait

* refactor: stop using extension trait, use initial Transcript trait

* feat: generic over CurveGroup sum-check verifier and algorithm

* feat: implement generic sum-check veriy

* bugfix: cargo clippy --fix

* chore: cargo fmt

* feat: (unstable) sum-check implementation

* feat: start benches

* chore: run clippy

* chore: run cargo fmt

* feat: add sum-check tests + benches

* chore: clippy + fmt

* chore: remove unstable sumcheck

* chore: delete duplicated sum-check code

* chore: remove deleted sum-check code from lib.rs imports

* feat: remove non generic traits, implement sum-check with generic trait and add test

* chore: remove non-generic struct

* chore: remove non generic verifier

* feat: make nifms generic over transcript and update to use poseidon transcript

* chore: cargo fmt

* chore: remove tmp benches

* chore: update cargo.toml

* refactor: remove Generic suffix

* feat: prover state generic over CurveGroup

* chore: disable clippy type complexity warning

* refactor: remove Transcript type and espresso transcript dependency

* refactor: SumCheckProver generic over CurveGroup

* chore: add line to eof for `Cargo.toml`

* bugfix: add error handling on sum-check prove and verify

* chore: clippy fix

* chore: add line at eof

* fix: use `map_err` and call `to_string()` on `PolyIOPErrors`

* Port HyperNova's multifolding from https://github.com/privacy-scaling-explorations/multifolding-poc adapting and refactoring some of its methods and structs.

Note: adapted mle.rs methods from dense to sparse repr.

Co-authored-by: George Kadianakis <desnacked@riseup.net>

* HyperNova: move CCS struct outside of LCCCS & CCCS

HyperNova nimfs: move CCS structure outside of LCCCS & CCCS, to avoid
carrying around the whole CCS and duplicating data when is not needed.

Also add feature flags for the folding schemes.

---------

Co-authored-by: George Kadianakis <desnacked@riseup.net>