Small value sumcheck

[wu-s-john]

Note: This PR builds on the Barrett reduction infrastructure from #111. The changes span src/big_num/, src/lagrange_accumulator/, and src/small_sumcheck.rs. Reviewers are encouraged to focus on src/lagrange_accumulator/ and src/small_sumcheck.rs — the src/big_num/ changes are covered by #111.

Summary

Implement the small-value sumcheck optimization (Algorithm 6) from "Speeding Up Sum-Check Proving" (Bagad, Dao, Domb, Thaler, IACR 2025/1117). This replaces expensive field multiplications with native integer arithmetic during the first ℓ₀ rounds when polynomial values are guaranteed small.

This PR provides the core infrastructure for the optimization. Integration into the Spartan prover is planned for a follow-up PR.

Benchmarks

Measured on M1 Max MacBook Pro with jemalloc, BN254 scalar field, ℓ₀ = 3.

num_vars	n	baseline (DMR)	small-value	speedup
16	65,536	4.98 ms	3.43 ms	1.45×
17	131,072	8.83 ms	4.79 ms	1.84×
18	262,144	15.07 ms	7.54 ms	2.00×
19	524,288	27.81 ms	12.84 ms	2.17×
20	1,048,576	49.96 ms	26.37 ms	1.90×
21	2,097,152	190.66 ms*	50.73 ms	3.76×*
22	4,194,304	152.75 ms	96.09 ms	1.59×
23	8,388,608	285.86 ms	181.02 ms	1.58×
24	16,777,216	546.06 ms	333.13 ms	1.64×
25	33,554,432	1,095.2 ms	653.01 ms	1.68×
26	67,108,864	2,150.1 ms	1,314.5 ms	1.64×

These results are consistent with the ~1.6-2× speedup and has achieved better overall performance than #98.

Key Components

Lagrange accumulator infrastructure (src/lagrange_accumulator/):

• LagrangeAccumulators: precomputed A_i(v, u) values for all rounds
• LagrangeIndex/LagrangePoint/LagrangeHatPoint: type-safe domain indices
• LagrangeBasisFactory: barycentric Lagrange basis with O(D) evaluation
• extend_to_lagrange_domain: batch extension from {0,1}^ℓ₀ to U_D^ℓ₀
• EqRoundFactor: tracks α = eq(τ_{<i}, r_{<i}) across rounds

Performance optimizations:

• Thread-local SpartanThreadState eliminates per-iteration allocations
• Delayed modular reduction via SignedWideLimbs accumulators
• Skip binary betas (Az·Bz = Cz on {0,1}^n for satisfying witnesses)
• Batched eq-weighted binding in transition phase

API:

• SmallValue trait: WideMul + Copy + Zero + Add + Sub + Send + Sync
• SmallValueEngine<SV>: blanket impl consolidates field requirements
• prove_cubic_small_value<E, SV, const LB>: main entry point

Test Plan

• prove_cubic_small_value produces identical proofs to prove_cubic_with_three_inputs (equivalence tests)
• cargo test -- --skip test_msm_ux
• cargo clippy

[Copilot]

Pull request overview

This PR adds the core implementation of the “small-value sumcheck” optimization (Algorithm 6 from IACR 2025/1117) by introducing Lagrange-accumulator infrastructure and a new small-value sumcheck prover path that uses native integer arithmetic and delayed reduction during the first ℓ₀ rounds.

Changes:

• Add src/lagrange_accumulator/ with domain types, Lagrange basis/extension logic, accumulator structures, and an optimized accumulator builder (including thread-local scratch state).
• Add src/small_sumcheck.rs implementing prove_cubic_small_value and a batched transition binding step from small-integer polynomials into field polynomials.
• Extend big-number infrastructure to support field×small delayed reduction via Barrett reduction (plus provider wiring and tests), and update sumcheck types to support equality comparisons.

Reviewed changes

Copilot reviewed 29 out of 29 changed files in this pull request and generated 5 comments.

Show a summary per file

File	Description
src/sumcheck.rs	Adjust EqSumCheckInstance API to accept tau slices; add EqSumCheckInstance::from_pyramids; make SumcheckProof comparable and expose compressed polys internally.
src/small_sumcheck.rs	New small-value sumcheck prover (ℓ₀ rounds via accumulators + transition + remaining rounds via eq-sumcheck instance).
src/provider/pt256.rs	Wire Barrett + small-value + delayed-reduction traits/tests for P256/T256 scalar fields.
src/provider/pasta.rs	Wire Barrett + small-value + delayed-reduction traits/tests for Pasta scalar fields.
src/provider/bn254.rs	Wire Barrett + small-value + delayed-reduction traits/tests for BN254 scalar field.
src/polys/univariate.rs	Make CompressedUniPoly derive PartialEq, Eq for proof equality checks.
src/polys/multilinear.rs	Generalize MultilinearPolynomial over coefficient type; move binding ops behind Field; add small-value helpers.
src/polys/eq.rs	Add eq “pyramid” builders used for accumulator building and pyramid reuse.
src/lib.rs	Register new internal modules (lagrange_accumulator, small_sumcheck).
src/lagrange_accumulator/mod.rs	New module root and re-exports for accumulator infrastructure.
src/lagrange_accumulator/thread_state.rs	Add per-rayon-thread scratch buffers to reduce allocations in accumulator building.
src/lagrange_accumulator/index.rs	Implement β→(round,v,u,y) index mapping (Definition A.5) for scatter.
src/lagrange_accumulator/extension.rs	Implement boolean→Lagrange-domain extension (Procedure 6).
src/lagrange_accumulator/evals.rs	Add compact evaluation containers for U_d and Û_d points.
src/lagrange_accumulator/eq_round.rs	Add per-round eq-factor tracking and t(1) derivation helper.
src/lagrange_accumulator/domain.rs	Add type-safe domain/index types for U_d/Û_d.
src/lagrange_accumulator/csr.rs	Add CSR container for compact storage of per-β index lists.
src/lagrange_accumulator/basis.rs	Add barycentric Lagrange basis factory and tensor coefficients.
src/lagrange_accumulator/accumulator.rs	Add round accumulators and accumulator collection types.
src/lagrange_accumulator/accumulator_builder.rs	Implement optimized accumulator builder for Spartan’s cubic relation and return reusable eq pyramids.
src/errors.rs	Add new error variants for small-value overflow and invalid ℓ₀ selection.
src/big_num/wide_mul.rs	Add widening multiplication trait for small integers.
src/big_num/small_value_field.rs	Add small-integer↔field conversion traits/helpers and tests.
src/big_num/mod.rs	Export Barrett/small-value modules and re-export new traits/types.
src/big_num/macros.rs	Add macros for Barrett constants, SmallValueField impls, and delayed-reduction impls for small ints.
src/big_num/limbs.rs	Add signed wide accumulators, magnitude subtraction helper, and limb utilities used by Barrett/DMR.
src/big_num/field_reduction_constants.rs	Introduce BarrettReductionConstants and corresponding test helpers.
src/big_num/delayed_reduction.rs	Add accumulation helpers for field×small and field×i128; add tests for small-value delayed reduction.
src/big_num/barrett.rs	Implement μ-Barrett reduction for 6- and 7-limb inputs plus test macros.

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