perf: share precomputed zeta-powers across ComputeEvaluationsAtZeta

[gbotrel]

Summary

Step 5 of `plan_optims.md`. The Horner loop in `EvaluateAtExt` / `ExtEvaluateAtExt` is sequential per task — n ext-mul + n ext-add + a `bits.Reverse64` per row. pprof showed it at 10–16% of total CPU on the synth bench, second only to the Merkle-node permutation.

Replace with one SIMD-accelerated inner product:

• Precompute the bit-reversed powers of zeta once, shared across every task that evaluates an n-row polynomial at the same point. Build cost is amortised over all tasks (typically 1–2 distinct (n, zeta) pairs per ComputeEvaluationsAtZeta call); construction itself is row-chunked via `powUint64Ext` binary-exponentiation seeding.
• Per-task work becomes `d.FFTInverse(_p, fft.DIF, ...)` followed by `ext.Vector(zPow).InnerProductByElement(_p)` (or `InnerProduct` on the ext rail). Both have AVX-512 paths in gnark-crypto.

Powers are stored in bit-reversed-canonical order so the DIF FFT output can feed straight into the inner product — no extra `BitReverse` pass.

Numbers

32-core EPYC 9R45, synth bench at 100 M cells, median of 3:

shape	before	after	speedup
tall (2²² × 24)	5.55 s	4.82 s	1.15×
wide (2¹⁶ × 1536)	1.25 s	1.20 s	1.04×

`evaluations-at-zeta` phase:

shape	before	after	speedup
tall	450 ms	149 ms	3.0×
wide	57 ms	7 ms	8.1×

The phase-level speedup overshoots prior estimates because Horner was paying for both ext-mul-chain and per-row `bits.Reverse64` indexing; the SIMD inner product retires both.

Test plan

• `go test ./... -count=1` — all green
• `go test -race ./internal/poly ./prover` — no races
• Every existing prover end-to-end test exercises `Verify` — an off-by-one in the power-table order or a wrong-rail inner product would surface

🤖 Generated with Claude Code

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Note

Medium Risk
Changes core polynomial evaluation math on the proving hot path; wrong power ordering would break proofs, but new tests parity-check against the prior Horner implementation.

Overview
Speeds up evaluations-at-zeta by replacing the per-column Horner loop after FFTInverse with a shared, bit-reversed ζ-power table and a single SIMD inner product (InnerProductByElement / InnerProduct).

poly gains BuildZPowBitReversed (parallel O(n) build, bit-reversed order to match DIF output) and EvaluateAtExtWithZPow / ExtEvaluateAtExtWithZPow. ComputeEvaluationsAtZeta caches one table per distinct (n, evalPoint) and uses the fast path for n > 1, keeping the old evaluators only for length-1 polynomials.

Tests assert the new path matches EvaluateAtExt / ExtEvaluateAtExt across sizes and ζ values.

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[Copilot]

Pull request overview

This PR aims to improve prover performance by adding shared zeta-power evaluation and broadening parallelism across several prover hot paths. It also adds benchmarking/profiling infrastructure to measure phase-level performance.

Changes:

• Adds shared bit-reversed zeta-power tables and SIMD inner-product evaluation for evaluations at zeta.
• Introduces parallelism/capped FFT task plumbing in Merkle, FRI, RS commitment, quotient, and DEEP quotient paths.
• Adds phase callbacks, synthetic benchmark tooling, and optimisation planning documentation.

Reviewed changes

Copilot reviewed 13 out of 13 changed files in this pull request and generated 2 comments.

Show a summary per file

File	Description
prover/prover.go	Reworks zeta evaluations, AIR quotient FFT task caps, DEEP quotient accumulation, and phase timing callbacks.
prover/bench_deep_quotient_test.go	Adds synthetic wide/tall prover benchmarks.
plan_optims.md	Adds optimisation roadmap and measurement guidance.
internal/reedsolomon/rs.go	Threads optional FFT options through RS encoding.
internal/poly/utils.go	Adds base-field binary exponentiation helper.
internal/poly/utils_test.go	Adds tests for PowUint64.
internal/poly/ext.go	Adds zeta-power table construction and WithZPow evaluation paths.
internal/poly/compute_quotient.go	Adds quotient parallel thresholds and parallel FFT/scaling loops.
internal/parallel/parallel.go	Adds task-budget and thresholded execution helpers.
internal/merkle/tree.go	Parallelizes internal-node tree construction by level.
internal/fri/fri.go	Parallelizes FRI folding and reuses parallel leaf hashing.
internal/commitment/commitment.go	Parallelizes RS encoding and exposes parallel leaf hashing helper.
bench/synth/main.go	Adds a synthetic proving benchmark/profiling command.

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