feat: added barrett_reduction implementation into uintx

[zac-williamson]

This PR adds a barrett_reduction method into unitx, a fast division algorithm when the divisor is known ahead of time such that precomputed factors can be determined.

barrett_reduction is used to speed up divmod for some important hardcoded moduli. Or particular relevance is the prime field associated with BN254 curve arithmetic, as expensive 1024-bit divmod operations are performed when computing witnesses within stdlib::bitfield - commonly used to perform non-native BN254 curve arithmetic.

Speeds up biggroup batch_mul 4x

[AztecBot]

Benchmark results

Metrics with a significant change:

• proof_construction_time_poseidon_hash_ms (1): 236 (+203%)
• proof_construction_time_poseidon_hash_ms (4): 80.0 (+135%)
• proof_construction_time_poseidon_hash_ms (16): 58.0 (+71%)
• proof_construction_time_poseidon_hash_ms (32): 94.0 (+62%)
• proof_construction_time_poseidon_hash_ms (64): 149 (+69%)
• proof_construction_time_poseidon_hash_100_ms (1): 11,358 (+98%)
• proof_construction_time_poseidon_hash_100_ms (4): 3,125 (+98%)
• proof_construction_time_poseidon_hash_100_ms (16): 1,395 (+93%)
• proof_construction_time_poseidon_hash_100_ms (32): 1,412 (+83%)
• proof_construction_time_poseidon_hash_100_ms (64): 1,450 (+82%)
• proof_construction_time_poseidon_hash_30_ms (1): 5,740 (+279%)
• proof_construction_time_poseidon_hash_30_ms (4): 1,575 (+277%)
• proof_construction_time_poseidon_hash_30_ms (16): 725 (+261%)
• proof_construction_time_poseidon_hash_30_ms (32): 776 (+253%)
• proof_construction_time_poseidon_hash_30_ms (64): 797 (+194%)
• app_circuit_witness_generation_time_in_ms (MultiCallEntrypoint:entrypoint): 1,528 (-18%)
• app_circuit_witness_generation_time_in_ms (SchnorrAccount:constructor): 1,110 (-29%)
• app_circuit_witness_generation_time_in_ms (SchnorrAccount:entrypoint): 2,195 (-17%)
• app_circuit_witness_generation_time_in_ms (Token:transfer): 3,990 (-20%)
• app_circuit_witness_generation_time_in_ms (Benchmarking:create_note): 985 (-30%)
• protocol_circuit_simulation_time_in_ms (base-rollup): 774 (-91%)
• protocol_circuit_simulation_time_in_ms (public-kernel-setup): 642 (-25%)
• batch_insert_into_indexed_tree_20_depth_ms (4096): 4,935 (+80%)
• batch_insert_into_indexed_tree_20_depth_hash_ms (4096): 0.880 (+84%)
• protocol_circuit_witness_generation_time_in_ms (base-parity): 1,156 (+80%)
• protocol_circuit_proving_time_in_ms (root-parity): 48,721 (+16%)
• protocol_circuit_proving_time_in_ms (root-rollup): 20,632 (-16%)
• l2_block_public_tx_process_time_in_ms (8): 25,373 (-33%)
• l2_block_public_tx_process_time_in_ms (32): 112,279 (-31%)
• l2_block_public_tx_process_time_in_ms (64): 222,259 (-34%)
• l2_block_processing_time_in_ms (64): 12,552 (+32%)

Detailed results

All benchmarks are run on txs on the Benchmarking contract on the repository. Each tx consists of a batch call to create_note and increment_balance, which guarantees that each tx has a private call, a nested private call, a public call, and a nested public call, as well as an emitted private note, an unencrypted log, and public storage read and write.

This benchmark source data is available in JSON format on S3 here.

Proof generation

Each column represents the number of threads used in proof generation.

Metric	1 threads	4 threads	16 threads	32 threads	64 threads
proof_construction_time_sha256_ms	5,717	1,544 (-1%)	689 (-3%)	755	780 (+1%)
proof_construction_time_sha256_30_ms	11,738 (+3%)	3,145 (+3%)	1,410 (+3%)	1,448 (+2%)	1,467 (+2%)
proof_construction_time_sha256_100_ms	44,727 (+2%)	12,055 (+3%)	5,719 (+6%)	5,806 (+8%)	5,493 (+3%)
proof_construction_time_poseidon_hash_ms	⚠️ 236 (+203%)	⚠️ 80.0 (+135%)	⚠️ 58.0 (+71%)	⚠️ 94.0 (+62%)	⚠️ 149 (+69%)
proof_construction_time_poseidon_hash_30_ms	⚠️ 5,740 (+279%)	⚠️ 1,575 (+277%)	⚠️ 725 (+261%)	⚠️ 776 (+253%)	⚠️ 797 (+194%)
proof_construction_time_poseidon_hash_100_ms	⚠️ 11,358 (+98%)	⚠️ 3,125 (+98%)	⚠️ 1,395 (+93%)	⚠️ 1,412 (+83%)	⚠️ 1,450 (+82%)

L2 block published to L1

Each column represents the number of txs on an L2 block published to L1.

Metric	8 txs	32 txs	64 txs
l1_rollup_calldata_size_in_bytes	1,412	1,412	1,412
l1_rollup_calldata_gas	9,464	9,476	9,464
l1_rollup_execution_gas	616,105	616,117	616,105
l2_block_processing_time_in_ms	1,287	4,794	⚠️ 12,552 (+32%)
l2_block_building_time_in_ms	46,962 (+8%)	192,827 (+12%)	388,595 (+13%)
l2_block_rollup_simulation_time_in_ms	46,789 (+8%)	192,177 (+13%)	387,307 (+13%)
l2_block_public_tx_process_time_in_ms	⚠️ 25,373 (-33%)	⚠️ 112,279 (-31%)	⚠️ 222,259 (-34%)

L2 chain processing

Each column represents the number of blocks on the L2 chain where each block has 16 txs.

Metric	3 blocks	5 blocks
node_history_sync_time_in_ms	9,475 (+1%)	14,452 (+1%)
node_database_size_in_bytes	14,475,344	21,348,432
pxe_database_size_in_bytes	18,071	29,868

Circuits stats

Stats on running time and I/O sizes collected for every kernel circuit run across all benchmarks.

Circuit	simulation_time_in_ms	witness_generation_time_in_ms	proving_time_in_ms	input_size_in_bytes	output_size_in_bytes	proof_size_in_bytes	num_public_inputs	size_in_gates
private-kernel-init	140 (+4%)	478 (+1%)	13,269 (-4%)	20,630	64,614	89,536	2,731	524,288
private-kernel-inner	431 (+5%)	945 (-1%)	45,491 (-8%)	92,318	64,614	89,536	2,731	2,097,152
private-kernel-tail	596	2,702 (+1%)	40,342 (-12%)	96,541	77,732	11,648	297	2,097,152
base-parity	6.56 (+2%)	⚠️ 1,156 (+80%)	2,857 (+2%)	128	64.0	2,208	2.00	131,072
root-parity	50.0 (+2%)	60.5 (-11%)	⚠️ 48,721 (+16%)	27,084	64.0	2,720	18.0	2,097,152
base-rollup	⚠️ 774 (-91%)	2,444 (+2%)	86,610 (+8%)	119,734	756	3,648	47.0	4,194,304
root-rollup	110	78.1 (+3%)	⚠️ 20,632 (-16%)	25,297	620	3,456	41.0	1,048,576
public-kernel-app-logic	553 (+1%)	3,107 (-14%)	50,811 (+10%)	105,253 (-1%)	86,550	116,768	3,582	2,097,152
public-kernel-tail	1,162 (+3%)	24,527 (+1%)	188,382 (+5%)	401,002	7,646	11,648	297	8,388,608
private-kernel-reset-small	616 (+3%)	1,947 (-14%)	50,277 (-2%)	120,733	64,614	89,536	2,731	2,097,152
merge-rollup	28.7 (-8%)	N/A	N/A	16,534	756	N/A	N/A	N/A
public-kernel-setup	⚠️ 642 (-25%)	N/A	N/A	105,253 (-1%)	86,550	N/A	N/A	N/A
public-kernel-teardown	542 (-15%)	N/A	N/A	105,253 (-1%)	86,550	N/A	N/A	N/A
private-kernel-tail-to-public	N/A	8,223 (-1%)	101,751 (-3%)	N/A	N/A	116,768	3,582	4,194,304

Stats on running time collected for app circuits

Function	input_size_in_bytes	output_size_in_bytes	witness_generation_time_in_ms	proof_size_in_bytes	proving_time_in_ms	size_in_gates	num_public_inputs
ContractClassRegisterer:register	1,344	9,944	466 (-1%)	N/A	N/A	N/A	N/A
ContractInstanceDeployer:deploy	1,408	9,944	41.1 (-2%)	N/A	N/A	N/A	N/A
MultiCallEntrypoint:entrypoint	1,920	9,944	⚠️ 1,528 (-18%)	N/A	N/A	N/A	N/A
SchnorrAccount:constructor	1,312	9,944	⚠️ 1,110 (-29%)	N/A	N/A	N/A	N/A
SchnorrAccount:entrypoint	2,304	9,944	⚠️ 2,195 (-17%)	16,768	54,095 (-1%)	2,097,152	457
Token:privately_mint_private_note	1,280	9,944	1,409 (-10%)	N/A	N/A	N/A	N/A
Token:transfer	1,376	9,944	⚠️ 3,990 (-20%)	16,768	57,416 (-1%)	2,097,152	457
Benchmarking:create_note	1,312 (-2%)	9,944	⚠️ 985 (-30%)	N/A	N/A	N/A	N/A
FPC:fee_entrypoint_public	1,344	9,944	224 (-7%)	N/A	N/A	N/A	N/A
SchnorrAccount:spend_private_authwit	1,280	9,944	77.3	N/A	N/A	N/A	N/A
Token:unshield	1,376	9,944	3,249 (-14%)	N/A	N/A	N/A	N/A
FPC:fee_entrypoint_private	1,376	9,944	4,018 (-14%)	N/A	N/A	N/A	N/A

Tree insertion stats

The duration to insert a fixed batch of leaves into each tree type.

Metric	1 leaves	16 leaves	64 leaves	128 leaves	512 leaves	1024 leaves	2048 leaves	4096 leaves	32 leaves
batch_insert_into_append_only_tree_16_depth_ms	10.5 (+1%)	17.1 (+1%)	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_count	16.7	31.8	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_ms	0.607 (+1%)	0.524 (+1%)	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_32_depth_ms	N/A	N/A	50.0 (+3%)	76.3 (+1%)	247	476	929	1,848 (+1%)	N/A
batch_insert_into_append_only_tree_32_depth_hash_count	N/A	N/A	95.9	159	543	1,055	2,079	4,127	N/A
batch_insert_into_append_only_tree_32_depth_hash_ms	N/A	N/A	0.510 (+3%)	0.470 (+1%)	0.449	0.444	0.440	0.441 (+1%)	N/A
batch_insert_into_indexed_tree_20_depth_ms	N/A	N/A	60.0 (+3%)	113 (+1%)	357 (+1%)	701	1,386	⚠️ 4,935 (+80%)	N/A
batch_insert_into_indexed_tree_20_depth_hash_count	N/A	N/A	106	208	692	1,363	2,707	5,395	N/A
batch_insert_into_indexed_tree_20_depth_hash_ms	N/A	N/A	0.520 (+3%)	0.504	0.483 (+1%)	0.481	0.480	⚠️ 0.880 (+84%)	N/A
batch_insert_into_indexed_tree_40_depth_ms	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	65.0 (+4%)
batch_insert_into_indexed_tree_40_depth_hash_count	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	107
batch_insert_into_indexed_tree_40_depth_hash_ms	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	0.576 (+4%)

Miscellaneous

Transaction sizes based on how many contract classes are registered in the tx.

Metric	0 registered classes	1 registered classes
tx_size_in_bytes	84,050	665,267

Transaction size based on fee payment method

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