feat: specific membership witness functions in aztec-nr

[benesjan]

Fixes #3663

Checklist:

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• If the pull request requires a cryptography review (e.g. cryptographic algorithm implementations) I have added the 'crypto' tag.
• I have reviewed my diff in github, line by line and removed unexpected formatting changes, testing logs, or commented-out code.
• Every change is related to the PR description.
• I have linked this pull request to relevant issues (if any exist).

[AztecBot]

Benchmark results

Metrics with a significant change:

• node_database_size_in_bytes (10): 6,696,504 (+71%)
• note_history_trial_decrypting_time_in_ms (10): 219 (+44%)
• note_trial_decrypting_time_in_ms (32): 76.6 (+93%)

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.

Values are compared against data from master at commit 9ccbbd96 and shown if the difference exceeds 1%.

L2 block published to L1

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

Metric	8 txs	32 txs	128 txs
l1_rollup_calldata_size_in_bytes	45,444	179,588	716,132
l1_rollup_calldata_gas	222,924	868,280	3,448,160
l1_rollup_execution_gas	842,011	3,595,388	22,203,529
l2_block_processing_time_in_ms	2,197 (-1%)	8,425 (-1%)	34,676 (-2%)
note_successful_decrypting_time_in_ms	343 (-4%)	1,039 (-1%)	3,802 (-2%)
note_trial_decrypting_time_in_ms	53.1 (-25%)	⚠️ 76.6 (+93%)	138 (-1%)
l2_block_building_time_in_ms	20,157	80,316	322,294
l2_block_rollup_simulation_time_in_ms	16,607 (-1%)	66,224	266,305
l2_block_public_tx_process_time_in_ms	3,519 (+1%)	14,023 (+1%)	55,721

L2 chain processing

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

Metric	5 blocks	10 blocks
node_history_sync_time_in_ms	24,458 (-2%)	47,696 (-1%)
note_history_successful_decrypting_time_in_ms	2,503 (-6%)	5,162 (-2%)
note_history_trial_decrypting_time_in_ms	79.9 (-12%)	⚠️ 219 (+44%)
node_database_size_in_bytes	3,410,848 (-6%)	⚠️ 6,696,504 (+71%)
pxe_database_size_in_bytes	29,940	59,499

Circuits stats

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

Circuit	circuit_simulation_time_in_ms	circuit_input_size_in_bytes	circuit_output_size_in_bytes
private-kernel-init	199	43,109	20,441
private-kernel-ordering	115	25,833	9,689
base-rollup	2,968	667,692	873
root-rollup	95.2 (+5%)	4,072	881
private-kernel-inner	260	64,516	20,441
public-kernel-private-input	171 (-1%)	25,203	20,441
public-kernel-non-first-iteration	170 (+1%)	25,245	20,441
merge-rollup	10.9	2,592	873

Miscellaneous

Transaction sizes based on how many contracts are deployed in the tx.

Metric	0 deployed contracts	1 deployed contracts
tx_size_in_bytes	10,323	26,359 (+2%)

[LHerskind]

Looks good. A small comment on unifying it, but this can also do.