feat: Poseidon2 gates for Ultra arithmetisation

[maramihali]

Add Poseidon2 gates to the UltraCircuitBuilder which now ensures that recursive verifier instantiated with the Ultra arithmetisation produce the correct number of constraints.

Updates required:

• change verification key length and constant proof length constants across the codebase (two selectors from the new gate whose commitments need to be in the vk and the poseidon relation becomes the one with the highest degree); changes to Prover.toml accordingly
• ensure the ultra recursive verifier still stays constant size now that hashing produces gates
• small modification to solidity verifer to reflect the ones in cpp with the caveat that the UltraKeccak flavor still doesnt support Poseidon gate (changes coming in a followup PR)

Tube circuit changes in # of gates (post finalisation):

• number of gates prior this change, in master: 13947018
• number of gates post this change: 14038982

Closes https://github.com/AztecProtocol/barretenberg/issues/1041

[github-actions[bot]]

Changes to circuit sizes

Generated at commit: 8d6a989ea79f0c445b115239c3507c703bd15e85, compared to commit: 70e61f973de063a972c726303f579ef34441d85f

🧾 Summary (100% most significant diffs)

Program	ACIR opcodes (+/-)	%	Circuit size (+/-)	%
public_kernel_setup	0 ➖	0.00%	+87,604 ❌	+4.88%
public_kernel_app_logic	0 ➖	0.00%	+87,605 ❌	+4.88%
public_kernel_teardown	0 ➖	0.00%	+87,604 ❌	+4.88%
rollup_merge	0 ➖	0.00%	+94,698 ❌	+3.63%
parity_root	0 ➖	0.00%	+185,085 ❌	+3.55%
rollup_root	0 ➖	0.00%	+100,051 ❌	+2.53%
public_kernel_tail	0 ➖	0.00%	-5,242,812 ✅	-56.27%
rollup_base	0 ➖	0.00%	-5,453,580 ✅	-60.19%

---

Full diff report 👇

Program	ACIR opcodes (+/-)	%	Circuit size (+/-)	%
public_kernel_setup	214,252 (0)	0.00%	1,882,602 (+87,604)	+4.88%
public_kernel_app_logic	216,339 (0)	0.00%	1,882,864 (+87,605)	+4.88%
public_kernel_teardown	217,364 (0)	0.00%	1,883,586 (+87,604)	+4.88%
rollup_merge	319 (0)	0.00%	2,703,744 (+94,698)	+3.63%
parity_root	374 (0)	0.00%	5,393,026 (+185,085)	+3.55%
rollup_root	744 (0)	0.00%	4,051,186 (+100,051)	+2.53%
public_kernel_tail	947,390 (0)	0.00%	4,073,918 (-5,242,812)	-56.27%
rollup_base	347,730 (0)	0.00%	3,607,548 (-5,453,580)	-60.19%

[AztecBot]

Benchmark results

Metrics with a significant change:

• protocol_circuit_simulation_time_in_ms (private-kernel-inner): 225 (+46%)
• protocol_circuit_simulation_time_in_ms (private-kernel-tail-to-public): 673 (-39%)
• avm_simulation_time_ms (Token:transfer_public): 18.9 (-31%)
• protocol_circuit_proving_time_in_ms (undefined): 78,488 (+18%)

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	5,780 (-1%)	1,565	709	750	779 (+2%)
proof_construction_time_sha256_30_ms	11,552 (-3%)	3,080 (-4%)	1,375 (-2%)	1,431	1,462
proof_construction_time_sha256_100_ms	44,163 (-3%)	11,803 (-1%)	5,416	5,401 (-1%)	5,682 (+6%)
proof_construction_time_poseidon_hash_ms	79.0	34.0	34.0	59.0	88.0
proof_construction_time_poseidon_hash_30_ms	1,534 (-1%)	424	204	235 (+6%)	270 (+1%)
proof_construction_time_poseidon_hash_100_ms	5,664 (-1%)	1,517	677	738 (+1%)	751 (+1%)

L2 block published to L1

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

Metric	4 txs	8 txs	16 txs
l1_rollup_calldata_size_in_bytes	4,324	7,844	14,852
l1_rollup_calldata_gas	49,720	92,414	177,632
l1_rollup_execution_gas	1,373,748	2,107,338	3,892,612
l2_block_processing_time_in_ms	254 (-2%)	440 (-9%)	798 (-4%)
l2_block_building_time_in_ms	8,894 (-1%)	17,355 (-1%)	34,822
l2_block_rollup_simulation_time_in_ms	8,893 (-1%)	17,354 (-1%)	34,821
l2_block_public_tx_process_time_in_ms	7,505	15,859 (-1%)	33,271

L2 chain processing

Each column represents the number of blocks on the L2 chain where each block has 8 txs.

Metric	3 blocks	5 blocks
node_history_sync_time_in_ms	2,939 (-6%)	3,883 (-5%)
node_database_size_in_bytes	12,636,240	16,719,952
pxe_database_size_in_bytes	16,254	26,813

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	input_size_in_bytes	output_size_in_bytes	proving_time_in_ms
private-kernel-init	91.1 (+1%)	401 (+6%)	21,755 (+1%)	44,860	N/A
private-kernel-inner	:warning: 225 (+46%)	702 (+4%)	72,566 (+1%)	45,007	N/A
private-kernel-reset-tiny	464 (-1%)	869 (+4%)	65,675	44,846	N/A
private-kernel-tail	195 (-1%)	157 (+3%)	50,686 (+1%)	52,257	N/A
base-parity	5.62 (+2%)	N/A	160	96.0	N/A
root-parity	35.2 (+7%)	N/A	73,948 (+7%)	96.0	N/A
base-rollup	2,728 (-1%)	N/A	189,136 (+1%)	664	N/A
root-rollup	40.2 (+5%)	N/A	58,173 (+7%)	716	N/A
public-kernel-setup	83.9 (+2%)	N/A	105,085 (+1%)	71,222	N/A
public-kernel-app-logic	94.7	N/A	104,911 (+1%)	71,222	N/A
public-kernel-tail	551	N/A	410,534	16,414	N/A
private-kernel-reset-small	467 (+2%)	N/A	66,341	45,629	N/A
private-kernel-tail-to-public	:warning: 673 (-39%)	646 (+2%)	460,796	1,825 (+8%)	N/A
public-kernel-teardown	82.6 (+2%)	N/A	105,349 (+1%)	71,222	N/A
merge-rollup	19.9 (+4%)	N/A	38,174 (+7%)	664	N/A
undefined	N/A	N/A	N/A	N/A	:warning: 78,488 (+18%)

Stats on running time collected for app circuits

Function	input_size_in_bytes	output_size_in_bytes	witness_generation_time_in_ms
ContractClassRegisterer:register	1,344	11,731	342
ContractInstanceDeployer:deploy	1,408	11,731	18.3 (+1%)
MultiCallEntrypoint:entrypoint	1,920	11,731	405
FeeJuice:deploy	1,376	11,731	388 (+2%)
SchnorrAccount:constructor	1,312	11,731	74.4 (-1%)
SchnorrAccount:entrypoint	2,304	11,731	410
Token:privately_mint_private_note	1,280	11,731	104 (+3%)
FPC:fee_entrypoint_public	1,344	11,731	28.1 (+3%)
Token:transfer	1,312	11,731	232 (+3%)
Benchmarking:create_note	1,344	11,731	88.7
SchnorrAccount:verify_private_authwit	1,280	11,731	27.5
Token:unshield	1,376	11,731	524 (+1%)
FPC:fee_entrypoint_private	1,376	11,731	705 (+1%)

AVM Simulation

Time to simulate various public functions in the AVM.

Function	time_ms	bytecode_size_in_bytes
FeeJuice:_increase_public_balance	54.6 (-2%)	7,739
FeeJuice:set_portal	10.0 (-21%)	2,354
Token:constructor	79.5 (-2%)	26,051
FPC:constructor	52.7 (+1%)	18,001
FeeJuice:mint_public	39.3 (+3%)	5,877
Token:mint_public	69.7 (+2%)	10,917
Token:assert_minter_and_mint	40.9 (+3%)	7,512
AuthRegistry:set_authorized	38.8 (+9%)	4,391
FPC:prepare_fee	231 (+1%)	7,043
Token:transfer_public	:warning: 18.9 (-31%)	39,426
FPC:pay_refund	52.6 (+3%)	10,234
Benchmarking:increment_balance	928 (-1%)	6,563
Token:_increase_public_balance	40.6 (+6%)	8,433
FPC:pay_refund_with_shielded_rebate	63.3 (-4%)	10,783

Public DB Access

Time to access various public DBs.

Function	time_ms
get-nullifier-index	0.155 (-5%)

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	256 leaves	512 leaves	1024 leaves
batch_insert_into_append_only_tree_16_depth_ms	2.17 (-4%)	3.86 (-3%)	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_count	16.8	31.7	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_ms	0.112 (-4%)	0.109 (-3%)	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_32_depth_ms	N/A	N/A	11.5	17.5 (-5%)	30.4 (-3%)	58.4 (-10%)	112 (-3%)
batch_insert_into_append_only_tree_32_depth_hash_count	N/A	N/A	95.9	159	287	543	1,055
batch_insert_into_append_only_tree_32_depth_hash_ms	N/A	N/A	0.110	0.102 (-5%)	0.0992 (-2%)	0.101 (-10%)	0.100 (-3%)
batch_insert_into_indexed_tree_20_depth_ms	N/A	N/A	14.5 (-1%)	25.2 (-8%)	43.0 (-3%)	81.4 (-14%)	159 (-3%)
batch_insert_into_indexed_tree_20_depth_hash_count	N/A	N/A	109	207	355	691	1,363
batch_insert_into_indexed_tree_20_depth_hash_ms	N/A	N/A	0.110	0.102 (-8%)	0.104 (-2%)	0.100 (-16%)	0.102 (-3%)
batch_insert_into_indexed_tree_40_depth_ms	N/A	N/A	16.4 (-5%)	N/A	N/A	N/A	N/A
batch_insert_into_indexed_tree_40_depth_hash_count	N/A	N/A	132	N/A	N/A	N/A	N/A
batch_insert_into_indexed_tree_40_depth_hash_ms	N/A	N/A	0.105 (-5%)	N/A	N/A	N/A	N/A

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	64,779	668,997

Transaction size based on fee payment method

| Metric | | | - | |