feat: protogalaxy combiner quotient

[maramihali]

This PR implements the computation of combiner quotient ($K$ polynomial in the paper) together with the necessary updates in various parts of the codebase:

• In the Barycentric and Univariate classes , we can now provide evaluations with custom domain_start and domain_end rather than assuming the list of evaluations is for points starting from 0 onwards
• In the common functionality shared between Sumcheck and Protogalaxy, I ensure the version of pow polynomial from PG paper can be used.

Checklist:

Remove the checklist to signal you've completed it. Enable auto-merge if the PR is ready to merge.

• 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).

[maramihali]

Until we handle alpha properly, this test doesn't really make sense so I decided to remove it.

[AztecBot]

Benchmark results

Metrics with a significant change:

• note_trial_decrypting_time_in_ms (8): 97.2 (+1701%)

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 ede7a47f 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	223,008	867,860	3,449,612
l1_rollup_execution_gas	842,095	3,594,968	22,204,981
l2_block_processing_time_in_ms	2,046 (+1%)	7,619 (-1%)	29,882 (-1%)
note_successful_decrypting_time_in_ms	295 (-2%)	870 (-2%)	3,175 (-1%)
note_trial_decrypting_time_in_ms	⚠️ 97.2 (+1701%)	77.1 (-20%)	133 (+1%)
l2_block_building_time_in_ms	13,367 (-1%)	52,949 (-1%)	210,081 (-2%)
l2_block_rollup_simulation_time_in_ms	12,000	47,633 (-1%)	189,024 (-2%)
l2_block_public_tx_process_time_in_ms	1,321 (-1%)	5,182 (-1%)	20,560 (-2%)

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	22,147 (+2%)	42,682
note_history_successful_decrypting_time_in_ms	2,096 (+2%)	4,083 (+3%)
note_history_trial_decrypting_time_in_ms	121 (+1%)	186 (+27%)
node_database_size_in_bytes	1,629,840	1,098,457
pxe_database_size_in_bytes	29,748	59,307

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	762 (-1%)	61,697	18,905
private-kernel-ordering	125 (-1%)	24,297	8,153
base-rollup	1,768 (-1%)	656,311	814
root-rollup	169	4,072	1,097
private-kernel-inner	787 (-1%)	81,568	18,905
public-kernel-private-input	41.3	41,519	18,841
public-kernel-non-first-iteration	26.0 (-1%)	41,497	18,841
merge-rollup	15.2	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	8,787	27,547

[codygunton]

Two little things + a merge then we're good to go.