Private Kernel Circuit - Tail

Requirements

The tail circuit abstains from processing individual private function calls. Instead, it incorporates the outcomes of a private kernel circuit and conducts additional processing essential for generating the final public inputs suitable for submission to the transaction pool, subsequently undergoing processing by Sequencers and Provers. The final public inputs must safeguard against revealing any private information unnecessary for the execution of public kernel circuits and rollup circuits.

Verification of the Previous Iteration

Verifying the previous kernel proof.

It verifies that the previous iteration was executed successfully with the given proof data, verification key, and public inputs, sourced from private_inputs.previous_kernel.

The preceding proof can be:

• Initial private kernel proof.
• Inner private kernel proof.
• Reset private kernel proof.

An inner iteration may be omitted when there's only a single private function call for the transaction. And a reset iteration can be skipped if there are no read requests and transient notes in the public inputs from the last iteration.

Ensuring the previous iteration is the last.

It checks the data within private_inputs.previous_kernel.public_inputs.transient_accumulated_data to ensure that no further private kernel iteration is needed.

1. The following must be empty to ensure all the private function calls are processed:

   • private_call_request_stack

2. The following must be empty to ensure a comprehensive final reset:

   • note_hash_read_requests
   • nullifier_read_requests
   • key_validation_request_contexts
   • The nullifier_counter associated with each note hash in note_hash_contexts.
   • The note_hash_counter associated with each nullifier in nullifier_contexts.

   A reset iteration should ideally precede this step. Although it doesn't have to be executed immediately before the tail circuit, as long as it effectively clears the specified values.

Processing Final Outputs

Siloing values.

Siloing a value with the address of the contract generating the value ensures that data produced by a contract is accurately attributed to the correct contract and cannot be misconstrued as data created in a different contract. This circuit guarantees the following siloed values:

1. Silo nullifiers:

   For each nullifier at index i > 0 in the nullifier_contexts within private_inputs, if nullifier.value != 0:

   nullifier_contexts[i].value = hash(nullifier.contract_address, nullifier.value)

   This process does not apply to nullifier_contexts[0], which is the hash of the transaction request created by the initial private kernel circuit.

2. Silo note_hashes:

   For each note_hash at index i in the note_hash_contexts within private_inputs, if note_hash.value != 0:

   note_hash_contexts[i].value = hash(note_nonce, siloed_hash)

   Where:

   • note_nonce = hash(first_nullifier, index)
     • first_nullifier = nullifier_contexts[0].value.
     • index = note_hash_hints[i], which is the index of the same note hash within public_inputs.note_hashes. Where note_hash_hints is provided as hints via private_inputs.
   • siloed_hash = hash(note_hash.contract_address, note_hash.value)

   Siloing with a note_nonce guarantees that each final note hash is a unique value in the note hash tree.

3. Silo l2_to_l1_messages:

   For each l2_to_l1_message at index i in l2_to_l1_message_contexts within [private_inputs], if l2_to_l1_message.value != 0:

   l2_to_l1_message_contexts[i].value = hash(l2_to_l1_message.contract_address, version_id, l2_to_l1_message.portal_contract_address, chain_id, l2_to_l1_message.value)

   Where version_id and chain_id are defined in public_inputs.constant_data.tx_context.

4. Silo unencrypted_log_hashes:

   For each log_hash at index i in the unencrypted_log_hash_contexts within private_inputs, if log_hash.hash != 0:

   unencrypted_log_hash_contexts[i].value = hash(log_hash.hash, log_hash.contract_address)

5. Silo encrypted_log_hashes:

   For each log_hash at index i in the encrypted_log_hash_contexts within private_inputs, if log_hash.hash != 0:

   encrypted_log_hash_contexts[i].value = hash(log_hash.hash, contract_address_tag)

   Where contract_address_tag = hash(log_hash.contract_address, log_hash.randomness)

Verifying and splitting ordered data.

The initial and inner kernel iterations may produce values in an unordered state due to the serial nature of the kernel, contrasting with the stack-based nature of code execution.

This circuit ensures the correct ordering of the following:

• note_hashes
• nullifiers
• l2_to_l1_messages
• unencrypted_log_hashes
• encrypted_log_hashes
• encrypted_note_preimage_hashes
• public_call_requests

In addition, the circuit split the ordered data into non_revertible_accumulated_data and revertible_accumulated_data using min_revertible_side_effect_counter.

1. Verify ordered public_call_requests:

   Initialize num_non_revertible and num_revertible to 0.

   For each request at index i in the unordered public_call_request_contexts within private_inputs.previous_kernel.public_inputs.transient_accumulated_data:

   • Find its associated mapped_request in public_call_requests[public_call_request_hints[i]] within public_inputs.
     • If request.counter < min_revertible_side_effect_counter:
       • The public_call_requests is in non_revertible_accumulated_data.
       • num_added = num_non_revertible.
     • If request.counter >= min_revertible_side_effect_counter:
       • The public_call_requests is in revertible_accumulated_data.
       • num_added = num_revertible.
   • If request.call_stack_item_hash != 0, verify that:
     • request == mapped_request
     • If num_added > 0, verify that:
       • public_call_requests[num_added].counter < public_call_requests[num_added - 1].counter
     • Increment num_added by 1: num_(non_)revertible += 1
   • Else:
     • All the subsequent requests (index >= i) in public_call_request_contexts must be empty.
     • All the subsequent requests (index >= num_non_revertible) in non_revertible_accumulated_data.public_call_requests must be empty.
     • All the subsequent requests (index >= num_revertible) in revertible_accumulated_data.public_call_requests must be empty.

   Note that requests in public_call_requests must be arranged in descending order to ensure the calls are executed in chronological order.

2. Verify the rest of the ordered arrays:

   Initialize num_non_revertible and num_revertible to 0.

   For each note_hash_context at index i in the unordered note_hash_contexts within private_inputs.previous_kernel.public_inputs.transient_accumulated_data:

   • Find its associated note_hash in note_hashes[note_hash_hints[i].index] within public_inputs.
     • If note_hash_context.counter < min_revertible_side_effect_counter:
       • The note_hashes is in non_revertible_accumulated_data.
       • num_added = num_non_revertible.
     • If note_hash_context.counter >= min_revertible_side_effect_counter:
       • The note_hashes is in revertible_accumulated_data.
       • num_added = num_revertible.
   • If note_hash_context.value != 0, verify that:
     • note_hash == note_hash_context.value
     • note_hash_hints[note_hash_hints[i].index].counter_(non_)revertible == note_hash_context.counter
     • If num_added > 0, verify that:
       • note_hash_hints[num_added].counter_(non_)revertible > note_hash_hints[num_added - 1].counter_(non_)revertible
     • Increment num_added by 1: num_(non_)revertible += 1
   • Else:
     • All the subsequent elements (index >= i) in note_hash_contexts must be empty.
     • All the subsequent elements (index >= num_non_revertible) in non_revertible_accumulated_data.note_hashes must be empty.
     • All the subsequent elements (index >= num_revertible) in revertible_accumulated_data.note_hashes must be empty.

   Repeat the same process for nullifiers, l2_to_l1_messages, unencrypted_log_hashes, encrypted_log_hashes, and encrypted_note_preimage_hashes, where:

   • Ordered nullifiers and l2_to_l1_messages are within public_inputs.
   • ordered_unencrypted_log_hashes_(non_)revertible, ordered_encrypted_log_hashes_(non_)revertible, and ordered_encrypted_note_preimage_hashes_(non_)revertible are provided as hints through private_inputs.

While ordering could occur gradually in each kernel iteration, the implementation is much simpler and typically more efficient to be done once in the tail circuit.

Recalibrating counters.

While the counter of a public_call_request is initially assigned in the private function circuit to ensure proper ordering within the transaction, it should be modified in this step. As using counter values obtained from private function circuits may leak information.

The requests in the public_call_requests within public_inputs have been sorted in descending order in the previous step. This circuit recalibrates their counters through the following steps:

• The counter of the last non-empty request is set to 1.
• The counters of the other non-empty requests are continuous values in descending order:
  • public_call_requests[i].counter = public_call_requests[i + 1].counter + 1

It's crucial for the counter of the last request to be 1, as it's assumed in the tail public kernel circuit that no storage writes have a counter 1.

Validating Public Inputs

Verifying the (non-)revertible accumulated data.

1. The following must align with the results after ordering, as verified in a previous step:

   • note_hashes
   • nullifiers
   • l2_to_l1_messages

2. The public_call_requests must adhere to a specific order with recalibrated counters, as verified in the previous steps.

3. The hashes and lengths for all logs are accumulated as follows:

   For each non-empty log_hash at index i in ordered_unencrypted_log_hashes_(non_)revertible, which is provided as hints, and the ordering was verified against the siloed hashes in previous steps:

   • accumulated_logs_hash = hash(accumulated_logs_hash, log_hash.hash)
     • If i == 0: accumulated_logs_hash = log_hash.hash
   • accumulated_logs_length += log_hash.length

   Check the values in the public_inputs are correct:

   • unencrypted_logs_hash == accumulated_logs_hash
   • unencrypted_log_preimages_length == accumulated_logs_length

   Repeat the same process for encrypted_logs_hashes and encrypted_note_preimages_hashes.

Verifying the transient accumulated data.

It ensures that all data in the transient_accumulated_data within public_inputs is empty.

Verifying other data.

This section follows the same process as outlined in the inner private kernel circuit.

In addition, it checks that the following are empty:

• old_public_data_tree_snapshot
• new_public_data_tree_snapshot

PrivateInputs

PreviousKernel

The format aligns with the PreviousKernel of the inner private kernel circuit.

Hints

Data that aids in the verifications carried out in this circuit:

Field	Type	Description
note_hash_hints	[OrderHint; MAX_NOTE_HASHES_PER_TX]	Hints for ordering note_hash_contexts.
nullifier_hints	[OrderHint; MAX_NULLIFIERS_PER_TX]	Hints for ordering nullifier_contexts.
public_call_request_hints	[field; MAX_PUBLIC_CALL_STACK_LENGTH_PER_TX]	Indices of ordered public_call_request_contexts.
unencrypted_log_hash_hints	[OrderHint; MAX_UNENCRYPTED_LOG_HASHES_PER_TX]	Hints for ordering unencrypted_log_hash_contexts.
ordered_unencrypted_log_hashes_revertible	[field; MAX_UNENCRYPTED_LOG_HASHES_PER_TX]	Ordered revertible unencrypted_log_hashes.
ordered_unencrypted_log_hashes_non_revertible	[field; MAX_UNENCRYPTED_LOG_HASHES_PER_TX]	Ordered non-revertible unencrypted_log_hashes.
encrypted_log_hash_hints	[OrderHint; MAX_ENCRYPTED_LOG_HASHES_PER_TX]	Hints for ordering encrypted_log_hash_contexts.
ordered_encrypted_log_hashes_revertible	[field; MAX_ENCRYPTED_LOG_HASHES_PER_TX]	Ordered revertible encrypted_log_hashes.
ordered_encrypted_log_hashes_non_revertible	[field; MAX_ENCRYPTED_LOG_HASHES_PER_TX]	Ordered non-revertible encrypted_log_hashes.
encrypted_note_preimage_hints	[OrderHint; MAX_ENCRYPTED_NOTE_PREIMAGE_HASHES_PER_TX]	Hints for ordering encrypted_note_preimage_hash_contexts.
ordered_encrypted_note_preimage_hashes_revertible	[field; MAX_ENCRYPTED_NOTE_PREIMAGE_HASHES_PER_TX]	Ordered revertible encrypted_note_preimage_hashes.
ordered_encrypted_note_preimage_hashes_non_revertible	[field; MAX_ENCRYPTED_NOTE_PREIMAGE_HASHES_PER_TX]	Ordered non-revertible encrypted_note_preimage_hashes.

OrderHint

Field	Type	Description
index	field	Index of the mapped element in the ordered array.
counter_revertible	u32	Counter of the element at index i in the revertible ordered array.
counter_non_revertible	u32	Counter of the element at index i in the non-revertible ordered array.

PublicInputs

The format aligns with the Public Inputs of the tail public kernel circuit.