eth2.0-specs/specs/core/1_custody-game.md

Ethereum 2.0 Phase 1 -- Custody Game

Notice: This document is a work-in-progress for researchers and implementers.

Table of contents

• Ethereum 2.0 Phase 1 -- Custody Game
  • Table of contents
  • Introduction
  • Terminology
  • Constants
    • Misc
    • Custody game parameters
    • Time parameters
    • Max operations per block
    • Reward and penalty quotients
    • Signature domain types
    • TODO PLACEHOLDER
  • Data structures
    • Custody objects
      • CustodyChunkChallenge
      • CustodyBitChallenge
      • CustodyChunkChallengeRecord
      • CustodyBitChallengeRecord
      • CustodyResponse
    • New beacon operations
      • CustodyKeyReveal
      • EarlyDerivedSecretReveal
    • Phase 0 container updates
      • Validator
      • BeaconState
      • BeaconBlockBody
  • Helpers
    • ceillog2
    • is_valid_merkle_branch_with_mixin
    • get_crosslink_chunk_count
    • legendre_bit
    • custody_subchunkify
    • get_custody_chunk_bit
    • get_chunk_bits_root
    • get_randao_epoch_for_custody_period
    • get_custody_period_for_validator
    • replace_empty_or_append
  • Per-block processing
    • Operations
      • Custody key reveals
      • Early derived secret reveals
      • Chunk challenges
      • Bit challenges
      • Custody responses
  • Per-epoch processing
    • Handling of custody-related deadlines

Introduction

This document details the beacon chain additions and changes in Phase 1 of Ethereum 2.0 to support the shard data custody game, building upon the Phase 0 specification.

Terminology

• Custody game—
• Custody period—
• Custody chunk—
• Custody chunk bit—
• Custody chunk challenge—
• Custody bit—
• Custody bit challenge—
• Custody key—
• Custody key reveal—
• Custody key mask—
• Custody response—
• Custody response deadline—

Constants

Misc

Name	Value
BLS12_381_Q	4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787
MINOR_REWARD_QUOTIENT	2**8 (= 256)
MAX_EPOCHS_PER_CROSSLINK	2**6 (= 64)

Custody game parameters

Name	Value
BYTES_PER_SHARD_BLOCK	2**14 (= 16,384)
BYTES_PER_CUSTODY_CHUNK	2**9 (= 512)
BYTES_PER_CUSTODY_SUBCHUNK	48
CHUNKS_PER_EPOCH	2 * BYTES_PER_SHARD_BLOCK * SLOTS_PER_EPOCH // BYTES_PER_CUSTODY_CHUNK
MAX_CUSTODY_CHUNKS	MAX_EPOCHS_PER_CROSSLINK * CHUNKS_PER_EPOCH
CUSTODY_DATA_DEPTH	ceillog2(MAX_CUSTODY_CHUNKS) + 1
CUSTODY_CHUNK_BIT_DEPTH	ceillog2(MAX_EPOCHS_PER_CROSSLINK * CHUNKS_PER_EPOCH // 256) + 2

Time parameters

Name	Value	Unit	Duration
MAX_CHUNK_CHALLENGE_DELAY	2**11 (= 2,048)	epochs	~9 days
CUSTODY_RESPONSE_DEADLINE	2**14 (= 16,384)	epochs	~73 days
RANDAO_PENALTY_EPOCHS	2**1 (= 2)	epochs	12.8 minutes
EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS	2**14	epochs	~73 days
EPOCHS_PER_CUSTODY_PERIOD	2**11 (= 2,048)	epochs	~9 days
CUSTODY_PERIOD_TO_RANDAO_PADDING	2**11 (= 2,048)	epochs	~9 days
MAX_REVEAL_LATENESS_DECREMENT	2**7 (= 128)	epochs	~14 hours

Max operations per block

Name	Value
MAX_CUSTODY_KEY_REVEALS	2**4 (= 16)
MAX_EARLY_DERIVED_SECRET_REVEALS	1
MAX_CUSTODY_CHUNK_CHALLENGES	2**2 (= 4)
MAX_CUSTODY_BIT_CHALLENGES	2**2 (= 4)
MAX_CUSTODY_RESPONSES	2**5 (= 32)

Reward and penalty quotients

Name	Value
EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE	2**1 (= 2)

Signature domain types

The following types are defined, mapping into DomainType (little endian):

Name	Value
DOMAIN_CUSTODY_BIT_CHALLENGE	6

TODO PLACEHOLDER

Name	Value
PLACEHOLDER	2**32

Data structures

Custody objects

CustodyChunkChallenge

class CustodyChunkChallenge(Container):
    responder_index: ValidatorIndex
    attestation: Attestation
    chunk_index: uint64

CustodyBitChallenge

class CustodyBitChallenge(Container):
    responder_index: ValidatorIndex
    attestation: Attestation
    challenger_index: ValidatorIndex
    responder_key: BLSSignature
    chunk_bits: Bitlist[MAX_CUSTODY_CHUNKS]
    signature: BLSSignature

CustodyChunkChallengeRecord

class CustodyChunkChallengeRecord(Container):
    challenge_index: uint64
    challenger_index: ValidatorIndex
    responder_index: ValidatorIndex
    inclusion_epoch: Epoch
    data_root: Hash
    depth: uint64
    chunk_index: uint64

CustodyBitChallengeRecord

class CustodyBitChallengeRecord(Container):
    challenge_index: uint64
    challenger_index: ValidatorIndex
    responder_index: ValidatorIndex
    inclusion_epoch: Epoch
    data_root: Hash
    chunk_count: uint64
    chunk_bits_merkle_root: Hash
    responder_key: BLSSignature

CustodyResponse

class CustodyResponse(Container):
    challenge_index: uint64
    chunk_index: uint64
    chunk: BytesN[BYTES_PER_CUSTODY_CHUNK]
    data_branch: List[Hash, CUSTODY_DATA_DEPTH]
    chunk_bits_branch: List[Hash, CUSTODY_CHUNK_BIT_DEPTH]
    chunk_bits_leaf: Bitvector[256]

New beacon operations

CustodyKeyReveal

class CustodyKeyReveal(Container):
    # Index of the validator whose key is being revealed
    revealer_index: ValidatorIndex
    # Reveal (masked signature)
    reveal: BLSSignature

EarlyDerivedSecretReveal

Represents an early (punishable) reveal of one of the derived secrets, where derived secrets are RANDAO reveals and custody reveals (both are part of the same domain).

class EarlyDerivedSecretReveal(Container):
    # Index of the validator whose key is being revealed
    revealed_index: ValidatorIndex
    # RANDAO epoch of the key that is being revealed
    epoch: Epoch
    # Reveal (masked signature)
    reveal: BLSSignature
    # Index of the validator who revealed (whistleblower)
    masker_index: ValidatorIndex
    # Mask used to hide the actual reveal signature (prevent reveal from being stolen)
    mask: Hash

Phase 0 container updates

Add the following fields to the end of the specified container objects. Fields with underlying type uint64 are initialized to 0 and list fields are initialized to [].

Validator

class Validator(Container):
    # next_custody_secret_to_reveal is initialised to the custody period
    # (of the particular validator) in which the validator is activated
    # = get_custody_period_for_validator(...)
    next_custody_secret_to_reveal: uint64
    max_reveal_lateness: Epoch

BeaconState

class BeaconState(Container):
    custody_chunk_challenge_records: List[CustodyChunkChallengeRecord, PLACEHOLDER]
    custody_bit_challenge_records: List[CustodyBitChallengeRecord, PLACEHOLDER]
    custody_challenge_index: uint64

    # Future derived secrets already exposed; contains the indices of the exposed validator
    # at RANDAO reveal period % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
    exposed_derived_secrets: Vector[List[ValidatorIndex, PLACEHOLDER],
                                    EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS]

BeaconBlockBody

class BeaconBlockBody(Container):
    custody_chunk_challenges: List[CustodyChunkChallenge, PLACEHOLDER]
    custody_bit_challenges: List[CustodyBitChallenge, PLACEHOLDER]
    custody_responses: List[CustodyResponse, PLACEHOLDER]
    custody_key_reveals: List[CustodyKeyReveal, PLACEHOLDER]
    early_derived_secret_reveals: List[EarlyDerivedSecretReveal, PLACEHOLDER]

Helpers

ceillog2

def ceillog2(x: uint64) -> int:
    return (x - 1).bit_length()

is_valid_merkle_branch_with_mixin

def is_valid_merkle_branch_with_mixin(leaf: Hash,
                                      branch: Sequence[Hash],
                                      depth: uint64,
                                      index: uint64,
                                      root: Hash,
                                      mixin: uint64) -> bool:
    value = leaf
    for i in range(depth):
        if index // (2**i) % 2:
            value = hash(branch[i] + value)
        else:
            value = hash(value + branch[i])
    value = hash(value + mixin.to_bytes(32, "little"))
    return value == root

get_crosslink_chunk_count

def get_custody_chunk_count(crosslink: Crosslink) -> int:
    crosslink_length = min(MAX_EPOCHS_PER_CROSSLINK, crosslink.end_epoch - crosslink.start_epoch)
    return crosslink_length * CHUNKS_PER_EPOCH

legendre_bit

Returns the Legendre symbol (a/q) normalizes as a bit (i.e. ((a/q) + 1) // 2). In a production implementation, a well-optimized library (e.g. GMP) should be used for this.

def legendre_bit(a: int, q: int) -> int:
    if a >= q:
        return legendre_bit(a % q, q)
    if a == 0:
        return 0
    assert(q > a > 0 and q % 2 == 1)
    t = 1
    n = q
    while a != 0:
        while a % 2 == 0:
            a //= 2
            r = n % 8
            if r == 3 or r == 5:
                t = -t
        a, n = n, a
        if a % 4 == n % 4 == 3:
            t = -t
        a %= n
    if n == 1:
        return (t + 1) // 2
    else:
        return 0

custody_subchunkify

Given one proof of custody chunk, returns the proof of custody subchunks of the correct sizes.

def custody_subchunkify(bytez: bytes) -> Sequence[bytes]:
    bytez += b'\x00' * (-len(bytez) % BYTES_PER_CUSTODY_SUBCHUNK)
    return [bytez[i:i + BYTES_PER_CUSTODY_SUBCHUNK]
            for i in range(0, len(bytez), BYTES_PER_CUSTODY_SUBCHUNK)]

get_custody_chunk_bit

def get_custody_chunk_bit(key: BLSSignature, chunk: bytes) -> bool:
    full_G2_element = bls_signature_to_G2(key)
    s = full_G2_element[0].coeffs
    bits = [legendre_bit((i + 1) * s[i % 2] + int.from_bytes(subchunk, "little"), BLS12_381_Q)
            for i, subchunk in enumerate(custody_subchunkify(chunk))]

    return bool(sum(bits) % 2)

get_chunk_bits_root

def get_chunk_bits_root(chunk_bits: Bitlist[MAX_CUSTODY_CHUNKS]) -> bit:
    aggregated_bits = 0
    for i, b in enumerate(chunk_bits):
        aggregated_bits += 2**i * b
    return legendre_bit(aggregated_bits, BLS12_381_Q)

get_randao_epoch_for_custody_period

def get_randao_epoch_for_custody_period(period: uint64, validator_index: ValidatorIndex) -> Epoch:
    next_period_start = (period + 1) * EPOCHS_PER_CUSTODY_PERIOD - validator_index % EPOCHS_PER_CUSTODY_PERIOD
    return Epoch(next_period_start + CUSTODY_PERIOD_TO_RANDAO_PADDING)

get_custody_period_for_validator

def get_custody_period_for_validator(state: BeaconState, validator_index: ValidatorIndex, epoch: Epoch=None) -> int:
    '''
    Return the reveal period for a given validator.
    '''
    epoch = get_current_epoch(state) if epoch is None else epoch
    return (epoch + validator_index % EPOCHS_PER_CUSTODY_PERIOD) // EPOCHS_PER_CUSTODY_PERIOD

replace_empty_or_append

def replace_empty_or_append(list: MutableSequence[Any], new_element: Any) -> int:
    for i in range(len(list)):
        if is_zero(list[i]):
            list[i] = new_element
            return i
    list.append(new_element)
    return len(list) - 1

Per-block processing

Operations

Add the following operations to the per-block processing, in the order given below and after all other operations in Phase 0.

Custody key reveals

Verify that len(block.body.custody_key_reveals) <= MAX_CUSTODY_KEY_REVEALS.

For each reveal in block.body.custody_key_reveals, run the following function:

def process_custody_key_reveal(state: BeaconState, reveal: CustodyKeyReveal) -> None:
    """
    Process ``CustodyKeyReveal`` operation.
    Note that this function mutates ``state``.
    """
    revealer = state.validators[reveal.revealer_index]
    epoch_to_sign = get_randao_epoch_for_custody_period(revealer.next_custody_secret_to_reveal, reveal.revealer_index)

    assert revealer.next_custody_secret_to_reveal < get_custody_period_for_validator(state, reveal.revealer_index)

    # Revealed validator is active or exited, but not withdrawn
    assert is_slashable_validator(revealer, get_current_epoch(state))

    # Verify signature
    assert bls_verify(
        pubkey=revealer.pubkey,
        message_hash=hash_tree_root(epoch_to_sign),
        signature=reveal.reveal,
        domain=get_domain(
            state=state,
            domain_type=DOMAIN_RANDAO,
            message_epoch=epoch_to_sign,
        ),
    )

    # Decrement max reveal lateness if response is timely
    if epoch_to_sign + EPOCHS_PER_CUSTODY_PERIOD >= get_current_epoch(state):
        if revealer.max_reveal_lateness >= MAX_REVEAL_LATENESS_DECREMENT:
            revealer.max_reveal_lateness -= MAX_REVEAL_LATENESS_DECREMENT
        else:
            revealer.max_reveal_lateness = 0
    else:
        revealer.max_reveal_lateness = max(
            revealer.max_reveal_lateness,
            get_current_epoch(state) - epoch_to_sign - EPOCHS_PER_CUSTODY_PERIOD
        )

    # Process reveal
    revealer.next_custody_secret_to_reveal += 1

    # Reward Block Preposer
    proposer_index = get_beacon_proposer_index(state)
    increase_balance(
        state,
        proposer_index,
        Gwei(get_base_reward(state, reveal.revealer_index) // MINOR_REWARD_QUOTIENT)
    )

Early derived secret reveals

Verify that len(block.body.early_derived_secret_reveals) <= MAX_EARLY_DERIVED_SECRET_REVEALS.

For each reveal in block.body.early_derived_secret_reveals, run the following function:

def process_early_derived_secret_reveal(state: BeaconState, reveal: EarlyDerivedSecretReveal) -> None:
    """
    Process ``EarlyDerivedSecretReveal`` operation.
    Note that this function mutates ``state``.
    """
    revealed_validator = state.validators[reveal.revealed_index]
    derived_secret_location = reveal.epoch % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS

    assert reveal.epoch >= get_current_epoch(state) + RANDAO_PENALTY_EPOCHS
    assert reveal.epoch < get_current_epoch(state) + EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
    assert not revealed_validator.slashed
    assert reveal.revealed_index not in state.exposed_derived_secrets[derived_secret_location]

    # Verify signature correctness
    masker = state.validators[reveal.masker_index]
    pubkeys = [revealed_validator.pubkey, masker.pubkey]
    message_hashes = [
        hash_tree_root(reveal.epoch),
        reveal.mask,
    ]

    assert bls_verify_multiple(
        pubkeys=pubkeys,
        message_hashes=message_hashes,
        signature=reveal.reveal,
        domain=get_domain(
            state=state,
            domain_type=DOMAIN_RANDAO,
            message_epoch=reveal.epoch,
        ),
    )

    if reveal.epoch >= get_current_epoch(state) + CUSTODY_PERIOD_TO_RANDAO_PADDING:
        # Full slashing when the secret was revealed so early it may be a valid custody
        # round key
        slash_validator(state, reveal.revealed_index, reveal.masker_index)
    else:
        # Only a small penalty proportional to proposer slot reward for RANDAO reveal
        # that does not interfere with the custody period
        # The penalty is proportional to the max proposer reward

        # Calculate penalty
        max_proposer_slot_reward = (
            get_base_reward(state, reveal.revealed_index)
            * SLOTS_PER_EPOCH
            // len(get_active_validator_indices(state, get_current_epoch(state)))
            // PROPOSER_REWARD_QUOTIENT
        )
        penalty = Gwei(
            max_proposer_slot_reward
            * EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE
            * (len(state.exposed_derived_secrets[derived_secret_location]) + 1)
        )

        # Apply penalty
        proposer_index = get_beacon_proposer_index(state)
        whistleblower_index = reveal.masker_index
        whistleblowing_reward = Gwei(penalty // WHISTLEBLOWER_REWARD_QUOTIENT)
        proposer_reward = Gwei(whistleblowing_reward // PROPOSER_REWARD_QUOTIENT)
        increase_balance(state, proposer_index, proposer_reward)
        increase_balance(state, whistleblower_index, whistleblowing_reward - proposer_reward)
        decrease_balance(state, reveal.revealed_index, penalty)

        # Mark this derived secret as exposed so validator cannot be punished repeatedly
        state.exposed_derived_secrets[derived_secret_location].append(reveal.revealed_index)

Chunk challenges

Verify that len(block.body.custody_chunk_challenges) <= MAX_CUSTODY_CHUNK_CHALLENGES.

For each challenge in block.body.custody_chunk_challenges, run the following function:

def process_chunk_challenge(state: BeaconState, challenge: CustodyChunkChallenge) -> None:
    # Verify the attestation
    assert is_valid_indexed_attestation(state, get_indexed_attestation(state, challenge.attestation))
    # Verify it is not too late to challenge
    assert (compute_epoch_at_slot(challenge.attestation.data.slot)
            >= get_current_epoch(state) - MAX_CHUNK_CHALLENGE_DELAY)
    responder = state.validators[challenge.responder_index]
    assert responder.exit_epoch >= get_current_epoch(state) - MAX_CHUNK_CHALLENGE_DELAY
    # Verify the responder participated in the attestation
    attesters = get_attesting_indices(state, challenge.attestation.data, challenge.attestation.aggregation_bits)
    assert challenge.responder_index in attesters
    # Verify the challenge is not a duplicate
    for record in state.custody_chunk_challenge_records:
        assert (
            record.data_root != challenge.attestation.data.crosslink.data_root or
            record.chunk_index != challenge.chunk_index
        )
    # Verify depth
    depth = ceillog2(get_custody_chunk_count(challenge.attestation.data.crosslink))
    assert challenge.chunk_index < 2**depth
    # Add new chunk challenge record
    new_record = CustodyChunkChallengeRecord(
        challenge_index=state.custody_challenge_index,
        challenger_index=get_beacon_proposer_index(state),
        responder_index=challenge.responder_index,
        inclusion_epoch=get_current_epoch(state),
        data_root=challenge.attestation.data.crosslink.data_root,
        depth=depth,
        chunk_index=challenge.chunk_index,
    )
    replace_empty_or_append(state.custody_chunk_challenge_records, new_record)

    state.custody_challenge_index += 1
    # Postpone responder withdrawability
    responder.withdrawable_epoch = FAR_FUTURE_EPOCH

Bit challenges

Verify that len(block.body.custody_bit_challenges) <= MAX_CUSTODY_BIT_CHALLENGES.

For each challenge in block.body.custody_bit_challenges, run the following function:

def process_bit_challenge(state: BeaconState, challenge: CustodyBitChallenge) -> None:
    attestation = challenge.attestation
    epoch = attestation.data.target.epoch
    shard = attestation.data.crosslink.shard

    # Verify challenge signature
    challenger = state.validators[challenge.challenger_index]
    domain = get_domain(state, DOMAIN_CUSTODY_BIT_CHALLENGE, get_current_epoch(state))
    assert bls_verify(challenger.pubkey, signing_root(challenge), challenge.signature, domain)
    # Verify challenger is slashable
    assert is_slashable_validator(challenger, get_current_epoch(state))
    # Verify attestation
    assert is_valid_indexed_attestation(state, get_indexed_attestation(state, attestation))
    # Verify attestation is eligible for challenging
    responder = state.validators[challenge.responder_index]
    assert get_current_epoch(state) <= get_randao_epoch_for_custody_period(
        get_custody_period_for_validator(state, challenge.responder_index, epoch),
        challenge.responder_index
    ) + 2 * EPOCHS_PER_CUSTODY_PERIOD + responder.max_reveal_lateness

    # Verify the responder participated in the attestation
    attesters = get_attesting_indices(state, attestation.data, attestation.aggregation_bits)
    assert challenge.responder_index in attesters
    # Verifier challenger is not already challenging
    for record in state.custody_bit_challenge_records:
        assert record.challenger_index != challenge.challenger_index
    # Verify the responder custody key
    epoch_to_sign = get_randao_epoch_for_custody_period(
        get_custody_period_for_validator(state, challenge.responder_index, epoch),
        challenge.responder_index,
    )
    domain = get_domain(state, DOMAIN_RANDAO, epoch_to_sign)
    assert bls_verify(responder.pubkey, hash_tree_root(epoch_to_sign), challenge.responder_key, domain)
    # Verify the chunk count
    chunk_count = get_custody_chunk_count(attestation.data.crosslink)
    assert chunk_count == len(challenge.chunk_bits)
    # Verify custody bit is incorrect
    committee = get_beacon_committee(state, epoch, shard)
    custody_bit = attestation.custody_bits[committee.index(challenge.responder_index)]
    assert custody_bit != get_chunk_bits_root(challenge.chunk_bits)
    # Add new bit challenge record
    new_record = CustodyBitChallengeRecord(
        challenge_index=state.custody_challenge_index,
        challenger_index=challenge.challenger_index,
        responder_index=challenge.responder_index,
        inclusion_epoch=get_current_epoch(state),
        data_root=attestation.data.crosslink.data_root,
        chunk_count=chunk_count,
        chunk_bits_merkle_root=hash_tree_root(challenge.chunk_bits),
        responder_key=challenge.responder_key,
    )
    replace_empty_or_append(state.custody_bit_challenge_records, new_record)
    state.custody_challenge_index += 1
    # Postpone responder withdrawability
    responder.withdrawable_epoch = FAR_FUTURE_EPOCH

Custody responses

Verify that len(block.body.custody_responses) <= MAX_CUSTODY_RESPONSES.

For each response in block.body.custody_responses, run the following function:

def process_custody_response(state: BeaconState, response: CustodyResponse) -> None:
    chunk_challenge = next((record for record in state.custody_chunk_challenge_records
                            if record.challenge_index == response.challenge_index), None)
    if chunk_challenge is not None:
        return process_chunk_challenge_response(state, response, chunk_challenge)

    bit_challenge = next((record for record in state.custody_bit_challenge_records
                          if record.challenge_index == response.challenge_index), None)
    if bit_challenge is not None:
        return process_bit_challenge_response(state, response, bit_challenge)

    assert False

def process_chunk_challenge_response(state: BeaconState,
                                     response: CustodyResponse,
                                     challenge: CustodyChunkChallengeRecord) -> None:
    # Verify chunk index
    assert response.chunk_index == challenge.chunk_index
    # Verify bit challenge data is null
    assert response.chunk_bits_branch == [] and response.chunk_bits_leaf == Hash()
    # Verify minimum delay
    assert get_current_epoch(state) >= challenge.inclusion_epoch + MAX_SEED_LOOKAHEAD
    # Verify the chunk matches the crosslink data root
    assert is_valid_merkle_branch(
        leaf=hash_tree_root(response.chunk),
        branch=response.data_branch,
        depth=challenge.depth,
        index=response.chunk_index,
        root=challenge.data_root,
    )
    # Clear the challenge
    records = state.custody_chunk_challenge_records
    records[records.index(challenge)] = CustodyChunkChallengeRecord()
    # Reward the proposer
    proposer_index = get_beacon_proposer_index(state)
    increase_balance(state, proposer_index, Gwei(get_base_reward(state, proposer_index) // MINOR_REWARD_QUOTIENT))

def process_bit_challenge_response(state: BeaconState,
                                   response: CustodyResponse,
                                   challenge: CustodyBitChallengeRecord) -> None:
    # Verify chunk index
    assert response.chunk_index < challenge.chunk_count
    # Verify responder has not been slashed
    responder = state.validators[challenge.responder_index]
    assert not responder.slashed
    # Verify the chunk matches the crosslink data root
    assert is_valid_merkle_branch(
        leaf=hash_tree_root(response.chunk),
        branch=response.data_branch,
        depth=ceillog2(challenge.chunk_count),
        index=response.chunk_index,
        root=challenge.data_root,
    )
    # Verify the chunk bit leaf matches the challenge data
    assert is_valid_merkle_branch_with_mixin(
        leaf=hash_tree_root(response.chunk_bits_leaf),
        branch=response.chunk_bits_branch,
        depth=ceillog2(MAX_CUSTODY_CHUNKS // 256),
        index=response.chunk_index // 256,
        root=challenge.chunk_bits_merkle_root,
        mixin=challenge.chunk_count,
    )
    # Verify the chunk bit does not match the challenge chunk bit
    assert (get_custody_chunk_bit(challenge.responder_key, response.chunk)
            != response.chunk_bits_leaf[response.chunk_index % 256])
    # Clear the challenge
    records = state.custody_bit_challenge_records
    records[records.index(challenge)] = CustodyBitChallengeRecord()
    # Slash challenger
    slash_validator(state, challenge.challenger_index, challenge.responder_index)

Per-epoch processing

Handling of custody-related deadlines

Run process_reveal_deadlines(state) immediately after process_registry_updates(state):

# begin insert @process_reveal_deadlines
    process_reveal_deadlines(state)
# end insert @process_reveal_deadlines
def process_reveal_deadlines(state: BeaconState) -> None:
    for index, validator in enumerate(state.validators):
        deadline = validator.next_custody_secret_to_reveal + (CUSTODY_RESPONSE_DEADLINE // EPOCHS_PER_CUSTODY_PERIOD)
        if get_custody_period_for_validator(state, ValidatorIndex(index)) > deadline:
            slash_validator(state, ValidatorIndex(index))

Run process_challenge_deadlines(state) immediately after process_reveal_deadlines(state):

# begin insert @process_challenge_deadlines
    process_challenge_deadlines(state)
# end insert @process_challenge_deadlines
def process_challenge_deadlines(state: BeaconState) -> None:
    for custody_chunk_challenge in state.custody_chunk_challenge_records:
        if get_current_epoch(state) > custody_chunk_challenge.inclusion_epoch + CUSTODY_RESPONSE_DEADLINE:
            slash_validator(state, custody_chunk_challenge.responder_index, custody_chunk_challenge.challenger_index)
            records = state.custody_chunk_challenge
            records[records.index(custody_chunk_challenge)] = CustodyChunkChallengeRecord()

    for custody_bit_challenge in state.custody_bit_challenge_records:
        if get_current_epoch(state) > custody_bit_challenge.inclusion_epoch + CUSTODY_RESPONSE_DEADLINE:
            slash_validator(state, custody_bit_challenge.responder_index, custody_bit_challenge.challenger_index)
            records = state.custody_bit_challenge_records
            records[records.index(custody_bit_challenge)] = CustodyBitChallengeRecord()

Append this to process_final_updates(state):

# begin insert @after_process_final_updates
    after_process_final_updates(state)
# end insert @after_process_final_updates
def after_process_final_updates(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    # Clean up exposed RANDAO key reveals
    state.exposed_derived_secrets[current_epoch % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS] = []
    # Reset withdrawable epochs if challenge records are empty
    records = state.custody_chunk_challenge_records + state.custody_bit_challenge_records
    validator_indices_in_records = set(
        [record.challenger_index for record in records] + [record.responder_index for record in records]
    )
    for index, validator in enumerate(state.validators):
        if index not in validator_indices_in_records:
            if validator.exit_epoch != FAR_FUTURE_EPOCH and validator.withdrawable_epoch == FAR_FUTURE_EPOCH:
                validator.withdrawable_epoch = Epoch(validator.exit_epoch + MIN_VALIDATOR_WITHDRAWABILITY_DELAY)