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Ethereum 2.0 Phase 1 -- Crosslinks and Shard Data

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

Table of contents

Introduction

This document describes the shard transition function (data layer only) and the shard fork choice rule as part of Phase 1 of Ethereum 2.0.

Configuration

Misc

Name Value Unit Duration
MAX_SHARDS 2**10 (= 1024)
ACTIVE_SHARDS 2**6 (= 64)
ONLINE_PERIOD 2**3 (= 8) epochs ~51 min
LIGHT_CLIENT_COMMITTEE_SIZE 2**7 (= 128)
LIGHT_CLIENT_COMMITTEE_PERIOD 2**8 (= 256) epochs ~27 hours
SHARD_BLOCK_CHUNK_SIZE 2**18 (= 262,144)
MAX_SHARD_BLOCK_CHUNKS 2**2 (= 4)
BLOCK_SIZE_TARGET 3 * 2**16 (= 196,608)
SHARD_BLOCK_OFFSETS [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233]
MAX_SHARD_BLOCKS_PER_ATTESTATION len(SHARD_BLOCK_OFFSETS)
EMPTY_CHUNK_ROOT hash_tree_root(BytesN[SHARD_BLOCK_CHUNK_SIZE]())
MAX_GASPRICE 2**14 (= 16,384) Gwei
GASPRICE_ADJUSTMENT_COEFFICIENT 2**3 (= 8)
DOMAIN_SHARD_LIGHT_CLIENT 192
DOMAIN_SHARD_PROPOSAL 193

Containers

ShardBlockWrapper

class ShardBlockWrapper(Container):
    shard_parent_root: Hash
    beacon_parent_root: Hash
    slot: Slot
    body: BytesN[SHARD_BLOCK_CHUNK_SIZE]
    signature: BLSSignature

ShardSignedHeader

class ShardSignedHeader(Container):
    shard_parent_root: Hash
    beacon_parent_root: Hash
    slot: Slot
    body_root: Hash

ShardState

class ShardState(Container):
    slot: Slot
    gasprice: Gwei
    root: Hash
    latest_block_hash: Hash

AttestationData

class AttestationData(Container):
    slot: Slot
    index: CommitteeIndex
    # LMD GHOST vote
    beacon_block_root: Hash
    # FFG vote
    source: Checkpoint
    target: Checkpoint
    # Shard transition root
    shard_transition_root: Hash

ShardTransition

class ShardTransition(Container):
    # Starting from slot
    start_slot: Slot
    # Shard block lengths
    shard_block_lengths: List[uint64, MAX_SHARD_BLOCKS_PER_ATTESTATION]
    # Shard data roots
    shard_data_roots: List[List[Hash, MAX_SHARD_BLOCK_CHUNKS], MAX_SHARD_BLOCKS_PER_ATTESTATION]
    # Intermediate state roots
    shard_state_roots: List[ShardState, MAX_SHARD_BLOCKS_PER_ATTESTATION]
    # Proposer signature aggregate
    proposer_signature_aggregate: BLSSignature

Attestation

class Attestation(Container):
    aggregation_bits: Bitlist[MAX_VALIDATORS_PER_COMMITTEE]
    data: AttestationData
    custody_bits: List[Bitlist[MAX_VALIDATORS_PER_COMMITTEE], MAX_SHARD_BLOCKS_PER_ATTESTATION]
    signature: BLSSignature

IndexedAttestation

class IndexedAttestation(Container):
    committee: List[ValidatorIndex, MAX_VALIDATORS_PER_COMMITTEE]
    attestation: Attestation

CompactCommittee

class CompactCommittee(Container):
    pubkeys: List[BLSPubkey, MAX_VALIDATORS_PER_COMMITTEE]
    compact_validators: List[uint64, MAX_VALIDATORS_PER_COMMITTEE]

AttestationCustodyBitWrapper

class AttestationCustodyBitWrapper(Container):
    attestation_root: Hash
    block_index: uint64
    bit: bool

PendingAttestation

class PendingAttestation(Container):
    aggregation_bits: Bitlist[MAX_VALIDATORS_PER_COMMITTEE]
    data: AttestationData
    inclusion_delay: Slot
    proposer_index: ValidatorIndex
    crosslink_success: bool

Helpers

get_online_validators

def get_online_indices(state: BeaconState) -> Set[ValidatorIndex]:
    active_validators = get_active_validator_indices(state, get_current_epoch(state))
    return set([i for i in active_validators if state.online_countdown[i] != 0])

pack_compact_validator

def pack_compact_validator(index: int, slashed: bool, balance_in_increments: int) -> int:
    """
    Creates a compact validator object representing index, slashed status, and compressed balance.
    Takes as input balance-in-increments (// EFFECTIVE_BALANCE_INCREMENT) to preserve symmetry with
    the unpacking function.
    """
    return (index << 16) + (slashed << 15) + balance_in_increments

committee_to_compact_committee

def committee_to_compact_committee(state: BeaconState, committee: Sequence[ValidatorIndex]) -> CompactCommittee:
    """
    Given a state and a list of validator indices, outputs the CompactCommittee representing them.
    """
    validators = [state.validators[i] for i in committee]
    compact_validators = [
        pack_compact_validator(i, v.slashed, v.effective_balance // EFFECTIVE_BALANCE_INCREMENT)
        for i, v in zip(committee, validators)
    ]
    pubkeys = [v.pubkey for v in validators]
    return CompactCommittee(pubkeys=pubkeys, compact_validators=compact_validators)

get_light_client_committee

def get_light_client_committee(beacon_state: BeaconState, epoch: Epoch) -> Sequence[ValidatorIndex]:
    source_epoch = epoch - epoch % LIGHT_CLIENT_COMMITTEE_PERIOD 
    if source_epoch > 0:
        source_epoch -= LIGHT_CLIENT_COMMITTEE_PERIOD
    active_validator_indices = get_active_validator_indices(beacon_state, source_epoch)
    seed = get_seed(beacon_state, source_epoch, DOMAIN_SHARD_LIGHT_CLIENT)
    return compute_committee(active_validator_indices, seed, 0, ACTIVE_SHARDS)[:TARGET_COMMITTEE_SIZE]

get_indexed_attestation

def get_indexed_attestation(beacon_state: BeaconState, attestation: Attestation) -> IndexedAttestation:
    committee = get_beacon_committee(beacon_state, attestation.data.slot, attestation.data.index)
    return IndexedAttestation(committee, attestation)

update_gasprice

def update_gasprice(prev_gasprice: Gwei, length: uint8) -> Gwei:
    if length > BLOCK_SIZE_TARGET:
        delta = prev_gasprice * (length - BLOCK_SIZE_TARGET) // BLOCK_SIZE_TARGET // GASPRICE_ADJUSTMENT_COEFFICIENT
        return min(prev_gasprice + delta, MAX_GASPRICE)
    else:
        delta = prev_gasprice * (BLOCK_SIZE_TARGET - length) // BLOCK_SIZE_TARGET // GASPRICE_ADJUSTMENT_COEFFICIENT
        if delta > prev_gasprice - GASPRICE_ADJUSTMENT_COEFFICIENT:
            return GASPRICE_ADJUSTMENT_COEFFICIENT
        else:
            return prev_gasprice - delta

is_valid_indexed_attestation

def is_valid_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> bool:
    """
    Check if ``indexed_attestation`` has valid indices and signature.
    """

    # Verify aggregate signature
    all_pubkeys = []
    all_message_hashes = []
    aggregation_bits = indexed_attestation.attestation.aggregation_bits
    assert len(aggregation_bits) == len(indexed_attestation.committee)
    for i, custody_bits in enumerate(indexed_attestation.attestation.custody_bits):
        assert len(custody_bits) == len(indexed_attestation.committee)
        for participant, abit, cbit in zip(indexed_attestation.committee, aggregation_bits, custody_bits):
            if abit:
                all_pubkeys.append(state.validators[participant].pubkey)
                # Note: only 2N distinct message hashes
                all_message_hashes.append(hash_tree_root(
                    AttestationCustodyBitWrapper(hash_tree_root(indexed_attestation.data), i, cbit)
                ))
            else:
                assert cbit == False
        
    return bls_verify_multiple(
        pubkeys=all_pubkeys,
        message_hashes=all_message_hashes,
        signature=indexed_attestation.signature,
        domain=get_domain(state, DOMAIN_BEACON_ATTESTER, indexed_attestation.data.target.epoch),
    )

get_attestation_shard

def get_shard(state: BeaconState, attestation: Attestation) -> Shard:
    return (attestation.data.index + get_start_shard(state, data.slot)) % ACTIVE_SHARDS

Beacon Chain Changes

New beacon state fields

    shard_states: Vector[ShardState, MAX_SHARDS]
    online_countdown: Bytes[VALIDATOR_REGISTRY_LIMIT]
    current_light_committee: CompactCommittee
    next_light_committee: CompactCommittee

New beacon block data fields

    shard_transitions: Vector[ShardTransition, MAX_SHARDS]
    light_client_signature_bitfield: Bitlist[LIGHT_CLIENT_COMMITTEE_SIZE]
    light_client_signature: BLSSignature

Attestation processing

validate_attestation

def validate_attestation(state: BeaconState, attestation: Attestation) -> None:
    data = attestation.data
    assert data.index < ACTIVE_SHARDS
    shard = get_shard(state, attestation)
    proposer_index = get_beacon_proposer_index(state)

    # Signature check
    assert is_valid_indexed_attestation(state, get_indexed_attestation(state, attestation))
    # Type 1: on-time attestations
    if data.custody_bits != []:
        # Correct slot
        assert data.slot == state.slot
        # Correct data root count
        start_slot = state.shard_next_slots[shard]
        offset_slots = [start_slot + x for x in SHARD_BLOCK_OFFSETS if start_slot + x < state.slot]
        assert len(attestation.custody_bits) == len(offset_slots)
        # Correct parent block root
        assert data.beacon_block_root == get_block_root_at_slot(state, state.slot - 1)
    # Type 2: delayed attestations
    else:
        assert state.slot - slot_to_epoch(data.slot) < EPOCH_LENGTH
        assert data.shard_transition_root == Hash()
        assert len(attestation.custody_bits) == 0

apply_shard_transition

def apply_shard_transition(state: BeaconState, shard: Shard, transition: ShardTransition) -> None:
    # Slot the attestation starts counting from
    start_slot = state.shard_next_slots[shard]

    # Correct data root count
    offset_slots = [start_slot + x for x in SHARD_BLOCK_OFFSETS if start_slot + x < state.slot]
    assert len(transition.shard_data_roots) == len(transition.shard_states) == len(transition.shard_block_lengths) == len(offset_slots)
    assert transition.start_slot == start_slot

    def chunks_to_body_root(chunks):
        return hash_tree_root(chunks + [EMPTY_CHUNK_ROOT] * (MAX_SHARD_BLOCK_CHUNKS - len(chunks)))

    # Reonstruct shard headers
    headers = []
    proposers = []
    shard_parent_root = state.shard_states[shard].latest_block_hash
    for i in range(len(offset_slots)):
        if any(transition.shard_data_roots):
            headers.append(ShardSignedHeader(
                shard_parent_root=shard_parent_root,
                parent_hash=get_block_root_at_slot(state, state.slot-1),
                slot=offset_slots[i],
                body_root=chunks_to_body_root(transition.shard_data_roots[i])
            ))
            proposers.append(get_shard_proposer(state, shard, offset_slots[i]))
            shard_parent_root = hash_tree_root(headers[-1])

    # Verify correct calculation of gas prices and slots and chunk roots
    prev_gasprice = state.shard_states[shard].gasprice
    for i in range(len(offset_slots)):
        shard_state, block_length, chunks = transition.shard_states[i], transition.shard_block_lengths[i], transition.shard_data_roots[i]
        assert shard_state.gasprice == update_gasprice(prev_gasprice, block_length)
        assert shard_state.slot == offset_slots[i]
        assert len(chunks) == block_length // SHARD_BLOCK_CHUNK_SIZE
        prev_gasprice = shard_state.gasprice

    # Verify combined proposer signature
    assert bls_verify_multiple(
        pubkeys=[state.validators[proposer].pubkey for proposer in proposers],
        message_hashes=[hash_tree_root(header) for header in headers],
        signature=proposer.proposer_signature_aggregate,
        domain=DOMAIN_SHARD_PROPOSAL
    )

    # Save updated state
    state.shard_states[shard] = transition.shard_states[-1]
    state.shard_states[shard].slot = state.slot - 1

process_attestations

def process_attestations(state: BeaconState, block: BeaconBlock, attestations: Sequence[Attestation]) -> None:
    pending_attestations = []
    # Basic validation
    for attestation in attestations:
       validate_attestation(state, attestation)
    # Process crosslinks
    online_indices = get_online_indices(state)
    winners = set()
    for shard in range(ACTIVE_SHARDS):
        success = False
        # All attestations in the block for this shard
        this_shard_attestations = [attestation for attestation in attestations if get_shard(state, attestation) == shard and attestation.data.slot == state.slot]
        # The committee for this shard
        this_shard_committee = get_beacon_committee(state, get_current_epoch(state), shard)
        # Loop over all shard transition roots
        for shard_transition_root in sorted(set([attestation.data.shard_transition_root for attestation in this_shard_attestations])):
            all_participants = set()
            participating_attestations = []
            for attestation in this_shard_attestations:
                participating_attestations.append(attestation)
                if attestation.data.shard_transition_root == shard_transition_root:
                    all_participants = all_participants.union(get_attesting_indices(state, attestation.data, attestation.aggregation_bits))
            if (
                get_total_balance(state, online_indices.intersection(all_participants)) * 3 >=
                get_total_balance(state, online_indices.intersection(this_shard_committee)) * 2
                and success is False
            ):
                assert shard_transition_root == hash_tree_root(block.shard_transition)
                process_crosslink(state, shard, block.shard_transition)
                # Apply proposer reward and cost
                estimated_attester_reward = sum([get_base_reward(state, attester) for attester in all_participants])
                increase_balance(state, proposer, estimated_attester_reward // PROPOSER_REWARD_COEFFICIENT)
                for shard_state, slot, length in zip(block.shard_transition.shard_states, offset_slots, block.shard_transition.shard_block_lengths):
                    decrease_balance(state, get_shard_proposer(state, shard, slot), shard_state.gasprice * length)
                winners.add((shard, shard_transition_root))
                success = True
        if not success:
            assert block.shard_transition == ShardTransition()
    for attestation in attestations:
        pending_attestation = PendingAttestation(
            aggregation_bits=attestation.aggregation_bits,
            data=attestation.data,
            inclusion_delay=state.slot - data.slot,
            crosslink_success=(attestation.shard, attestation.shard_transition_root) in winners and attestation.data.slot == state.slot,
            proposer_index=proposer_index
        )
        if attestation.data.target.epoch == get_current_epoch(state):
            assert attestation.data.source == state.current_justified_checkpoint
            state.current_epoch_attestations.append(pending_attestation)
        else:
            assert attestation.data.source == state.previous_justified_checkpoint
            state.previous_epoch_attestations.append(pending_attestation)

Misc block post-processing

def misc_block_post_process(state: BeaconState, block: BeaconBlock):
    # Verify that a `shard_transition` in a block is empty if an attestation was not processed for it
    for shard in range(MAX_SHARDS):
        if state.shard_states[shard].slot != state.slot - 1:
            assert block.shard_transition[shard] == ShardTransition()
    for pending_attestation in state.current_epoch_attestations + state.previous_epoch_attestations:
        for index in get_attesting_indices(state, pending_attestation.data, pending_attestation.aggregation_bits):
            online_countdown[index] = ONLINE_PERIOD

Light client processing

def process_light_client_signatures(state: BeaconState, block: BeaconBlock):
    committee = get_light_client_committee(state, get_current_epoch(state))
    assert len(block.light_client_signature_bitfield) == len(committee)
    tot_reward = 0
    signer_keys = []
    for i, bit in enumerate(block.light_client_signature_bitfield):
        if bit:
            signer_keys.append(state.validators[committee[i]].pubkey)
            increase_balance(state, committee[i], get_base_reward(state, committee[i]))
            tot_reward += get_base_reward(state, committee[i])

    increase_balance(state, get_beacon_proposer_index(state), tot_reward // PROPOSER_REWARD_COEFFICIENT)
    
    assert bls_verify(
        pubkey=bls_aggregate_pubkeys(signer_keys),
        message_hash=get_block_root_at_slot(state, state.slot - 1),
        signature=block.light_client_signature,
        domain=DOMAIN_LIGHT_CLIENT
    )

Epoch transition

def phase_1_epoch_transition(state):
    # Slowly remove validators from the "online" set if they do not show up
    for index in range(len(state.validators)):
        if state.online_countdown[index] != 0:
            state.online_countdown[index] = state.online_countdown[index] - 1
    
    # Update light client committees
    if get_current_epoch(state) % LIGHT_CLIENT_COMMITTEE_PERIOD == 0:
        state.current_light_committee = state.next_light_committee
        new_committee = get_light_client_committee(state, get_current_epoch(state) + LIGHT_CLIENT_COMMITTEE_PERIOD)
        state.next_light_committee = committee_to_compact_committee(state, new_committee)

Fraud proofs

TODO. The intent is to have a single universal fraud proof type, which contains the following parts:

  1. An on-time attestation on some shard signing a ShardTransition
  2. An index i of a particular position to focus on
  3. The ShardTransition itself
  4. The full body of the block
  5. A Merkle proof to the shard_states in the parent block the attestation is referencing

The proof verifies that one of the two conditions is false:

  1. custody_bits[i][j] != generate_custody_bit(subkey, block_contents) for any j
  2. execute_state_transition(shard, slot, transition.shard_states[i-1].root, hash_tree_root(parent), get_shard_proposer(state, shard, slot), block_contents) != transition.shard_states[i].root (if i=0 then instead use parent.shard_states[shard][-1].root)

Shard state transition function

def shard_state_transition(shard: Shard, slot: Slot, pre_state: Hash, previous_beacon_root: Hash, proposer_pubkey: BLSPubkey, block_data: BytesN[MAX_SHARD_BLOCK_CHUNKS * SHARD_BLOCK_CHUNK_SIZE]) -> Hash:
    # We will add something more substantive in phase 2
    return hash(pre_state + hash_tree_root(previous_beacon_root) + hash_tree_root(block_data))

Honest committee member behavior

Suppose you are a committee member on shard shard at slot current_slot. Let state be the head beacon state you are building on. Three seconds into slot slot, run the following procedure:

  • Initialize proposals = [], shard_states = [], shard_state = state.shard_states[shard][-1], start_slot = shard_state.slot.
  • Let offset_slots = [start_slot + x for x in SHARD_BLOCK_OFFSETS if start_slot + x < state.slot]
  • For slot in offset_slots, do the following:
    • Look for all valid proposals for slot; that is, a Bytes proposal where shard_state_transition(shard, slot, shard_state, get_block_root_at_slot(state, state.slot - 1), get_shard_proposer(state, shard, slot), proposal) returns a result and does not throw an exception. Let choices be the set of non-empty valid proposals you discover.
    • If len(choices) == 0, do proposals.append(make_empty_proposal(shard_state, slot))
    • If len(choices) == 1, do proposals.append(choices[0])
    • If len(choices) > 1, let winning_proposal be the proposal with the largest number of total attestations from slots in state.shard_next_slots[shard]....slot-1 supporting it or any of its descendants, breaking ties by choosing the first proposal locally seen. Do proposals.append(winning_proposal).
    • Set shard_state = shard_state_transition(shard, slot, shard_state, get_block_root_at_slot(state, state.slot - 1), get_shard_proposer(state, shard, slot), proposals[-1]) and do shard_states.append(shard_state).

Make an attestation using shard_data_roots = [hash_tree_root(proposal) for proposal in proposals] and shard_state_roots = shard_states.