eth2.0-specs/specs/core/0_beacon-chain.md

Ethereum 2.0 Phase 0 -- The Beacon Chain

tags: spec, eth2.0, casper, sharding, beacon

NOTICE: This document is a work-in-progress for researchers and implementers. It reflects recent spec changes and takes precedence over the Python proof-of-concept implementation.

Introduction

This document represents the specification for Phase 0 of Ethereum 2.0 -- The Beacon Chain.

At the core of Ethereum 2.0 is a system chain called the "beacon chain". The beacon chain stores and manages the set of active proof-of-stake validators. In the initial deployment phases of Ethereum 2.0 the only mechanism to become a validator is to make a fixed-size one-way ETH deposit to a registration contract on the Ethereum 1.0 PoW chain. Induction as a validator happens after registration transaction receipts are processed by the beacon chain and after a queuing process. Deregistration is either voluntary or done forcibly as a penalty for misbehavior.

The primary source of load on the beacon chain are "attestations". Attestations simultaneously attest to a shard block and a corresponding beacon chain block. A sufficient number of attestations for the same shard block create a "crosslink", confirming the shard segment up to that shard block into the beacon chain. Crosslinks also serve as infrastructure for asynchronous cross-shard communication.

Terminology

• Validator - a participant in the Casper/sharding consensus system. You can become one by depositing 32 ETH into the Casper mechanism.
• Active validator set - those validators who are currently participating, and which the Casper mechanism looks to produce and attest to blocks, crosslinks and other consensus objects.
• Committee - a (pseudo-) randomly sampled subset of the active validator set. When a committee is referred to collectively, as in "this committee attests to X", this is assumed to mean "some subset of that committee that contains enough validators that the protocol recognizes it as representing the committee".
• Proposer - the validator that creates a beacon chain block
• Attester - a validator that is part of a committee that needs to sign off on a beacon chain block while simultaneously creating a link (crosslink) to a recent shard block on a particular shard chain.
• Beacon chain - the central PoS chain that is the base of the sharding system.
• Shard chain - one of the chains on which user transactions take place and account data is stored.
• Crosslink - a set of signatures from a committee attesting to a block in a shard chain, which can be included into the beacon chain. Crosslinks are the main means by which the beacon chain "learns about" the updated state of shard chains.
• Slot - a period of SLOT_DURATION seconds, during which one proposer has the ability to create a beacon chain block and some attesters have the ability to make attestations
• Cycle - a span of slots during which all validators get exactly one chance to make an attestation
• Finalized, justified - see Casper FFG finalization here: https://arxiv.org/abs/1710.09437
• Withdrawal period - number of slots between a validator exit and the validator balance being withdrawable
• Genesis time - the Unix time of the genesis beacon chain block at slot 0

Constants

Constant	Value	Unit	Approximation
SHARD_COUNT	2**10 (= 1,024)	shards
DEPOSIT_SIZE	2**5 (= 32)	ETH
MIN_ONLINE_DEPOSIT_SIZE	2**4 (= 16)	ETH
GWEI_PER_ETH	10**9	Gwei/ETH
TARGET_COMMITTEE_SIZE	2**8 (= 256)	validators
GENESIS_TIME	TBD	seconds
SLOT_DURATION	2**4 (= 16)	seconds
CYCLE_LENGTH	2**6 (= 64)	slots	~17 minutes
MIN_VALIDATOR_SET_CHANGE_INTERVAL	2**8 (= 256)	slots	~1.1 hours
RANDAO_SLOTS_PER_LAYER	2**12 (= 4096)	slots	~18 hours
SQRT_E_DROP_TIME	2**16 (= 65,536)	slots	~12 days
WITHDRAWAL_PERIOD	2**19 (= 524,288)	slots	~97 days
SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD	2**16 (= 65,536)	slots	~12 days
BASE_REWARD_QUOTIENT	2**15 (= 32,768)	—
MAX_VALIDATOR_CHURN_QUOTIENT	2**5 (= 32)	—
LOGOUT_MESSAGE	"LOGOUT"	—
INITIAL_FORK_VERSION	0	—

Notes

• See a recommended min committee size of 111 here https://vitalik.ca/files/Ithaca201807_Sharding.pdf); our algorithm will generally ensure the committee size is at least half the target.
• The SQRT_E_DROP_TIME constant is the amount of time it takes for the quadratic leak to cut deposits of non-participating validators by ~39.4%.
• The BASE_REWARD_QUOTIENT constant is the per-slot interest rate assuming all validators are participating, assuming total deposits of 1 ETH. It corresponds to ~3.88% annual interest assuming 10 million participating ETH.
• At most 1/MAX_VALIDATOR_CHURN_QUOTIENT of the validators can change during each validator set change.

Validator status codes

Name	Value
PENDING_ACTIVATION	0
ACTIVE	1
PENDING_EXIT	2
PENDING_WITHDRAW	3
WITHDRAWN	4
PENALIZED	127

Special record types

Name	Value
LOGOUT	0
CASPER_SLASHING	1
RANDAO_CHANGE	2

Validator set delta flags

Name	Value
ENTRY	0
EXIT	1

PoW chain registration contract

The initial deployment phases of Ethereum 2.0 are implemented without consensus changes to the PoW chain. A registration contract is added to the PoW chain to deposit ETH. This contract has a registration function which takes as arguments pubkey, withdrawal_shard, withdrawal_address, randao_commitment as defined in a ValidatorRecord below. A BLS proof_of_possession of types bytes is given as a final argument.

The registration contract emits a log with the various arguments for consumption by the beacon chain. It does not do validation, pushing the registration logic to the beacon chain. In particular, the proof of possession (based on the BLS12-381 curve) is not verified by the registration contract.

Data structures

Beacon chain blocks

A BeaconBlock has the following fields:

{
    # Slot number
    'slot': 'uint64',
    # Proposer RANDAO reveal
    'randao_reveal': 'hash32',
    # Recent PoW chain reference (block hash)
    'pow_chain_reference': 'hash32',
    # Skip list of previous beacon block hashes
    # i'th item is the most recent ancestor whose slot is a multiple of 2**i for i = 0, ..., 31
    'ancestor_hashes': ['hash32'],
    # Active state root
    'active_state_root': 'hash32',
    # Crystallized state root
    'crystallized_state_root': 'hash32',
    # Attestations
    'attestations': [AttestationRecord],
    # Specials (e.g. logouts, penalties)
    'specials': [SpecialRecord]
}

An AttestationRecord has the following fields:

{
    # Slot number
    'slot': 'uint64',
    # Shard number
    'shard': 'uint16',
    # Beacon block hashes not part of the current chain, oldest to newest
    'oblique_parent_hashes': ['hash32'],
    # Shard block hash being attested to
    'shard_block_hash': 'hash32',
    # Last crosslink hash
    'last_crosslink_hash': 'hash32',
    # Root of data between last hash and this one
    'shard_block_combined_data_root': 'hash32',
    # Attester participation bitfield (1 bit per attester)
    'attester_bitfield': 'bytes',
    # Slot of last justified beacon block
    'justified_slot': 'uint64',
    # Hash of last justified beacon block
    'justified_block_hash': 'hash32',
    # BLS aggregate signature
    'aggregate_sig': ['uint256']
}

An AttestationSignedData has the following fields:

{
    # Fork version
    'fork_version': 'uint64',
    # Slot number
    'slot': 'uint64',
    # Shard number
    'shard': 'uint16',
    # CYCLE_LENGTH parent hashes
    'parent_hashes': ['hash32'],
    # Shard block hash
    'shard_block_hash': 'hash32',
    # Last crosslink hash
    'last_crosslink_hash': 'hash32',
    # Root of data between last hash and this one
    'shard_block_combined_data_root': 'hash32',
    # Slot of last justified beacon block referenced in the attestation
    'justified_slot': 'uint64'
}

A SpecialRecord has the following fields:

{
    # Kind
    'kind': 'uint8',
    # Data
    'data': ['bytes']
}

Beacon chain state

For convenience we define the beacon chain state in two parts: "active state" and "crystallized state".

The ActiveState has the following fields:

{
    # Attestations not yet processed
    'pending_attestations': [AttestationRecord],
    # Specials not yet been processed
    'pending_specials': [SpecialRecord]
    # recent beacon block hashes needed to process attestations, older to newer
    'recent_block_hashes': ['hash32'],
    # RANDAO state
    'randao_mix': 'hash32'
}

The CrystallizedState has the following fields:

{
    # Slot of last validator set change
    'validator_set_change_slot': 'uint64',
    # List of validators
    'validators': [ValidatorRecord],
    # Most recent crosslink for each shard
    'crosslinks': [CrosslinkRecord],
    # Last crystallized state recalculation
    'last_state_recalculation_slot': 'uint64',
    # Last finalized slot
    'last_finalized_slot': 'uint64',
    # Last justified slot
    'last_justified_slot': 'uint64',
    # Number of consecutive justified slots
    'justified_streak': 'uint64',
    # Committee members and their assigned shard, per slot
    'shard_and_committee_for_slots': [[ShardAndCommittee]],
    # Persistent shard committees
    'persistent_committees': [['uint24']],
    'persistent_committee_reassignments': [ShardReassignmentRecord],
    # Randao seed used for next shuffling
    'next_shuffling_seed': 'hash32',
    # Total deposits penalized in the given withdrawal period
    'deposits_penalized_in_period': ['uint32'],
    # Hash chain of validator set changes (for light clients to easily track deltas)
    'validator_set_delta_hash_chain': 'hash32'
    # Parameters relevant to hard forks / versioning.
    # Should be updated only by hard forks.
    'pre_fork_version': 'uint32',
    'post_fork_version': 'uint32',
    'fork_slot_number': 'uint64',
}

A ValidatorRecord has the following fields:

{
    # BLS public key
    'pubkey': 'uint256',
    # Withdrawal shard number
    'withdrawal_shard': 'uint16',
    # Withdrawal address
    'withdrawal_address': 'address',
    # RANDAO commitment
    'randao_commitment': 'hash32',
    # Slot the RANDAO commitment was last changed
    'randao_last_change': 'uint64',
    # Balance in Gwei
    'balance': 'uint64',
    # Status code
    'status': 'uint8',
    # Slot when validator exited (or 0)
    'exit_slot': 'uint64'
}

A CrosslinkRecord has the following fields:

{
    # Slot number
    'slot': 'uint64',
    # Shard chain block hash
    'shard_block_hash': 'hash32'
}

A ShardAndCommittee object has the following fields:

{
    # Shard number
    'shard': 'uint16',
    # Validator indices
    'committee': ['uint24']
}

A ShardReassignmentRecord object has the following fields:

{
    # Which validator to reassign
    'validator_index': 'uint24',
    # To which shard
    'shard': 'uint16',
    # When
    'slot': 'uint64'
}

Beacon chain processing

The beacon chain is the "main chain" of the PoS system. The beacon chain's main responsibilities are:

• Store and maintain the set of active, queued and exited validators
• Process crosslinks (see above)
• Process its own block-by-block consensus, as well as the finality gadget

Processing the beacon chain is fundamentally similar to processing a PoW chain in many respects. Clients download and process blocks, and maintain a view of what is the current "canonical chain", terminating at the current "head". However, because of the beacon chain's relationship with the existing PoW chain, and because it is a PoS chain, there are differences.

For a block on the beacon chain to be processed by a node, four conditions have to be met:

• The parent pointed to by the ancestor_hashes[0] has already been processed and accepted
• An attestation from the proposer of the block (see later for definition) is included along with the block in the network message object
• The PoW chain block pointed to by the pow_chain_reference has already been processed and accepted
• The node's local clock time is greater than or equal to the minimum timestamp as computed by GENESIS_TIME + block.slot * SLOT_DURATION

If these conditions are not met, the client should delay processing the beacon block until the conditions are all satisfied.

Beacon block production is significantly different because of the proof of stake mechanism. A client simply checks what it thinks is the canonical chain when it should create a block, and looks up what its slot number is; when the slot arrives, it either proposes or attests to a block as required. Note that this requires each node to have a clock that is roughly (ie. within SLOT_DURATION seconds) synchronized with the other nodes.

Beacon chain fork choice rule

The beacon chain uses the Casper FFG fork choice rule of "favor the chain containing the highest-slot-number justified block". To choose between chains that are all descended from the same justified block, the chain uses "immediate message driven GHOST" (IMD GHOST) to choose the head of the chain.

For a description see: https://ethresear.ch/t/beacon-chain-casper-ffg-rpj-mini-spec/2760

For an implementation with a network simulator see: https://github.com/ethereum/research/blob/master/clock_disparity/ghost_node.py

Here's an example of its working (green is finalized blocks, yellow is justified, grey is attestations):

Beacon chain state transition function

We now define the state transition function. At the high level, the state transition is made up of two parts:

1. The per-block processing, which happens every block, and affects the ActiveState only.
2. The crystallized state recalculation, which happens only if block.slot >= last_state_recalculation_slot + CYCLE_LENGTH, and affects the CrystallizedState and ActiveState.

The crystallized state recalculation generally focuses on changes to the validator set, including adjusting balances and adding and removing validators, as well as processing crosslinks and managing block justification/finalization, and the per-block processing generally focuses on verifying aggregate signatures and saving temporary records relating to the in-block activity in the ActiveState.

Helper functions

Below are various helper functions.

The following is a function that gets active validator indices from the validator list:

def get_active_validator_indices(validators)
    return [i for i, v in enumerate(validators) if v.status == ACTIVE]

The following is a function that shuffles the validator list:

def shuffle(values: List[Any],
            seed: Hash32) -> List[Any]:
    """
    Returns the shuffled ``values`` with seed as entropy.
    """
    values_count = len(values)

    # Entropy is consumed from the seed in 3-byte (24 bit) chunks.
    rand_bytes = 3
    # The highest possible result of the RNG.
    rand_max = 2 ** (rand_bytes * 8) - 1

    # The range of the RNG places an upper-bound on the size of the list that
    # may be shuffled. It is a logic error to supply an oversized list.
    assert values_count < rand_max

    output = [x for x in values]
    source = seed
    index = 0
    while index < values_count - 1:
        # Re-hash the `source` to obtain a new pattern of bytes.
        source = hash(source)
        # Iterate through the `source` bytes in 3-byte chunks.
        for position in range(0, 32 - (32 % rand_bytes), rand_bytes):
            # Determine the number of indices remaining in `values` and exit
            # once the last index is reached.
            remaining = values_count - index
            if remaining == 1:
                break

            # Read 3-bytes of `source` as a 24-bit big-endian integer.
            sample_from_source = int.from_bytes(
                source[position:position + rand_bytes], 'big'
            )

            # Sample values greater than or equal to `sample_max` will cause
            # modulo bias when mapped into the `remaining` range.
            sample_max = rand_max - rand_max % remaining

            # Perform a swap if the consumed entropy will not cause modulo bias.
            if sample_from_source < sample_max:
                # Select a replacement index for the current index.
                replacement_position = (sample_from_source % remaining) + index
                # Swap the current index with the replacement index.
                output[index], output[replacement_position] = output[replacement_position], output[index]
                index += 1
            else:
                # The sample causes modulo bias. A new sample should be read.
                pass

    return output

Here's a function that splits a list into split_count pieces:

def split(seq: List[Any], split_count: int) -> List[Any]:
    """
    Returns the split ``seq`` in ``split_count`` pieces in protocol.
    """
    list_length = len(seq)
    return [
        seq[(list_length * i // split_count): (list_length * (i + 1) // split_count)]
        for i in range(split_count)
    ]

A helper method for readability:

def clamp(minval: int, maxval: int, x: int) -> int:
    if x <= minval:
        return minval
    elif x >= maxval:
        return maxval
    else:
        return x

Now, our combined helper method:

def get_new_shuffling(seed: Hash32,
                      validators: List[ValidatorRecord],
                      crosslinking_start_shard: int) -> List[List[ShardAndCommittee]]:
    active_validators = get_active_validator_indices(validators)

    committees_per_slot = clamp(
        1,
        SHARD_COUNT // CYCLE_LENGTH,
        len(active_validators) // CYCLE_LENGTH // TARGET_COMMITTEE_SIZE,
    )

    output = []

    # Shuffle with seed
    shuffled_active_validator_indices = shuffle(active_validators, seed)

    # Split the shuffled list into cycle_length pieces
    validators_per_slot = split(shuffled_active_validator_indices, CYCLE_LENGTH)

    for slot, slot_indices in enumerate(validators_per_slot):
        # Split the shuffled list into committees_per_slot pieces
        shard_indices = split(slot_indices, committees_per_slot)

        shard_id_start = crosslinking_start_shard + slot * committees_per_slot

        shards_and_committees_for_slot = [
            ShardAndCommittee(
                shard=(shard_id_start + shard_position) % SHARD_COUNT,
                committee=indices
            )
            for shard_position, indices in enumerate(shard_indices)
        ]
        output.append(shards_and_committees_for_slot)

    return output

Here's a diagram of what's going on:

We also define two functions for retrieving data from the state:

def get_shards_and_committees_for_slot(crystallized_state: CrystallizedState,
                                       slot: int) -> List[ShardAndCommittee]:
    earliest_slot_in_array = crystallized_state.last_state_recalculation - CYCLE_LENGTH
    assert earliest_slot_in_array <= slot < earliest_slot_in_array + CYCLE_LENGTH * 2
    return crystallized_state.shard_and_committee_for_slots[slot - earliest_slot_in_array]

def get_block_hash(active_state: ActiveState,
                   current_block: BeaconBlock,
                   slot: int) -> Hash32:
    earliest_slot_in_array = current_block.slot - len(active_state.recent_block_hashes)
    assert earliest_slot_in_array <= slot < current_block.slot
    return active_state.recent_block_hashes[slot - earliest_slot_in_array]

get_block_hash(_, _, s) should always return the block hash in the beacon chain at slot s, and get_shards_and_committees_for_slot(_, s) should not change unless the validator set changes.

We define another set of helpers to be used throughout: bytes1(x): return x.to_bytes(1, 'big'), bytes2(x): return x.to_bytes(2, 'big'), and so on for all integers, particularly 1, 2, 3, 4, 8, 32.

We define a function to "add a link" to the validator hash chain, used when a validator is added or removed:

def add_validator_set_change_record(crystallized_state: CrystallizedState,
                                    index: int,
                                    pubkey: int,
                                    flag: int) -> None:
    crystallized_state.validator_set_delta_hash_chain = \
        hash(crystallized_state.validator_set_delta_hash_chain +
             bytes1(flag) + bytes3(index) + bytes32(pubkey))

Finally, we abstractly define int_sqrt(n) for use in reward/penalty calculations as the largest integer k such that k**2 <= n. Here is one possible implementation, though clients are free to use their own including standard libraries for integer square root if available and meet the specification.

def int_sqrt(n: int) -> int:
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

On startup

Run the following code:

def on_startup(initial_validator_entries: List[Any]) -> Tuple[CrystallizedState, ActiveState]:
    # Induct validators
    validators = []
    for pubkey, proof_of_possession, withdrawal_shard, withdrawal_address, \
            randao_commitment in initial_validator_entries:
        add_validator(
            validators=validators,
            pubkey=pubkey,
            proof_of_possession=proof_of_possession,
            withdrawal_shard=withdrawal_shard,
            withdrawal_address=withdrawal_address,
            randao_commitment=randao_commitment,
            current_slot=0,
            status=ACTIVE,
        )
    # Setup crystallized state
    x = get_new_shuffling(bytes([0] * 32), validators, 0)
    crosslinks = [
        CrosslinkRecord(
            slot=0,
            hash=bytes([0] * 32)
        )
        for i in range(SHARD_COUNT)
    ]
    crystallized_state = CrystallizedState(
        validator_set_change_slot=0,
        validators=validators,
        crosslinks=crosslinks,
        last_state_recalculation_slot=0,
        last_finalized_slot=0,
        last_justified_slot=0,
        justified_streak=0,
        shard_and_committee_for_slots=x + x,
        persistent_committees=split(shuffle(validators, bytes([0] * 32)), SHARD_COUNT),
        persistent_committee_reassignments=[],
        deposits_penalized_in_period=[],
        next_shuffling_seed=b'\x00'*32,
        validator_set_delta_hash_chain=bytes([0] * 32),  # stub
        pre_fork_version=INITIAL_FORK_VERSION,
        post_fork_version=INITIAL_FORK_VERSION,
        fork_slot_number=0
    )

    # Setup active state
    recent_block_hashes = [
        bytes([0] * 32)
        for _ in range(CYCLE_LENGTH * 2)
    ]
    active_state = ActiveState(
        pending_attestations=[],
        pending_specials=[],
        recent_block_hashes=recent_block_hashes,
        randao_mix=bytes([0] * 32)  # stub
    )

    return crystallized_state, active_state

The CrystallizedState() and ActiveState() constructors should initialize all values to zero bytes, an empty value or an empty array depending on context. The add_validator routine is defined below.

Routine for adding a validator

This routine should be run for every validator that is inducted as part of a log created on the PoW chain [TODO: explain where to check for these logs]. The status of the validators added after genesis is PENDING_ACTIVATION. These logs should be processed in the order in which they are emitted by the PoW chain.

First, a helper function:

def min_empty_validator(validators: List[ValidatorRecord]):
    for i, v in enumerate(validators):
        if v.status == WITHDRAWN:
            return i
    return None

Now, to add a validator:

def add_validator(validators: List[ValidatorRecord],
                  pubkey: int,
                  proof_of_possession: bytes,
                  withdrawal_shard: int,
                  withdrawal_address: Address,
                  randao_commitment: Hash32,
                  status: int,
                  current_slot: int) -> int:
    # if following assert fails, validator induction failed
    # move on to next validator registration log
    assert BLSVerify(pub=pubkey,
                     msg=hash(pubkey),
                     sig=proof_of_possession)
    rec = ValidatorRecord(
        pubkey=pubkey,
        withdrawal_shard=withdrawal_shard,
        withdrawal_address=withdrawal_address,
        randao_commitment=randao_commitment,
        randao_last_change=current_slot,
        balance=DEPOSIT_SIZE * GWEI_PER_ETH,
        status=status,
        exit_slot=0
    )
    index = min_empty_validator(validators)
    if index is None:
        validators.append(rec)
        return len(validators) - 1
    else:
        validators[index] = rec
        return index

Routine for removing a validator

def exit_validator(index, crystallized_state, penalize, current_slot):
    validator = crystallized_state.validators[index]
    validator.exit_slot = current_slot
    if penalize:
        validator.status = PENALIZED
        crystallized_state.deposits_penalized_in_period[current_slot // WITHDRAWAL_PERIOD] += validator.balance
    else:
        validator.status = PENDING_EXIT
    add_validator_set_change_record(crystallized_state, index, validator.pubkey, EXIT)

Per-block processing

This procedure should be carried out every beacon block.

• Let parent_hash be the hash of the immediate previous beacon block (ie. equal to ancestor_hashes[0]).
• Let parent be the beacon block with the hash parent_hash

First, set recent_block_hashes to the output of the following:

def append_to_recent_block_hashes(old_block_hashes: List[Hash32],
                                  parent_slot: int,
                                  current_slot: int,
                                  parent_hash: Hash32) -> List[Hash32]:
    d = current_slot - parent_slot
    return old_block_hashes + [parent_hash] * d

The output of get_block_hash should not change, except that it will no longer throw for current_slot - 1. Also, check that the block's ancestor_hashes array was correctly updated, using the following algorithm:

def update_ancestor_hashes(parent_ancestor_hashes: List[Hash32],
                           parent_slot_number: int,
                           parent_hash: Hash32) -> List[Hash32]:
    new_ancestor_hashes = copy.copy(parent_ancestor_hashes)
    for i in range(32):
        if parent_slot_number % 2**i == 0:
            new_ancestor_hashes[i] = parent_hash
    return new_ancestor_hashes

A beacon block can have 0 or more AttestationRecord objects

For each one of these attestations:

• Verify that slot <= parent.slot and slot >= max(parent.slot - CYCLE_LENGTH + 1, 0).
• Verify that justified_slot is equal to or earlier than last_justified_slot.
• Verify that justified_block_hash is the hash of the block in the current chain at the slot -- justified_slot.
• Verify that either last_crosslink_hash or shard_block_hash equals state.crosslinks[shard].shard_block_hash.
• Compute parent_hashes = [get_block_hash(active_state, block, slot - CYCLE_LENGTH + i) for i in range(1, CYCLE_LENGTH - len(oblique_parent_hashes) + 1)] + oblique_parent_hashes (eg, if CYCLE_LENGTH = 4, slot = 5, the actual block hashes starting from slot 0 are Z A B C D E F G H I J, and oblique_parent_hashes = [D', E'] then parent_hashes = [B, C, D' E']). Note that when creating an attestation for a block, the hash of that block itself won't yet be in the active_state, so you would need to add it explicitly.
• Let attestation_indices be get_shards_and_committees_for_slot(crystallized_state, slot)[x], choosing x so that attestation_indices.shard equals the shard value provided to find the set of validators that is creating this attestation record.
• Verify that len(attester_bitfield) == ceil_div8(len(attestation_indices)), where ceil_div8 = (x + 7) // 8. Verify that bits len(attestation_indices).... and higher, if present (i.e. len(attestation_indices) is not a multiple of 8), are all zero.
• Derive a group public key by adding the public keys of all of the attesters in attestation_indices for whom the corresponding bit in attester_bitfield (the ith bit is (attester_bitfield[i // 8] >> (7 - (i %8))) % 2) equals 1.
• Let fork_version = pre_fork_version if slot < fork_slot_number else post_fork_version.
• Verify that aggregate_sig verifies using the group pubkey generated and the serialized form of AttestationSignedData(fork_version, slot, shard, parent_hashes, shard_block_hash, last_crosslinked_hash, shard_block_combined_data_root, justified_slot) as the message.

Extend the list of AttestationRecord objects in the active_state with those included in the block, ordering the new additions in the same order as they came in the block. Similarly extend the list of SpecialRecord objects in the active_state with those included in the block.

Let curblock_proposer_index be the validator index of the block.slot % len(get_shards_and_committees_for_slot(crystallized_state, block.slot)[0].committee)'th attester in get_shards_and_committees_for_slot(crystallized_state, block.slot)[0], and parent_proposer_index be the validator index of the parent block, calculated similarly. Verify that an attestation from the parent_proposer_index'th validator is part of the first (ie. item 0 in the array) AttestationRecord object; this attester can be considered to be the proposer of the parent block. In general, when a beacon block is produced, it is broadcasted at the network layer along with the attestation from its proposer.

Additionally, verify and update the RANDAO reveal. This is done as follows:

• Let repeat_hash(x, n) = x if n == 0 else repeat_hash(hash(x), n-1).
• Let V = crystallized_state.validators[curblock_proposer_index].
• Verify that repeat_hash(block.randao_reveal, (block.slot - V.randao_last_change) // RANDAO_SLOTS_PER_LAYER + 1) == V.randao_commitment, and set active_state.randao_mix = xor(active_state.randao_mix, block.randao_reveal) and append to ActiveState.pending_specials a SpecialObject(kind=RANDAO_CHANGE, data=[bytes8(curblock_proposer_index), block.randao_reveal, bytes8(block.slot)]).

State recalculations (every CYCLE_LENGTH slots)

Repeat while slot - last_state_recalculation_slot >= CYCLE_LENGTH:

Adjust justified slots and crosslink status

For every slot s in the range last_state_recalculation_slot - CYCLE_LENGTH ... last_state_recalculation_slot - 1:

• Let total_balance be the total balance of active validators.
• Let total_balance_attesting_at_s be the total balance of validators that attested to the beacon block at slot s.
• If 3 * total_balance_attesting_at_s >= 2 * total_balance set last_justified_slot = max(last_justified_slot, s) and justified_streak += 1. Otherwise set justified_streak = 0.
• If justified_streak >= CYCLE_LENGTH + 1 set last_finalized_slot = max(last_finalized_slot, s - CYCLE_LENGTH - 1).

For every (shard, shard_block_hash) tuple:

• Let total_balance_attesting_to_h be the total balance of validators that attested to the shard block with hash shard_block_hash.
• Let total_committee_balance be the total balance in the committee of validators that could have attested to the shard block with hash shard_block_hash.
• If 3 * total_balance_attesting_to_h >= 2 * total_committee_balance, set crosslinks[shard] = CrosslinkRecord(slot=last_state_recalculation_slot + CYCLE_LENGTH, hash=shard_block_hash).

Balance recalculations related to FFG rewards

Note: When applying penalties in the following balance recalculations implementers should make sure the uint64 does not underflow.

• Let total_balance be the total balance of active validators.
• Let total_balance_in_eth = total_balance // GWEI_PER_ETH.
• Let reward_quotient = BASE_REWARD_QUOTIENT * int_sqrt(total_balance_in_eth). (The per-slot maximum interest rate is 1/reward_quotient.)
• Let quadratic_penalty_quotient = SQRT_E_DROP_TIME**2. (The portion lost by offline validators after D slots is about D*D/2/quadratic_penalty_quotient.)
• Let time_since_finality = block.slot - last_finalized_slot.

For every slot s in the range last_state_recalculation_slot - CYCLE_LENGTH ... last_state_recalculation_slot - 1:

• Let total_balance_participating be the total balance of validators that voted for the canonical beacon block at slot s. In the normal case every validator will be in one of the CYCLE_LENGTH slots following slot s and so can vote for a block at slot s.
• Let B be the balance of any given validator whose balance we are adjusting, not including any balance changes from this round of state recalculation.
• If time_since_finality <= 3 * CYCLE_LENGTH adjust the balance of participating and non-participating validators as follows:
  • Participating validators gain B // reward_quotient * (2 * total_balance_participating - total_balance) // total_balance. (Note that this value may be negative.)
  • Non-participating validators lose B // reward_quotient.
• Otherwise:
  • Participating validators gain nothing.
  • Non-participating validators lose B // reward_quotient + B * time_since_finality // quadratic_penalty_quotient.

In addition, validators with status == PENALIZED lose B // reward_quotient + B * time_since_finality // quadratic_penalty_quotient.

Balance recalculations related to crosslink rewards

For every shard number shard for which a crosslink committee exists in the cycle prior to the most recent cycle (last_state_recalculation_slot - CYCLE_LENGTH ... last_state_recalculation_slot - 1), let V be the corresponding validator set. Let B be the balance of any given validator whose balance we are adjusting, not including any balance changes from this round of state recalculation. For each shard, V:

• Let total_balance_of_v be the total balance of V.
• Let winning_shard_hash be the hash that the largest total deposits signed for the shard during the cycle.
• Define a "participating validator" as a member of V that signed a crosslink of winning_shard_hash.
• Let total_balance_of_v_participating be the total balance of the subset of V that participated.
• Let time_since_last_confirmation = block.slot - crosslinks[shard].slot.
• Adjust balances as follows:
  • Participating validators gain B // reward_quotient * (2 * total_balance_of_v_participating - total_balance_of_v) // total_balance_of_v.
  • Non-participating validators lose B // reward_quotient.

Process penalties, logouts and other special objects

For each SpecialRecord obj in active_state.pending_specials:

• [covers LOGOUT]: If obj.kind == LOGOUT, interpret data[0:3] as a uint24 and data[3:35] as a hash32, where validator_index = data[0:3] and signature = data[3:35]. If BLSVerify(pubkey=validators[validator_index].pubkey, msg=hash(LOGOUT_MESSAGE + bytes8(fork_version)), sig=signature), where fork_version = pre_fork_version if slot < fork_slot_number else post_fork_version, and validators[validator_index].status == ACTIVE, run exit_validator(validator_index, crystallized_state, penalize=False, current_slot=block.slot)
• [covers NO_DBL_VOTE, NO_SURROUND, NO_DBL_PROPOSE slashing conditions]: If obj.kind == CASPER_SLASHING, interpret data[0] as a list of concatenated uint32 values where each value represents an index into validators, data[1] as the data being signed and data[2] as an aggregate signature. Interpret data[3:6] similarly. Verify that both signatures are valid, that the two signatures are signing distinct data, and that they are either signing the same slot number, or that one surrounds the other (ie. source1 < source2 < target2 < target1). Let indices be the list of indices in both signatures; verify that its length is at least 1. For each validator index v in indices, if its status does not equal PENALIZED, then run exit_validator(v, crystallized_state, penalize=True, current_slot=block.slot)
• [covers RANDAO_CHANGE]: If obj.kind == RANDAO_CHANGE, interpret data[0:3] as a uint24, data[3:35] as a hash32, and data[35:43] as a uint64, where block_proposer_index = data[0:3], randao_commitment = data[3:35], and randao_last_change = data[35:43]. Set validators[block_proposer_index].randao_commitment = randao_commitment, and set validators[block_proposer_index].randao_last_change = randao_last_change.

Validator set change

A validator set change can happen after a state recalculation if all of the following criteria are satisfied:

• block.slot - crystallized_state.validator_set_change_slot >= MIN_VALIDATOR_SET_CHANGE_INTERVAL
• last_finalized_slot > crystallized_state.validator_set_change_slot
• For every shard number shard in shard_and_committee_for_slots, crosslinks[shard].slot > crystallized_state.validator_set_change_slot

Then, run the following algorithm to update the validator set:

def change_validators(validators: List[ValidatorRecord]) -> None:
    # The active validator set
    active_validators = get_active_validator_indices(validators)
    # The total balance of active validators
    total_balance = sum([v.balance for i, v in enumerate(validators) if i in active_validators])
    # The maximum total wei that can deposit+withdraw
    max_allowable_change = max(
        2 * DEPOSIT_SIZE * GWEI_PER_ETH,
        total_balance // MAX_VALIDATOR_CHURN_QUOTIENT
    )
    # Go through the list start to end depositing+withdrawing as many as possible
    total_changed = 0
    for i in range(len(validators)):
        if validators[i].status == PENDING_ACTIVATION:
            validators[i].status = ACTIVE
            total_changed += DEPOSIT_SIZE * GWEI_PER_ETH
            add_validator_set_change_record(
                crystallized_state=crystallized_state,
                index=i,
                pubkey=validators[i].pubkey,
                flag=ENTRY
            )
        if validators[i].status == PENDING_EXIT:
            validators[i].status = PENDING_WITHDRAW
            validators[i].exit_slot = current_slot
            total_changed += validators[i].balance
            add_validator_set_change_record(
                crystallized_state=crystallized_state,
                index=i,
                pubkey=validators[i].pubkey,
                flag=EXIT
            )
        if total_changed >= max_allowable_change:
            break

    # Calculate the total ETH that has been penalized in the last ~2-3 withdrawal periods
    period_index = current_slot // WITHDRAWAL_PERIOD
    total_penalties = (
        (crystallized_state.deposits_penalized_in_period[period_index]) +
        (crystallized_state.deposits_penalized_in_period[period_index - 1] if period_index >= 1 else 0) +
        (crystallized_state.deposits_penalized_in_period[period_index - 2] if period_index >= 2 else 0)
    )
    # Separate loop to withdraw validators that have been logged out for long enough, and
    # calculate their penalties if they were slashed
    for i in range(len(validators)):
        if validators[i].status in (PENDING_WITHDRAW, PENALIZED) and current_slot >= validators[i].exit_slot + WITHDRAWAL_PERIOD:
            if validators[i].status == PENALIZED:
                validators[i].balance -= validators[i].balance * min(total_penalties * 3, total_balance) // total_balance
            validators[i].status = WITHDRAWN

            withdraw_amount = validators[i].balance
            ...
            # STUB: withdraw to shard chain

• Set crystallized_state.validator_set_change_slot = crystallized_state.last_state_recalculation_slot
• Set shard_and_committee_for_slots[:CYCLE_LENGTH] = shard_and_committee_for_slots[CYCLE_LENGTH:]
• Let next_start_shard = (shard_and_committee_for_slots[-1][-1].shard + 1) % SHARD_COUNT
• Set shard_and_committee_for_slots[CYCLE_LENGTH:] = get_new_shuffling(crystallized_state.next_shuffling_seed, validators, next_start_shard)
• Set crystallized_state.next_shuffling_seed = active_state.randao_mix

If a validator set change does NOT happen

• Set shard_and_committee_for_slots[:CYCLE_LENGTH] = shard_and_committee_for_slots[CYCLE_LENGTH:]
• Let time_since_finality = block.slot - crystallized_state.validator_set_change_slot
• Let start_shard = shard_and_committee_for_slots[0][0].shard
• If time_since_finality * CYCLE_LENGTH <= MIN_VALIDATOR_SET_CHANGE_INTERVAL or time_since_finality is an exact power of 2, set shard_and_committee_for_slots[CYCLE_LENGTH:] = get_new_shuffling(crystallized_state.next_shuffling_seed, validators, start_shard) and set crystallized_state.next_shuffling_seed = active_state.randao_mix. Note that start_shard is not changed from last cycle.

Finally...

• Remove all attestation records older than slot crystallized_state.last_state_recalculation_slot
• Empty the active_state.pending_specials list
• For any validator with index v with balance less than MIN_ONLINE_DEPOSIT_SIZE and status ACTIVE, run exit_validator(v, crystallized_state, penalize=False, current_slot=block.slot)
• Set active_state.recent_block_hashes = active_state.recent_block_hashes[CYCLE_LENGTH:]
• Set crystallized_state.last_state_recalculation_slot += CYCLE_LENGTH

For any validator that was added or removed from the active validator list during this state recalculation:

• If the validator was removed, remove their index from the persistent_committees and remove any ShardReassignmentRecords containing their index from persistent_committee_reassignments.
• If the validator was added with index validator_index:
  • let assigned_shard = hash(active_state.randao_mix + bytes8(validator_index)) % SHARD_COUNT
  • let reassignment_record = ShardReassignmentRecord(validator_index=validator_index, shard=assigned_shard, slot=block.slot + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD)
  • Append reassignment_record to the end of persistent_committee_reassignments

Now run the following code to reshuffle a few proposers:

active_validator_indices = get_active_validator_indices(validators)
num_validators_to_reshuffle = len(active_validator_indices) // SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD
for i in range(num_validators_to_reshuffle):
    # Multiplying i to 2 to ensure we have different input to all the required hashes in the shuffling
    # and none of the hashes used for entropy in this loop will be the same
    vid = active_validator_indices[hash(active_state.randao_mix + bytes8(i * 2)) % len(active_validator_indices)]
    new_shard = hash(active_state.randao_mix + bytes8(i * 2 + 1)) % SHARD_COUNT
    shard_reassignment_record = ShardReassignmentRecord(
        validator_index=vid,
        shard=new_shard,
        slot=block.slot + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD
    )
    crystallized_state.persistent_committee_reassignments.append(shard_reassignment_record)

while len(crystallized_state.persistent_committee_reassignments) > 0 and crystallized_state.persistent_committee_reassignments[0].slot <= block.slot:
    rec = crystallized_state.persistent_committee_reassignments.pop(0)
    for committee in crystallized_state.persistent_committees:
        if rec.validator_index in committee:
            committee.pop(
                committee.index(rec.validator_index)
            )
    crystallized_state.persistent_committees[rec.shard].append(rec.validator_index)

TODO

Note: This spec is ~65% complete.

Missing

• Specify the Merklelisation rules for beacon state and blocks and merge crystallized_state_root and active_state_root (issue 54)
• Specify the rules around acceptable values for pow_chain_reference (issue 58)
• Specify the shard chain blocks, blobs, proposers, etc.
• Specify the deposit contract on the PoW chain in Vyper
• Specify the beacon chain genesis rules (issue 58)
• Specify the logic for proofs of custody, including slashing conditions
• Specify BLSVerify and rework the spec for BLS12-381 throughout
• Specify the constraints for SpecialRecords (issue 43)
• Undergo peer review, security audits and formal verification

Documentation

• Specify the various assumptions (global clock, networking latency, validator honesty, validator liveness, etc.)
• Add an appendix on gossip networks and the offchain signature aggregation logic
• Add a glossary (in a separate glossary.md) to comprehensively and precisely define all the terms
• Clearly document the various edge cases, e.g. with committee sizing
• Rework the document for readability

Possible modifications and additions

• Replace the IMD fork choice rule with LMD
• Homogenise types to uint64 (PR 36)
• Reduce the slot duration to 8 seconds
• Allow for the delayed inclusion of aggregated signatures
• Introduce a RANDAO slashing condition for early reveals
• Use a separate hash function for the proof of possession
• Rework the ShardAndCommittee data structures
• Add a double-batched Merkle accumulator for historical beacon chain blocks
• Allow for deposits larger than 32 ETH, as well as deposit top-ups
• Add penalties for deposits below 32 ETH (or some other threshold)
• Add a SpecialRecord to (re)register

Appendix

Appendix A - Hash function

We aim to have a STARK-friendly hash function hash(x) for the production launch of the beacon chain. While the standardisation process for a STARK-friendly hash function takes place—led by STARKware, who will produce a detailed report with recommendations—we use BLAKE2b-512 as a placeholder. Specifically, we set hash(x) := BLAKE2b-512(x)[0:32] where the BLAKE2b-512 algorithm is defined in RFC 7693 and the input x is of type bytes.

Copyright

Copyright and related rights waived via CC0.