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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
`DEPOSIT_CONTRACT_ADDRESS`	TBD	-
`TARGET_COMMITTEE_SIZE`	2**8 (= 256)	validators
`SLOT_DURATION`	6	seconds
`CYCLE_LENGTH`	2**6 (= 64)	slots	~6 minutes
`MIN_VALIDATOR_SET_CHANGE_INTERVAL`	2**8 (= 256)	slots	~25 minutes
`SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD`	2**17 (= 131,072)	slots	~9 days
`MIN_ATTESTATION_INCLUSION_DELAY`	2**2 (= 4)	slots	~24 seconds
`SQRT_E_DROP_TIME`	2**11 (= 1,024)	cycles	~9 days
`WITHDRAWALS_PER_CYCLE`	2**2 (=4)	validators	5.2m ETH in ~6 months
`MIN_WITHDRAWAL_PERIOD`	2**13 (= 8,192)	slots	~14 hours
`DELETION_PERIOD`	2**22 (= 4,194,304)	slots	~290 days
`COLLECTIVE_PENALTY_CALCULATION_PERIOD`	2**20 (= 1,048,576)	slots	~2.4 months
`POW_RECEIPT_ROOT_VOTING_PERIOD`	2**10 (= 1,024)	slots	~1.7 hours
`SLASHING_WHISTLEBLOWER_REWARD_DENOMINATOR`	2**9 (= 512)
`BASE_REWARD_QUOTIENT`	2**11 (= 2,048)	—
`INCLUDER_REWARD_SHARE_QUOTIENT`	2**3 (= 8)	—
`MAX_VALIDATOR_CHURN_QUOTIENT`	2**5 (= 32)	—
`POW_CONTRACT_MERKLE_TREE_DEPTH`	2**5 (= 32)	-
`MAX_ATTESTATION_COUNT`	2**7 (= 128)	-
`LOGOUT_MESSAGE`	`"LOGOUT"`	—
`INITIAL_FORK_VERSION`	0	—

Notes

See a recommended min committee size of 111 here; 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 dictates the per-cycle interest rate assuming all validators are participating, assuming total deposits of 1 ETH. It corresponds to ~2.57% 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	Maximum count
`LOGOUT`	`0`	`16`
`CASPER_SLASHING`	`1`	`16`
`PROPOSER_SLASHING`	`2`	`16`
`DEPOSIT_PROOF`	`3`	`16`

Validator set delta flags

Name	Value
`ENTRY`	`0`
`EXIT`	`1`

Domains for BLS signatures

Name	Value
`DOMAIN_DEPOSIT`	`0`
`DOMAIN_ATTESTATION`	`1`
`DOMAIN_PROPOSAL`	`2`
`DOMAIN_LOGOUT`	`3`

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_credentials, 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 receipt root
    'candidate_pow_receipt_root': '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'],
    # State root
    'state_root': 'hash32',
    # Attestations
    'attestations': [AttestationRecord],
    # Specials (e.g. logouts, penalties)
    'specials': [SpecialRecord],
    # Proposer signature
    'proposer_signature': ['uint384'],
}

An AttestationRecord has the following fields:

{
    data: AttestationSignedData
    # Attester participation bitfield
    'attester_bitfield': 'bytes',
    # Proof of custody bitfield
    'poc_bitfield': 'bytes',
    # BLS aggregate signature
    'aggregate_sig': ['uint384']
}

AttestationSignedData:

{
        # Slot number
        'slot': 'uint64',
        # Shard number
        'shard': 'uint64',
        # Hash of the block we're signing
        'block_hash': 'hash32',
        # Hash of the ancestor at the cycle boundary
        'cycle_boundary_hash': 'hash32',
        # Shard block hash being attested to
        'shard_block_hash': 'hash32',
        # Last crosslink hash
        'last_crosslink_hash': 'hash32',
        # Slot of last justified beacon block
        'justified_slot': 'uint64',
        # Hash of last justified beacon block
        'justified_block_hash': 'hash32',
}

A ProposalSignedData has the following fields:

{
    # Slot number
    'slot': 'uint64',
    # Shard number (or `2**64 - 1` for beacon chain)
    'shard': 'uint64',
    # Block hash
    'block_hash': 'hash32',
}

A SpecialRecord has the following fields:

{
    # Kind
    'kind': 'uint64',
    # Data
    'data': 'bytes'
}

Beacon chain state

The BeaconState 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 cycle-boundary state recalculation
    'last_state_recalculation_slot': 'uint64',
    # Last finalized slot
    'last_finalized_slot': 'uint64',
    # Justification source
    'justification_source': 'uint64',
    'prev_cycle_justification_source': 'uint64',
    # Recent justified slot bitmask
    'justified_slot_bitfield': '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': ['uint64'],
    # Hash chain of validator set changes (for light clients to easily track deltas)
    'validator_set_delta_hash_chain': 'hash32'
    # Current sequence number for withdrawals
    'current_exit_seq': 'uint64',
    # Genesis time
    'genesis_time': 'uint64',
    # PoW receipt root
    'processed_pow_receipt_root': 'hash32',
    'candidate_pow_receipt_roots': [CandidatePoWReceiptRootRecord],
    # Parameters relevant to hard forks / versioning.
    # Should be updated only by hard forks.
    'pre_fork_version': 'uint64',
    'post_fork_version': 'uint64',
    'fork_slot_number': 'uint64',
    # Attestations not yet processed
    'pending_attestations': [ProcessedAttestations],
    # recent beacon block hashes needed to process attestations, older to newer
    'recent_block_hashes': ['hash32'],
    # RANDAO state
    'randao_mix': 'hash32'
}

A ValidatorRecord has the following fields:

{
    # BLS public key
    'pubkey': 'uint384',
    # Withdrawal credentials
    'withdrawal_credentials': 'hash32',
    # RANDAO commitment
    'randao_commitment': 'hash32',
    # Slot the proposer has skipped (ie. layers of RANDAO expected)
    'randao_skips': 'uint64',
    # Balance in Gwei
    'balance': 'uint64',
    # Status code
    'status': 'uint64',
    # Slot when validator last changed status (or 0)
    'last_status_change_slot': 'uint64'
    # Sequence number when validator exited (or 0)
    'exit_seq': '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': 'uint64',
    # Validator indices
    'committee': ['uint24']
}

A ShardReassignmentRecord object has the following fields:

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

A CandidatePoWReceiptRootRecord object contains the following fields:

{
    # Candidate PoW receipt root
    'candidate_pow_receipt_root': 'hash32',
    # Vote count
    'votes': 'uint64'
}

A ProcessedAttestation object has the following fields:

{
    # Signed data
    'data': AttestationSignedData,
    # Attester participation bitfield (2 bits per attester)
    'attester_bitfield': 'bytes',
    # Proof of custody bitfield
    'poc_bitfield': 'bytes',
    # Slot in which it was included
    'slot_included': '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 processed_pow_receipt_root has already been processed and accepted
The node's local clock time is greater than or equal to the minimum timestamp as computed by state.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 fork choice rule is a hybrid that combines justification and finality with Latest Message Driven (LMD) Greediest Heaviest Observed SubTree (GHOST). At any point in time a validator v subjectively calculates the beacon chain head as follows.

Let store be the set of attestations and blocks that the validator v has observed and verified (in particular, block ancestors must be recursively verified). Attestations not part of any chain are still included in store.
Let finalized_head be the finalized block with the highest slot number. (A block B is finalized if there is a descendant of B in store the processing of which sets B as finalized.)
Let justified_head be the descendant of finalized_head with the highest slot number that has been justified for at least CYCLE_LENGTH slots. (A block B is justified is there is a descendant of B in store the processing of which sets B as justified.) If no such descendant exists set justified_head to finalized_head.
Let get_ancestor(store, block, slot) be the ancestor of block with slot number slot. The get_ancestor function can be defined recursively as def get_ancestor(store, block, slot): return block if block.slot == slot else get_ancestor(store, store.get_parent(block), slot).
Let get_latest_attestation(store, validator) be the attestation with the highest slot number in store from validator. If several such attestations exist use the one the validator v observed first.
Let get_latest_attestation_target(store, validator) be the target block in the attestation get_latest_attestation(store, validator).
The head is lmd_ghost(store, justified_head) where the function lmd_ghost is defined below. Note that the implementation below is suboptimal; there are implementations that compute the head in time logarithmic in slot count.

def lmd_ghost(store, start):
    validators = start.state.validators
    active_validators = [validators[i] for i in
                         get_active_validator_indices(validators, start.slot)]
    attestation_targets = [get_latest_attestation_target(store, validator)
                           for validator in active_validators]
    def get_vote_count(block):
        return len([target for target in attestation_targets if
                    get_ancestor(store, target, block.slot) == block])

    head = start
    while 1:
        children = get_children(head)
        if len(children) == 0:
            return head        
        head = max(children, key=get_vote_count)

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:

The per-block processing, which happens every block, and only affects a few parts of the state.
The inter-cycle state recalculation, which happens only if block.slot >= last_state_recalculation_slot + CYCLE_LENGTH, and affects the entire state.

The inter-cycle 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, while the per-block processing generally focuses on verifying aggregate signatures and saving temporary records relating to the per-block activity in the BeaconState.

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 any list; we primarily use it for 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(state: BeaconState,
                                       slot: int) -> List[ShardAndCommittee]:
    earliest_slot_in_array = state.last_state_recalculation_slot - CYCLE_LENGTH
    assert earliest_slot_in_array <= slot < earliest_slot_in_array + CYCLE_LENGTH * 2
    return state.shard_and_committee_for_slots[slot - earliest_slot_in_array]

def get_block_hash(state: BeaconState,
                   current_block: BeaconBlock,
                   slot: int) -> Hash32:
    earliest_slot_in_array = current_block.slot - len(state.recent_block_hashes)
    assert earliest_slot_in_array <= slot < current_block.slot
    return 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.

The following is a function that determines the proposer of a beacon block:

def get_beacon_proposer_index(state:BeaconState, slot: int) -> ValidatorRecord:
    first_committee = get_shards_and_committees_for_slot(state, slot)[0].committee
    index = first_committee[slot % len(first_committee)]
    return index

The following is a function that determines the validators that participated in an attestation:

def get_attestation_participants(state: State,
                                 attestation_data: AttestationSignedData,
                                 attester_bitfield: bytes) -> List[int]:
    sncs_for_slot = get_shards_and_committees_for_slot(state, attestation_data.slot)
    snc = [x for x in sncs_for_slot if x.shard == attestation_data.shard][0]
    assert len(attester_bitfield) == ceil_div8(len(snc.committee))
    participants = []
    for i, vindex in enumerate(snc.committee):
        bit = (attester_bitfield[i//8] >> (7 - (i % 8))) % 2
        if bit == 1:
            participants.append(vindex)
    return participants

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 determine the balance of a validator used for determining punishments and calculating stake:

def balance_at_stake(validator: ValidatorRecord) -> int:
   return min(validator.balance, DEPOSIT_SIZE)

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(state: BeaconState,
                                    index: int,
                                    pubkey: int,
                                    flag: int) -> None:
    state.validator_set_delta_hash_chain = \
        hash(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

PoW chain contract

The beacon chain is initialized when a condition is met inside a contract on the existing PoW chain. This contract's code in Vyper is as follows:

SECONDS_PER_DAY: constant(uint256) = 86400
POW_CONTRACT_MERKLE_TREE_DEPTH: constant(int128) = 32

HashChainValue: event({previous_receipt_root: bytes32, data: bytes[2064], total_deposit_count: int128})
ChainStart: event({receipt_root: bytes32, time: bytes[8]})

receipt_tree: bytes32[int128]
total_deposit_count: int128

@payable
@public
def deposit(deposit_params: bytes[2048]):
    index:int128 = self.total_deposit_count + 2**POW_CONTRACT_MERKLE_TREE_DEPTH
    msg_gwei_bytes8: bytes[8] = slice(concat("", convert(msg.value / 10**9, bytes32)), start=24, len=8)
    timestamp_bytes8: bytes[8] = slice(concat("", convert(block.timestamp, bytes32)), start=24, len=8)
    deposit_data: bytes[2064] = concat(deposit_params, msg_gwei_bytes8, timestamp_bytes8)

    log.HashChainValue(self.receipt_tree[1], deposit_data, self.total_deposit_count)    

    self.receipt_tree[index] = sha3(deposit_data)
    for i in range(POW_CONTRACT_MERKLE_TREE_DEPTH):
        index /= 2
        self.receipt_tree[index] = sha3(concat(self.receipt_tree[index * 2], self.receipt_tree[index * 2 + 1]))
    self.total_deposit_count += 1
    if self.total_deposit_count == 16384:
        timestamp_day_boundary: uint256 = as_unitless_number(block.timestamp) - as_unitless_number(block.timestamp) % SECONDS_PER_DAY + SECONDS_PER_DAY
        timestamp_day_boundary_bytes8: bytes[8] = slice(concat("", convert(timestamp_day_boundary, bytes32)), start=24, len=8)
        log.ChainStart(self.receipt_tree[1], timestamp_day_boundary_bytes8)

@public
@constant
def get_receipt_root() -> bytes32:
    return self.receipt_tree[1]

The contract is at address DEPOSIT_CONTRACT_ADDRESS. When a user wishes to become a validator by moving their ETH from the 1.0 chain to the 2.0 chain, they should call the deposit function, sending along 32 ETH and providing as deposit_params a SimpleSerialize'd DepositParams object of the form:

{
    'pubkey': 'int256',
    'proof_of_possession': ['int256'],
    'withdrawal_credentials`: 'hash32',
    'randao_commitment`: 'hash32'
}

If the user wishes to deposit more than DEPOSIT_SIZE ETH, they would need to make multiple calls. When the contract publishes a ChainStart log, this initializes the chain, calling on_startup with:

initial_validator_entries equal to the list of data records published as HashChainValue logs so far, in the order in which they were published (oldest to newest).
genesis_time equal to the time value published in the log
processed_pow_receipt_root equal to the receipt_root value published in the log

On startup

A valid block with slot 0 (the "genesis block") has the following values. Other validity rules (eg. requiring a signature) do not apply.

{
    'slot': 0,
    'randao_reveal': bytes32(0),
    'candidate_pow_receipt_roots': [],
    'ancestor_hashes': [bytes32(0) for i in range(32)],
    'state_root': STARTUP_STATE_ROOT,
    'attestations': [],
    'specials': [],
    'proposer_signature': [0, 0]
}

STARTUP_STATE_ROOT is the root of the initial state, computed by running the following code:

def on_startup(initial_validator_entries: List[Any], genesis_time: uint64, processed_pow_receipt_root: Hash32) -> BeaconState:
    # Induct validators
    validators = []
    for pubkey, proof_of_possession, withdrawal_credentials, \
            randao_commitment in initial_validator_entries:
        add_validator(
            validators=validators,
            pubkey=pubkey,
            proof_of_possession=proof_of_possession,
            withdrawal_credentials=withdrawal_credentials,
            randao_commitment=randao_commitment,
            current_slot=0,
            status=ACTIVE,
        )
    # Setup state
    x = get_new_shuffling(bytes([0] * 32), validators, 0)
    crosslinks = [
        CrosslinkRecord(
            slot=0,
            hash=bytes([0] * 32)
        )
        for i in range(SHARD_COUNT)
    ]
    state = BeaconState(
        validator_set_change_slot=0,
        validators=validators,
        crosslinks=crosslinks,
        last_state_recalculation_slot=0,
        last_finalized_slot=0,
        justification_source=0,
        prev_cycle_justification_source=0,
        justified_slot_bitfield=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=bytes([0] * 32),
        validator_set_delta_hash_chain=bytes([0] * 32),  # stub
        current_exit_seq=0,
        genesis_time=genesis_time,
        processed_pow_receipt_root=processed_pow_receipt_root,
        candidate_pow_receipt_roots=[],
        pre_fork_version=INITIAL_FORK_VERSION,
        post_fork_version=INITIAL_FORK_VERSION,
        fork_slot_number=0,
        pending_attestations=[],
        pending_specials=[],
        recent_block_hashes=[bytes([0] * 32) for _ in range(CYCLE_LENGTH * 2)],
        randao_mix=bytes([0] * 32)  # stub
    )

    return state

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, some helper functions:

def min_empty_validator_index(validators: List[ValidatorRecord], current_slot: int):
    for i, v in enumerate(validators):
        if v.status == WITHDRAWN and v.last_status_change_slot + DELETION_PERIOD <= current_slot:
            return i
    return None

def get_fork_version(state: State, slot: int) -> int:
    return state.pre_fork_version if slot < state.fork_slot_number else state.post_fork_version
    
def get_domain(state: State, slot: int, base_domain: int) -> int:
    return get_fork_version(state, slot) * 2**32 + base_domain

Now, to add a validator:

def add_validator(state: State,
                  pubkey: int,
                  proof_of_possession: bytes,
                  withdrawal_credentials: Hash32,
                  randao_commitment: Hash32,
                  status: int,
                  current_slot: int) -> int:
    # if following assert fails, validator induction failed
    # move on to next validator registration log
    signed_message = bytes32(pubkey) + bytes2(withdrawal_shard) + withdrawal_credentials + randao_commitment
    assert BLSVerify(pub=pubkey,
                     msg=hash(signed_message),
                     sig=proof_of_possession,
                     domain=get_domain(state, current_slot, DOMAIN_DEPOSIT))
    # Pubkey uniqueness
    assert pubkey not in [v.pubkey for v in state.validators]
    rec = ValidatorRecord(
        pubkey=pubkey,
        withdrawal_credentials=withdrawal_credentials,
        randao_commitment=randao_commitment,
        randao_skips=0,
        balance=DEPOSIT_SIZE * GWEI_PER_ETH,
        status=status,
        last_status_change_slot=current_slot,
        exit_seq=0
    )
    # Add the validator
    index = min_empty_validator_index(state.validators)
    if index is None:
        state.validators.append(rec)
        index = len(state.validators) - 1
    else:
        state.validators[index] = rec
    return index

BLSVerify is a function for verifying a BLS12-381 signature, defined in the BLS12-381 spec.

Routine for removing a validator

def exit_validator(index, state, block, penalize, current_slot):
    validator = state.validators[index]
    validator.last_status_change_slot = current_slot
    validator.exit_seq = state.current_exit_seq
    state.current_exit_seq += 1
    for committee in state.persistent_committees:
        for i, vindex in committee:
            if vindex == index:
                committee.pop(i)
                break
    if penalize:
        state.deposits_penalized_in_period[current_slot // COLLECTIVE_PENALTY_CALCULATION_PERIOD] += balance_at_stake(validator)
        validator.status = PENALIZED
        whistleblower_xfer_amount = validator.deposit // SLASHING_WHISTLEBLOWER_REWARD_DENOMINATOR
        validator.deposit -= whistleblower_xfer_amount
        state.validators[get_beacon_proposer_index(state, block.slot)].deposit += whistleblower_xfer_amount
    else:
        validator.status = PENDING_EXIT
    add_validator_set_change_record(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

Verify attestations

Verify that there are at most MAX_ATTESTATION_COUNT AttestationRecord objects.

For each AttestationRecord object obj:

Verify that obj.data.slot <= block.slot - MIN_ATTESTATION_INCLUSION_DELAY and obj.data.slot >= max(parent.slot - CYCLE_LENGTH + 1, 0).
Verify that obj.data.justified_slot is equal to justification_source if obj.data.slot >= state.last_state_recalculation_slot else prev_cycle_justification_source
Verify that obj.data.justified_block_hash is equal to get_block_hash(state, block, obj.data.justified_slot).
Verify that either obj.data.last_crosslink_hash or obj.data.shard_block_hash equals state.crosslinks[shard].shard_block_hash.
aggregate_sig verification:
- Let participants = get_attestation_participants(state, obj.data, obj.attester_bitfield)
- Let group_public_key = BLSAddPubkeys([state.validators[v].pubkey for v in participants])
- Check BLSVerify(pubkey=group_public_key, msg=obj.data, sig=aggregate_sig, domain=get_domain(state, slot, DOMAIN_ATTESTATION)).
[TO BE REMOVED IN PHASE 1] Verify that shard_block_hash == bytes([0] * 32).
Append ProcessedAttestation(data=obj.data, attester_bitfield=obj.attester_bitfield, poc_bitfield=obj.poc_bitfield, slot_included=block.slot) to state.pending_attestations.

Verify proposer signature

Let proposal_hash = hash(ProposalSignedData(block.slot, 2**64 - 1, block_hash_without_sig)) where block_hash_without_sig is the hash of the block except setting proposer_signature to [0, 0].

Verify that BLSVerify(pubkey=state.validators[get_beacon_proposer_index(state, block.slot)].pubkey, data=proposal_hash, sig=block.proposer_signature, domain=get_domain(state, block.slot, DOMAIN_PROPOSAL)) passes.

Verify and process RANDAO reveal

First run the following state transition to update randao_skips variables for the missing slots.

for slot in range(parent.slot + 1, block.slot):
    proposer_index = get_beacon_proposer_index(state, slot)
    state.validators[proposer_index].randao_skips += 1

Then:

Let repeat_hash(x, n) = x if n == 0 else repeat_hash(hash(x), n-1).
Let proposer = state.validators[get_beacon_proposer_index(state, block.slot)].
Verify that repeat_hash(block.randao_reveal, proposer.randao_skips + 1) == proposer.randao_commitment
Set state.randao_mix = xor(state.randao_mix, block.randao_reveal), proposer.randao_commitment = block.randao_reveal, proposer.randao_skips = 0

Process PoW receipt root

If block.candidate_pow_receipt_root is x.candidate_pow_receipt_root for some x in state.candidate_pow_receipt_roots, set x.votes += 1. Otherwise, append to state.candidate_pow_receipt_roots a new CandidatePoWReceiptRootRecord(candidate_pow_receipt_root=block.candidate_pow_receipt_root, votes=1).

Process penalties, logouts and other special objects

Verify that the quantity of each type of object in block.specials is less than or equal to its maximum (see table at the top). Verify that objects are sorted in order of kind (ie. block.specials[i+1].kind >= block.specials[i].kind for all 0 <= i < len(block.specials-1)).

For each SpecialRecord obj in block.specials, verify that its kind is one of the below values, and that obj.data deserializes according to the format for the given kind, then process it. The word "verify" when used below means that if the given verification check fails, the block containing that SpecialRecord is invalid.

LOGOUT

{
    'validator_index': 'uint64',
    'signature': '[uint384]'
}

Perform the following checks:

Verify that BLSVerify(pubkey=validators[data.validator_index].pubkey, msg=bytes([0] * 32), sig=data.signature, domain=get_domain(state, current_slot, DOMAIN_LOGOUT))
Verify that validators[validator_index].status == ACTIVE.
Verify that block.slot >= last_status_change_slot + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD

Run exit_validator(data.validator_index, state, block, penalize=False, current_slot=block.slot).

CASPER_SLASHING

{
    'vote1_aggregate_sig_indices': '[uint24]',
    'vote1_data': AttestationSignedData,
    'vote1_aggregate_sig': '[uint384]',
    'vote2_aggregate_sig_indices': '[uint24]',
    'vote2_data': AttestationSignedData,
    'vote2_aggregate_sig': '[uint384]',
}

Perform the following checks:

For each vote, verify that BLSVerify(pubkey=aggregate_pubkey([validators[i].pubkey for i in vote_aggregate_sig_indices]), msg=vote_data, sig=vote_aggregate_sig, domain=get_domain(state, vote_data.slot, DOMAIN_ATTESTATION)) passes.
Verify that vote1_data != vote2_data.
Let intersection = [x for x in vote1_aggregate_sig_indices if x in vote2_aggregate_sig_indices]. Verify that len(intersection) >= 1.
Verify that vote1_data.justified_slot < vote2_data.justified_slot < vote2_data.slot <= vote1_data.slot.

For each validator index v in intersection, if state.validators[v].status does not equal PENALIZED, then run exit_validator(v, state, block, penalize=True, current_slot=block.slot)

PROPOSER_SLASHING

{
    'proposer_index': 'uint24',
    'proposal1_data': ProposalSignedData,
    'proposal1_signature': '[uint384]',
    'proposal2_data': ProposalSignedData,
    'proposal1_signature': '[uint384]',
}

For each proposal_signature, verify that BLSVerify(pubkey=validators[proposer_index].pubkey, msg=hash(proposal_data), sig=proposal_signature, domain=get_domain(state, proposal_data.slot, DOMAIN_PROPOSAL)) passes. Verify that proposal1_data.slot == proposal2_data.slot but proposal1 != proposal2. If state.validators[proposer_index].status does not equal PENALIZED, then run exit_validator(proposer_index, state, penalize=True, current_slot=block.slot)

DEPOSIT_PROOF

{
    'merkle_branch': '[hash32]',
    'merkle_tree_index': 'uint64',
    'deposit_data': {
         'deposit_params': DepositParams,
         'msg_value': 'uint64',
         'timestamp': 'uint64'
    }
}

Note that deposit_data in serialized form should be the DepositParams followed by 8 bytes for the msg_value and 8 bytes for the timestamp, or exactly the deposit_data in the PoW contract of which the hash was placed into the Merkle tree.

Use the following procedure to verify the merkle_branch, setting leaf=serialized_deposit_data, depth=POW_CONTRACT_MERKLE_TREE_DEPTH and root=state.processed_pow_receipt_root:

def verify_merkle_branch(leaf: Hash32, branch: [Hash32], depth: int, index: int, root: Hash32) -> bool:
    value = leaf
    for i in range(depth):
        if index % 2:
            value = hash(branch[i], value)
        else:
            value = hash(value, branch[i])
    return value == root

Verify that deposit_data.msg_value == DEPOSIT_SIZE and block.slot - (deposit_data.timestamp - state.genesis_time) // SLOT_DURATION < DELETION_PERIOD.

Run add_validator(validators, deposit_data.deposit_params.pubkey, deposit_data.deposit_params.proof_of_possession, deposit_data.deposit_params.withdrawal_credentials, deposit_data.deposit_params.randao_commitment, PENDING_ACTIVATION, block.slot).

Cycle boundary processing

Repeat the steps in this section while block.slot - last_state_recalculation_slot >= CYCLE_LENGTH. For simplicity, we'll use s as last_state_recalculation_slot.

Note: last_state_recalculation_slot will always be a multiple of CYCLE_LENGTH. In the "happy case", this process will trigger, and loop once, every time block.slot passes a new exact multiple of CYCLE_LENGTH, but if a chain skips more than an entire cycle then the loop may run multiple times, incrementing last_state_recalculation_slot by CYCLE_LENGTH with each iteration.

Precomputation

All validators:

Let active_validators = [state.validators[i] for i in get_active_validator_indices(state.validators)].
Let total_balance = sum([balance_at_stake(v) for v in 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 2/reward_quotient.)

Validators justifying the cycle boundary block at the start of the current cycle:

Let this_cycle_attestations = [a for a in state.pending_attestations if s <= a.data.slot < s + CYCLE_LENGTH]. (note: this is the set of attestations of slots in the cycle s...s+CYCLE_LENGTH-1, not attestations that got included in the chain during the cycle s...s+CYCLE_LENGTH-1)
Let this_cycle_boundary_attestations = [a for a in this_cycle_attestations if a.data.cycle_boundary_hash == get_block_hash(state, block, s) and a.justified_slot == state.justification_source].
Let this_cycle_boundary_attesters be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.attester_bitfield) for a in this_cycle_boundary_attestations].
Let this_cycle_boundary_attesting_balance = sum([balance_at_stake(v) for v in this_cycle_boundary_attesters]).

Validators justifying the cycle boundary block at the start of the previous cycle:

Let prev_cycle_attestations = [a for a in state.pending_attestations if s - CYCLE_LENGTH <= a.slot < s].
Let prev_cycle_boundary_attestations = [a for a in this_cycle_attestations + prev_cycle_attestations if a.cycle_boundary_hash == get_block_hash(state, block, s - CYCLE_LENGTH) and a.justified_slot == state.prev_cycle_justification_source].
Let prev_cycle_boundary_attesters be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.attester_bitfield) for a in prev_cycle_boundary_attestations].
Let prev_cycle_boundary_attesting_balance = sum([balance_at_stake(v) for v in prev_cycle_boundary_attesters]).

For every ShardAndCommittee object obj in shard_and_committee_for_slots, let:

attesting_validators(obj, shard_block_hash) be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.attester_bitfield) for a in this_cycle_attestations + prev_cycle_attestations if a.shard == obj.shard and a.shard_block_hash == shard_block_hash]
attesting_validators(obj) be equal to attesting_validators(obj, shard_block_hash) for the value of shard_block_hash such that sum([balance_at_stake(v) for v in attesting_validators(obj, shard_block_hash)]) is maximized (ties broken by favoring lower shard_block_hash values)
total_attesting_balance(obj) be the sum of the balances-at-stake of attesting_validators(obj)
winning_hash(obj) be the winning shard_block_hash value
total_balance(obj) = sum([balance_at_stake(v) for v in obj.committee])

Let inclusion_slot(v) equal a.slot_included for the attestation a where v is in get_attestation_participants(state, a.data, a.attester_bitfield), and inclusion_distance(v) = a.slot_included - a.data.slot for the same attestation. We define a function adjust_for_inclusion_distance(magnitude, dist) which adjusts the reward of an attestation based on how long it took to get included (the longer, the lower the reward). Returns a value between 0 and magnitude

def adjust_for_inclusion_distance(magnitude: int, dist: int) -> int:
    return magnitude // 2 + (magnitude // 2) * MIN_ATTESTATION_INCLUSION_DELAY // dist

For any validator v, base_reward(v) = balance_at_stake(v) // reward_quotient

Adjust justified slots and crosslink status

Set state.justified_slot_bitfield = (state.justified_slot_bitfield * 2) % 2**64.
If 3 * prev_cycle_boundary_attesting_balance >= 2 * total_balance then set state.justified_slot_bitfield &= 2 (ie. flip the second lowest bit to 1) and new_justification_source = s - CYCLE_LENGTH.
If 3 * this_cycle_boundary_attesting_balance >= 2 * total_balance then set state.justified_slot_bitfield &= 1 (ie. flip the lowest bit to 1) and new_justification_source = s.
If state.justification_source == s - CYCLE_LENGTH and state.justified_slot_bitfield % 4 == 3, set last_finalized_slot = justification_source.
If state.justification_source == s - CYCLE_LENGTH - CYCLE_LENGTH and state.justified_slot_bitfield % 8 == 7, set state.last_finalized_slot = state.justification_source.
If state.justification_source == s - CYCLE_LENGTH - 2 * CYCLE_LENGTH and state.justified_slot_bitfield % 16 in (15, 14), set last_finalized_slot = justification_source.
Set state.prev_cycle_justification_source = state.justification_source and if new_justification_source has been set, set state.justification_source = new_justification_source.

For every ShardAndCommittee object obj:

If 3 * total_attesting_balance(obj) >= 2 * total_balance(obj), set crosslinks[shard] = CrosslinkRecord(slot=last_state_recalculation_slot + CYCLE_LENGTH, hash=winning_hash(obj)).

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

Let quadratic_penalty_quotient = SQRT_E_DROP_TIME**2. (The portion lost by offline validators after D cycles is about D*D/2/quadratic_penalty_quotient.)
Let time_since_finality = block.slot - state.last_finalized_slot.

Case 1: time_since_finality <= 4 * CYCLE_LENGTH:

Any validator v in prev_cycle_boundary_attesters gains adjust_for_inclusion_distance(base_reward(v) * prev_cycle_boundary_attesting_balance // total_balance, inclusion_distance(v)).
Any active validator v not in prev_cycle_boundary_attesters loses base_reward(v).

Case 2: time_since_finality > 4 * CYCLE_LENGTH:

Any validator in prev_cycle_boundary_attesters sees their balance unchanged.
Any active validator v not in prev_cycle_boundary_attesters, and any validator with status == PENALIZED, loses base_reward(v) + balance_at_stake(v) * time_since_finality // quadratic_penalty_quotient.

For each v in prev_cycle_boundary_attesters, we determine the proposer proposer_index = get_beacon_proposer_index(state, inclusion_slot(v)) and set state.validators[proposer_index].balance += base_reward(v) // INCLUDER_REWARD_SHARE_QUOTIENT.

For every ShardAndCommittee object obj in shard_and_committee_for_slots[:CYCLE_LENGTH] (ie. the objects corresponding to the cycle before the current one), for each v in [state.validators[index] for index in obj.committee], adjust balances as follows:

If v in attesting_validators(obj), v.balance += adjust_for_inclusion_distance(base_reward(v) * total_attesting_balance(obj) // total_balance(obj)), inclusion_distance(v)).
If v not in attesting_validators(obj), v.balance -= base_reward(v).

If last_state_recalculation_slot % POW_RECEIPT_ROOT_VOTING_PERIOD == 0, then:

If for any x in state.candidate_pow_receipt_root, x.votes * 2 >= POW_RECEIPT_ROOT_VOTING_PERIOD set state.processed_pow_receipt_root = x.receipt_root.
Set state.candidate_pow_receipt_roots = [].

Proposer reshuffling

Run the following code to update the shard proposer set:

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(state.randao_mix + bytes8(i * 2)) % len(active_validator_indices)]
    new_shard = hash(state.randao_mix + bytes8(i * 2 + 1)) % SHARD_COUNT
    shard_reassignment_record = ShardReassignmentRecord(
        validator_index=vid,
        shard=new_shard,
        slot=s + SHARD_PERSISTENT_COMMITTEE_CHANGE_PERIOD
    )
    state.persistent_committee_reassignments.append(shard_reassignment_record)

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

Validator set change

A validator set change can happen if all of the following criteria are satisfied:

last_finalized_slot > state.validator_set_change_slot
For every shard number shard in shard_and_committee_for_slots, crosslinks[shard].slot > state.validator_set_change_slot

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

def change_validators(validators: List[ValidatorRecord], current_slot: int) -> None:
    # The active validator set
    active_validators = get_active_validator_indices(validators)
    # The total balance of active validators
    total_balance = sum([balance_at_stake(v) 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(
                state=state,
                index=i,
                pubkey=validators[i].pubkey,
                flag=ENTRY
            )
        if validators[i].status == PENDING_EXIT:
            validators[i].status = PENDING_WITHDRAW
            validators[i].last_status_change_slot = current_slot
            total_changed += balance_at_stake(validators[i])
            add_validator_set_change_record(
                state=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 // COLLECTIVE_PENALTY_CALCULATION_PERIOD
    total_penalties = (
        (state.deposits_penalized_in_period[period_index]) +
        (state.deposits_penalized_in_period[period_index - 1] if period_index >= 1 else 0) +
        (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

    def withdrawable(v):
        return v.status in (PENDING_WITHDRAW, PENALIZED) and current_slot >= v.last_status_change_slot + MIN_WITHDRAWAL_PERIOD

    withdrawable_validators = sorted(filter(withdrawable, validators), key=lambda v: v.exit_seq)
    for v in withdrawable_validators[:WITHDRAWALS_PER_CYCLE]:
        if v.status == PENALIZED:
            v.balance -= balance_at_stake(v) * min(total_penalties * 3, total_balance) // total_balance
        v.status = WITHDRAWN
        v.last_status_change_slot = current_slot

        withdraw_amount = v.balance
        # STUB: withdraw to shard chain

And perform the following updates to the state:

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

If a validator set change does NOT happen

Set state.shard_and_committee_for_slots[:CYCLE_LENGTH] = state.shard_and_committee_for_slots[CYCLE_LENGTH:]
Let time_since_finality = block.slot - state.validator_set_change_slot
Let start_shard = state.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 state.shard_and_committee_for_slots[CYCLE_LENGTH:] = get_new_shuffling(state.next_shuffling_seed, validators, start_shard) and set state.next_shuffling_seed = state.randao_mix. Note that start_shard is not changed from last cycle.

Finally...

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

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.

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