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Ethereum 2.0 Phase 0 -- The Beacon Chain

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 [python-poc].

Table of contents

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 registry of validators. In the initial deployment phases of Ethereum 2.0 the only mechanism to become a validator is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a validator happens when Ethereum 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and after a queuing process. Exit is either voluntary or done forcibly as a penalty for misbehavior.

The primary source of load on the beacon chain is "attestations". Attestations are availability votes for a shard block, and simultaneously proof of stake votes for a beacon 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.

Notation

Code snippets appearing in this style are to be interpreted as Python code. Beacon blocks that trigger unhandled Python exceptions (e.g. out-of-range list accesses) and failed asserts are considered invalid.

Terminology

  • Validator - a participant in the Casper/sharding consensus system. You can become one by depositing 32 ETH into the Casper mechanism.
  • Active validator - a validator currently participating in the protocol which the Casper mechanism looks to produce and attest to blocks, crosslinks and other consensus objects.
  • Committee - a (pseudo-) randomly sampled subset of active validators. 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.
  • Block root - a 32-byte Merkle root of a beacon chain block or shard chain block. Previously called "block hash".
  • 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
  • Epoch - an aligned span of slots during which all validators get exactly one chance to make an attestation
  • Finalized, justified - see Casper FFG finalization [casper-ffg]
  • Withdrawal period - the 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

Misc

Name Value Unit
SHARD_COUNT 2**10 (= 1,024) shards
TARGET_COMMITTEE_SIZE 2**7 (= 128) validators
EJECTION_BALANCE 2**4 * 1e9 (= 16,000,000,000) Gwei
MAX_BALANCE_CHURN_QUOTIENT 2**5 (= 32) -
BEACON_CHAIN_SHARD_NUMBER 2**64 - 1 -
MAX_CASPER_VOTES 2**10 (= 1,024) votes
MAX_WITHDRAWALS_PER_EPOCH 2**2 (= 4) withdrawals
  • For the safety of crosslinks TARGET_COMMITTEE_SIZE exceeds the recommended minimum committee size of 111; with sufficient active validators (at least EPOCH_LENGTH * TARGET_COMMITTEE_SIZE), the shuffling algorithm ensures committee sizes at least TARGET_COMMITTEE_SIZE. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)

Deposit contract

Name Value Unit
DEPOSIT_CONTRACT_ADDRESS TBD
DEPOSIT_CONTRACT_TREE_DEPTH 2**5 (= 32) -
MIN_DEPOSIT_AMOUNT 2**0 * 1e9 (= 1,000,000,000) Gwei
MAX_DEPOSIT_AMOUNT 2**5 * 1e9 (= 32,000,000,000) Gwei

Initial values

Name Value
GENESIS_FORK_VERSION 0
GENESIS_SLOT 0
GENESIS_EPOCH slot_to_epoch(GENESIS_SLOT)
GENESIS_START_SHARD 0
FAR_FUTURE_EPOCH 2**64 - 1
ZERO_HASH int_to_bytes32(0)
EMPTY_SIGNATURE int_to_bytes96(0)
BLS_WITHDRAWAL_PREFIX_BYTE int_to_bytes1(0)

Time parameters

Name Value Unit Duration
SLOT_DURATION 6 seconds 6 seconds
MIN_ATTESTATION_INCLUSION_DELAY 2**2 (= 4) slots 24 seconds
EPOCH_LENGTH 2**6 (= 64) slots 6.4 minutes
SEED_LOOKAHEAD 2**0 (= 1) epochs 6.4 minutes
ENTRY_EXIT_DELAY 2**2 (= 4) epochs 25.6 minutes
ETH1_DATA_VOTING_PERIOD 2**4 (= 16) epochs ~1.7 hours
MIN_VALIDATOR_WITHDRAWAL_EPOCHS 2**8 (= 256) epochs ~27 hours

State list lengths

| Name | Value | Unit | | LATEST_BLOCK_ROOTS_LENGTH | 2**13 (= 8,192) | slots | | LATEST_RANDAO_MIXES_LENGTH | 2**13 (= 8,192) | epochs | | LATEST_INDEX_ROOTS_LENGTH | 2**13 (= 8,192) | epochs | | LATEST_PENALIZED_EXIT_LENGTH | 2**13 (= 8,192) | epochs |

Reward and penalty quotients

Name Value
BASE_REWARD_QUOTIENT 2**5 (= 32)
WHISTLEBLOWER_REWARD_QUOTIENT 2**9 (= 512)
INCLUDER_REWARD_QUOTIENT 2**3 (= 8)
INACTIVITY_PENALTY_QUOTIENT 2**24 (= 16,777,216)
  • The BASE_REWARD_QUOTIENT parameter dictates the per-epoch reward. It corresponds to ~2.54% annual interest assuming 10 million participating ETH in every epoch.
  • The INACTIVITY_PENALTY_QUOTIENT equals INVERSE_SQRT_E_DROP_TIME**2 where INVERSE_SQRT_E_DROP_TIME := 2**12 epochs (~18 days) is the time it takes the inactivity penalty to reduce the balance of non-participating validators to about 1/sqrt(e) ~= 60.6%. Indeed, the balance retained by offline validators after n epochs is about (1-1/INACTIVITY_PENALTY_QUOTIENT)**(n**2/2) so after INVERSE_SQRT_E_DROP_TIME epochs it is roughly (1-1/INACTIVITY_PENALTY_QUOTIENT)**(INACTIVITY_PENALTY_QUOTIENT/2) ~= 1/sqrt(e).

Status flags

Name Value
INITIATED_EXIT 2**0 (= 1)
WITHDRAWABLE 2**1 (= 2)

Max operations per block

Name Value
MAX_PROPOSER_SLASHINGS 2**4 (= 16)
MAX_CASPER_SLASHINGS 2**4 (= 16)
MAX_ATTESTATIONS 2**7 (= 128)
MAX_DEPOSITS 2**4 (= 16)
MAX_EXITS 2**4 (= 16)

Signature domains

Name Value
DOMAIN_DEPOSIT 0
DOMAIN_ATTESTATION 1
DOMAIN_PROPOSAL 2
DOMAIN_EXIT 3
DOMAIN_RANDAO 4

Data structures

The following data structures are defined as SimpleSerialize (SSZ) objects.

Beacon chain operations

Proposer slashings

ProposerSlashing
{
    # Proposer index
    'proposer_index': 'uint24',
    # First proposal data
    'proposal_data_1': ProposalSignedData,
    # First proposal signature
    'proposal_signature_1': 'bytes96',
    # Second proposal data
    'proposal_data_2': ProposalSignedData,
    # Second proposal signature
    'proposal_signature_2': 'bytes96',
}

Casper slashings

CasperSlashing
{
    # First batch of votes
    'slashable_vote_data_1': SlashableVoteData,
    # Second batch of votes
    'slashable_vote_data_2': SlashableVoteData,
}
SlashableVoteData
{
    # Validator indices with custody bit equal to 0
    'custody_bit_0_indices': ['uint24'],
    # Validator indices with custody bit equal to 1
    'custody_bit_1_indices': ['uint24'],
    # Attestation data
    'data': AttestationData,
    # Aggregate signature
    'aggregate_signature': 'bytes96',
}

Attestations

Attestation
{
    # Attestation data
    'data': AttestationData,
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # BLS aggregate signature
    'aggregate_signature': 'bytes96',
}
AttestationData
{
    # Slot number
    'slot': 'uint64',
    # Shard number
    'shard': 'uint64',
    # Hash of root of the signed beacon block
    'beacon_block_root': 'bytes32',
    # Hash of root of the ancestor at the epoch boundary
    'epoch_boundary_root': 'bytes32',
    # Shard block's hash of root
    'shard_block_root': 'bytes32',
    # Last crosslink's hash of root
    'latest_crosslink_root': 'bytes32',
    # Last justified epoch in the beacon state
    'justified_epoch': 'uint64',
    # Hash of the last justified beacon block
    'justified_block_root': 'bytes32',
}
AttestationDataAndCustodyBit
{
    # Attestation data
    data: AttestationData,
    # Custody bit
    custody_bit: bool,
}

Deposits

Deposit
{
    # Branch in the deposit tree
    'branch': ['bytes32'],
    # Index in the deposit tree
    'index': 'uint64',
    # Data
    'deposit_data': DepositData,
}
DepositData
{
    # Amount in Gwei
    'amount': 'uint64',
    # Timestamp from deposit contract
    'timestamp': 'uint64',
    # Deposit input
    'deposit_input': DepositInput,
}
DepositInput
{
    # BLS pubkey
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # A BLS signature of this `DepositInput`
    'proof_of_possession': 'bytes96',
}

Exits

Exit
{
    # Minimum epoch for processing exit
    'epoch': 'uint64',
    # Index of the exiting validator
    'validator_index': 'uint24',
    # Validator signature
    'signature': 'bytes96',
}

Beacon chain blocks

BeaconBlock

{
    ## Header ##
    'slot': 'uint64',
    'parent_root': 'bytes32',
    'state_root': 'bytes32',
    'randao_reveal': 'bytes96',
    'eth1_data': Eth1Data,
    'signature': 'bytes96',

    ## Body ##
    'body': BeaconBlockBody,
}

BeaconBlockBody

{
    'proposer_slashings': [ProposerSlashing],
    'casper_slashings': [CasperSlashing],
    'attestations': [Attestation],
    'custody_reseeds': [CustodyReseed],
    'custody_challenges': [CustodyChallenge],
    'custody_responses': [CustodyResponse],
    'deposits': [Deposit],
    'exits': [Exit],
}

CustodyReseed, CustodyChallenge, and CustodyResponse will be defined in phase 1; for now, put dummy classes as these lists will remain empty throughout phase 0.

ProposalSignedData

{
    # Slot number
    'slot': 'uint64',
    # Shard number (`BEACON_CHAIN_SHARD_NUMBER` for beacon chain)
    'shard': 'uint64',
    # Block's hash of root
    'block_root': 'bytes32',
}

Beacon chain state

BeaconState

{
    # Misc
    'slot': 'uint64',
    'genesis_time': 'uint64',
    'fork': Fork,  # For versioning hard forks

    # Validator registry
    'validator_registry': [Validator],
    'validator_balances': ['uint64'],
    'validator_registry_update_epoch': 'uint64',
    'validator_registry_exit_count': 'uint64',

    # Randomness and committees
    'latest_randao_mixes': ['bytes32'],
    'latest_vdf_outputs': ['bytes32'],
    'previous_epoch_start_shard': 'uint64',
    'current_epoch_start_shard': 'uint64',
    'previous_calculation_epoch': 'uint64',
    'current_calculation_epoch': 'uint64',
    'previous_epoch_seed': 'bytes32',
    'current_epoch_seed': 'bytes32',

    # Custody challenges
    'custody_challenges': [CustodyChallenge],

    # Finality
    'previous_justified_epoch': 'uint64',
    'justified_epoch': 'uint64',
    'justification_bitfield': 'uint64',
    'finalized_epoch': 'uint64',

    # Recent state
    'latest_crosslinks': [Crosslink],
    'latest_block_roots': ['bytes32'],  # Needed to process attestations, older to newer
    'latest_index_roots': ['bytes32'],
    'latest_penalized_balances': ['uint64'],  # Balances penalized at every withdrawal period
    'latest_attestations': [PendingAttestation],
    'batched_block_roots': ['bytes32'],

    # Ethereum 1.0 chain data
    'latest_eth1_data': Eth1Data,
    'eth1_data_votes': [Eth1DataVote],
}

Validator

{
    # BLS public key
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Epoch when validator activated
    'activation_epoch': 'uint64',
    # Epoch when validator exited
    'exit_epoch': 'uint64',
    # Epoch when validator withdrew
    'withdrawal_epoch': 'uint64',
    # Epoch when validator was penalized
    'penalized_epoch': 'uint64',
    # Exit counter when validator exited
    'exit_count': 'uint64',
    # Status flags
    'status_flags': 'uint64',
    # Slot of latest custody reseed
    'latest_custody_reseed_slot': 'uint64',
    # Slot of second-latest custody reseed
    'penultimate_custody_reseed_slot': 'uint64',
}

Crosslink

{
    # Epoch number
    'epoch': 'uint64',
    # Shard block root
    'shard_block_root': 'bytes32',
}

PendingAttestation

{
    # Signed data
    'data': AttestationData,
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # Slot the attestation was included
    'slot_included': 'uint64',
}

Fork

{
    # Previous fork version
    'previous_version': 'uint64',
    # Current fork version
    'current_version': 'uint64',
    # Fork epoch number
    'epoch': 'uint64',
}

Eth1Data

{
    # Root of the deposit tree
    'deposit_root': 'bytes32',
    # Block hash
    'block_hash': 'bytes32',
}

Eth1DataVote

{
    # Data being voted for
    'eth1_data': Eth1Data,
    # Vote count
    'vote_count': 'uint64',
}

Custom Types

We define the following Python custom types for type hinting and readability:

Name Type Description
SlotNumber unsigned 64-bit integer the number of a slot
EpochNumber unsigned 64-bit integer the number of an epoch
ShardNumber unsigned 64-bit integer the number of a shard
ValidatorIndex unsigned 24-bit integer the index number of a validator in the registry
Gwei unsigned 64-bit integer an amount in Gwei
Bytes32 32-byte data binary data with 32-byte length
BLSPubkey 48-byte data a public key in BLS signature scheme
BLSSignature 96-byte data a signature in BLS signature scheme

Ethereum 1.0 deposit contract

The initial deployment phases of Ethereum 2.0 are implemented without consensus changes to Ethereum 1.0. A deposit contract at address DEPOSIT_CONTRACT_ADDRESS is added to Ethereum 1.0 for deposits of ETH to the beacon chain. Validator balances will be withdrawable to the shards in phase 2, i.e. when the EVM2.0 is deployed and the shards have state.

Deposit arguments

The deposit contract has a single deposit function which takes as argument a SimpleSerialize'd DepositInput.

Withdrawal credentials

One of the DepositInput fields is withdrawal_credentials. It is a commitment to credentials for withdrawals to shards. The first byte of withdrawal_credentials is a version number. As of now the only expected format is as follows:

  • withdrawal_credentials[:1] == BLS_WITHDRAWAL_PREFIX_BYTE
  • withdrawal_credentials[1:] == hash(withdrawal_pubkey)[1:] where withdrawal_pubkey is a BLS pubkey

The private key corresponding to withdrawal_pubkey will be required to initiate a withdrawal. It can be stored separately until a withdrawal is required, e.g. in cold storage.

Deposit logs

Every Ethereum 1.0 deposit, of size between MIN_DEPOSIT_AMOUNT and MAX_DEPOSIT_AMOUNT, emits a Deposit log for consumption by the beacon chain. The deposit contract does little validation, pushing most of the validator onboarding logic to the beacon chain. In particular, the proof of possession (a BLS12 signature) is not verified by the deposit contract.

ChainStart log

When sufficiently many full deposits have been made the deposit contract emits the ChainStart log. The beacon chain state may then be initialized by calling the get_initial_beacon_state function (defined below) where:

  • genesis_time equals time in the ChainStart log
  • latest_eth1_data.deposit_root equals deposit_root in the ChainStart log, and latest_eth1_data.block_hash equals the hash of the block that included the log
  • initial_validator_deposits is a list of Deposit objects built according to the Deposit logs up to the deposit that triggered the ChainStart log, processed in the order in which they were emitted (oldest to newest)

Vyper code

## compiled with v0.1.0-beta.6 ##

MIN_DEPOSIT_AMOUNT: constant(uint256) = 1000000000  # Gwei
MAX_DEPOSIT_AMOUNT: constant(uint256) = 32000000000  # Gwei
GWEI_PER_ETH: constant(uint256) = 1000000000  # 10**9
CHAIN_START_FULL_DEPOSIT_THRESHOLD: constant(uint256) = 16384  # 2**14
DEPOSIT_CONTRACT_TREE_DEPTH: constant(uint256) = 32
TWO_TO_POWER_OF_TREE_DEPTH: constant(uint256) = 4294967296  # 2**32
SECONDS_PER_DAY: constant(uint256) = 86400

Deposit: event({previous_deposit_root: bytes32, data: bytes[2064], merkle_tree_index: bytes[8]})
ChainStart: event({deposit_root: bytes32, time: bytes[8]})

deposit_tree: map(uint256, bytes32)
deposit_count: uint256
full_deposit_count: uint256

@payable
@public
def deposit(deposit_input: bytes[2048]):
    assert msg.value >= as_wei_value(MIN_DEPOSIT_AMOUNT, "gwei")
    assert msg.value <= as_wei_value(MAX_DEPOSIT_AMOUNT, "gwei")

    index: uint256 = self.deposit_count + TWO_TO_POWER_OF_TREE_DEPTH
    deposit_amount: bytes[8] = slice(concat("", convert(msg.value / GWEI_PER_ETH, bytes32)), start=24, len=8)
    deposit_timestamp: bytes[8] = slice(concat("", convert(block.timestamp, bytes32)), start=24, len=8)
    deposit_data: bytes[2064] = concat(deposit_amount, deposit_timestamp, deposit_input)
    merkle_tree_index: bytes[8] = slice(concat("", convert(index, bytes32)), start=24, len=8)

    log.Deposit(self.deposit_tree[1], deposit_data, merkle_tree_index)

    # add deposit to merkle tree
    self.deposit_tree[index] = sha3(deposit_data)
    for i in range(DEPOSIT_CONTRACT_TREE_DEPTH):
        index /= 2
        self.deposit_tree[index] = sha3(concat(self.deposit_tree[index * 2], self.deposit_tree[index * 2 + 1]))

    self.deposit_count += 1
    if msg.value == as_wei_value(MAX_DEPOSIT_AMOUNT, "gwei"):
        self.full_deposit_count += 1
        if self.full_deposit_count == CHAIN_START_FULL_DEPOSIT_THRESHOLD:
            timestamp_day_boundary: uint256 = as_unitless_number(block.timestamp) - as_unitless_number(block.timestamp) % SECONDS_PER_DAY + SECONDS_PER_DAY
            chainstart_time: bytes[8] = slice(concat("", convert(timestamp_day_boundary, bytes32)), start=24, len=8)
            log.ChainStart(self.deposit_tree[1], chainstart_time)

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

@public
@constant
def get_branch(leaf: uint256) -> bytes32[32]: # size is DEPOSIT_CONTRACT_TREE_DEPTH (symbolic const not supported)
    branch: bytes32[32] # size is DEPOSIT_CONTRACT_TREE_DEPTH
    index: uint256 = leaf + TWO_TO_POWER_OF_TREE_DEPTH
    for i in range(DEPOSIT_CONTRACT_TREE_DEPTH):
        branch[i] = self.deposit_tree[bitwise_xor(index, 1)]
        index /= 2
    return branch

Beacon chain processing

The beacon chain is the system chain for Ethereum 2.0. The main responsibilities of the beacon chain are:

  • Store and maintain the registry of validators
  • Process crosslinks (see above)
  • Process its per-slot consensus, as well as the finality gadget

Processing the beacon chain is similar to processing the Ethereum 1.0 chain. 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 Ethereum 1.0, and because it is a proof-of-stake chain, there are differences.

For a beacon chain block, block, to be processed by a node, the following conditions must be met:

  • The parent block with root block.parent_root has been processed and accepted.
  • The node has processed its state up to slot, block.slot - 1.
  • An Ethereum 1.0 block pointed to by the state.latest_eth1_data.block_hash has been processed and accepted.
  • The node's local clock time is greater than or equal to 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 (i.e. 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.

  • Abstractly define Store as the type of storage object for the chain data and 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 yet included in any chain are still included in store.
  • Let finalized_head be the finalized block with the highest epoch. (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 epoch that has been justified for at least 1 epoch. (A block B is justified if 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: Store, block: BeaconBlock, slot: SlotNumber) -> BeaconBlock be the ancestor of block with slot number slot. The get_ancestor function can be defined recursively as def get_ancestor(store: Store, block: BeaconBlock, slot: SlotNumber) -> BeaconBlock: return block if block.slot == slot else get_ancestor(store, store.get_parent(block), slot).
  • Let get_latest_attestation(store: Store, validator: Validator) -> Attestation 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: Store, validator: Validator) -> BeaconBlock be the target block in the attestation get_latest_attestation(store, validator).
  • Let get_children(store: Store, block: BeaconBlock) -> List[BeaconBlock] returns the child blocks of the given block.
  • Let justified_head_state be the resulting BeaconState object from processing the chain up to the justified_head.
  • The head is lmd_ghost(store, justified_head_state, 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: Store, start_state: BeaconState, start_block: BeaconBlock) -> BeaconBlock:
    validators = start_state.validator_registry
    active_validators = [
        validators[i]
        for i in get_active_validator_indices(validators, start_state.slot)
    ]
    attestation_targets = [
        get_latest_attestation_target(store, validator)
        for validator in active_validators
    ]

    def get_vote_count(block: BeaconBlock) -> int:
        return len([
            target
            for target in attestation_targets
            if get_ancestor(store, target, block.slot) == block
        ])

    head = start_block
    while 1:
        children = get_children(store, 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 a high level the state transition is made up of two parts:

  1. The per-slot transitions, which happens every slot, and only affects a parts of the state.
  2. The per-epoch transitions, which happens in the last slot of every epoch (i.e. (state.slot + 1) % EPOCH_LENGTH == 0), and affects the entire state.

The per-slot transitions generally focus on verifying aggregate signatures and saving temporary records relating to the per-slot activity in the BeaconState. The per-epoch transitions focus on the validator registry, including adjusting balances and activating and exiting validators, as well as processing crosslinks and managing block justification/finalization.

Helper functions

Note: The definitions below are for specification purposes and are not necessarily optimal implementations.

hash

The hash function is denoted by hash. In Phase 0 the beacon chain is deployed with the same hash function as Ethereum 1.0, i.e. Keccak-256 (also incorrectly known as SHA3).

Note: We aim to migrate to a S[T/N]ARK-friendly hash function in a future Ethereum 2.0 deployment phase.

hash_tree_root

def hash_tree_root(object: SSZSerializable) -> Bytes32 is a function for hashing objects into a single root utilizing a hash tree structure. hash_tree_root is defined in the SimpleSerialize spec.

slot_to_epoch

def slot_to_epoch(slot: SlotNumber) -> EpochNumber:
    return slot // EPOCH_LENGTH

get_current_epoch

def get_current_epoch(state: BeaconState) -> EpochNumber:
    return slot_to_epoch(state.slot)

get_epoch_start_slot

def get_epoch_start_slot(epoch: EpochNumber) -> SlotNumber:
    return epoch * EPOCH_LENGTH

is_active_validator

def is_active_validator(validator: Validator, epoch: EpochNumber) -> bool:
    """
    Checks if ``validator`` is active.
    """
    return validator.activation_epoch <= epoch < validator.exit_epoch

get_active_validator_indices

def get_active_validator_indices(validators: List[Validator], epoch: EpochNumber) -> List[ValidatorIndex]:
    """
    Gets indices of active validators from ``validators``.
    """
    return [i for i, v in enumerate(validators) if is_active_validator(v, epoch)]

shuffle

def shuffle(values: List[Any], seed: Bytes32) -> 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

split

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

get_epoch_committee_count

def get_epoch_committee_count(active_validator_count: int) -> int:
    return max(
        1,
        min(
            SHARD_COUNT // EPOCH_LENGTH,
            active_validator_count // EPOCH_LENGTH // TARGET_COMMITTEE_SIZE,
        )
    ) * EPOCH_LENGTH

get_shuffling

def get_shuffling(seed: Bytes32,
                  validators: List[Validator],
                  epoch: EpochNumber) -> List[List[ValidatorIndex]]
    """
    Shuffles ``validators`` into crosslink committees seeded by ``seed`` and ``epoch``.
    Returns a list of ``committees_per_epoch`` committees where each
    committee is itself a list of validator indices.
    """

    active_validator_indices = get_active_validator_indices(validators, epoch)

    committees_per_epoch = get_epoch_committee_count(len(active_validator_indices))

    # Shuffle
    seed = xor(seed, int_to_bytes32(epoch))
    shuffled_active_validator_indices = shuffle(active_validator_indices, seed)

    # Split the shuffled list into committees_per_epoch pieces
    return split(shuffled_active_validator_indices, committees_per_epoch)

Invariant: if get_shuffling(seed, validators, epoch) returns some value x for some epoch <= get_current_epoch(state) + ENTRY_EXIT_DELAY, it should return the same value x for the same seed and epoch and possible future modifications of validators forever in phase 0, and until the ~1 year deletion delay in phase 2 and in the future.

Note: this definition and the next few definitions make heavy use of repetitive computing. Production implementations are expected to appropriately use caching/memoization to avoid redoing work.

get_previous_epoch_committee_count

def get_previous_epoch_committee_count(state: BeaconState) -> int:
    previous_active_validators = get_active_validator_indices(
        state.validator_registry,
        state.previous_calculation_epoch,
    )
    return get_epoch_committee_count(len(previous_active_validators))

get_current_epoch_committee_count

def get_current_epoch_committee_count(state: BeaconState) -> int:
    current_active_validators = get_active_validator_indices(
        state.validator_registry,
        state.current_calculation_epoch,
    )
    return get_epoch_committee_count(len(current_active_validators))

get_crosslink_committees_at_slot

def get_crosslink_committees_at_slot(state: BeaconState,
                                     slot: SlotNumber) -> List[Tuple[List[ValidatorIndex], ShardNumber]]:
    """
    Returns the list of ``(committee, shard)`` tuples for the ``slot``.
    """
    epoch = slot_to_epoch(slot)
    current_epoch = get_current_epoch(state)
    previous_epoch = current_epoch - 1 if epoch > GENESIS_EPOCH else current_epoch
    next_epoch = current_epoch + 1

    assert previous_epoch <= epoch < next_epoch

    if epoch < current_epoch:
        committees_per_epoch = get_previous_epoch_committee_count(state)
        seed = state.previous_epoch_seed
        shuffling_epoch = state.previous_calculation_epoch
        shuffling_start_shard = state.previous_epoch_start_shard
    else:
        committees_per_epoch = get_current_epoch_committee_count(state)
        seed = state.current_epoch_seed
        shuffling_epoch = state.current_calculation_epoch
        shuffling_start_shard = state.current_epoch_start_shard

    shuffling = get_shuffling(
        seed,
        state.validator_registry,
        shuffling_epoch,
    )
    offset = slot % EPOCH_LENGTH
    committees_per_slot = committees_per_epoch // EPOCH_LENGTH
    slot_start_shard = (shuffling_start_shard + committees_per_slot * offset) % SHARD_COUNT

    return [
        (
            shuffling[committees_per_slot * offset + i],
            (slot_start_shard + i) % SHARD_COUNT,
        )
        for i in range(committees_per_slot)
    ]

Note: we plan to replace the shuffling algorithm with a pointwise-evaluable shuffle (see https://github.com/ethereum/eth2.0-specs/issues/323), which will allow calculation of the committees for each slot individually.

get_block_root

def get_block_root(state: BeaconState,
                   slot: SlotNumber) -> Bytes32:
    """
    Returns the block root at a recent ``slot``.
    """
    assert state.slot <= slot + LATEST_BLOCK_ROOTS_LENGTH
    assert slot < state.slot
    return state.latest_block_roots[slot % LATEST_BLOCK_ROOTS_LENGTH]

get_block_root(_, s) should always return hash_tree_root of the block in the beacon chain at slot s, and get_crosslink_committees_at_slot(_, s) should not change unless the validator registry changes.

get_randao_mix

def get_randao_mix(state: BeaconState,
                   epoch: EpochNumber) -> Bytes32:
    """
    Returns the randao mix at a recent ``slot``.
    """
    assert get_current_epoch(state) < epoch + LATEST_RANDAO_MIXES_LENGTH
    assert epoch <= get_current_epoch(state)
    return state.latest_randao_mixes[epoch % LATEST_RANDAO_MIXES_LENGTH]

get_active_index_root

def get_active_index_root(state: BeaconState,
                          epoch: EpochNumber) -> Bytes32:
    """
    Returns the index root at a recent ``epoch``.
    """
    assert get_current_epoch(state) < epoch + LATEST_INDEX_ROOTS_LENGTH
    assert epoch <= get_current_epoch(state)
    return state.latest_index_roots[epoch % LATEST_INDEX_ROOTS_LENGTH]

generate_seed

def generate_seed(state: BeaconState,
                  epoch: EpochNumber) -> Bytes32:
    """
    Generate a seed for the given ``epoch``.
    """
    randao_mix_epoch = epoch + LATEST_RANDAO_MIXES_LENGTH - SEED_LOOKAHEAD

    return hash(
        get_randao_mix(state, randao_mix_epoch) +
        get_active_index_root(state, epoch)
    )

get_beacon_proposer_index

def get_beacon_proposer_index(state: BeaconState,
                              slot: SlotNumber) -> ValidatorIndex:
    """
    Returns the beacon proposer index for the ``slot``.
    """
    first_committee, _ = get_crosslink_committees_at_slot(state, slot)[0]
    return first_committee[slot % len(first_committee)]

merkle_root

def merkle_root(values: List[Bytes32]) -> Bytes32:
    """
    Merkleize ``values`` (where ``len(values)`` is a power of two) and return the Merkle root.
    """
    o = [0] * len(values) + values
    for i in range(len(values) - 1, 0, -1):
        o[i] = hash(o[i * 2] + o[i * 2 + 1])
    return o[1]

get_attestation_participants

def get_attestation_participants(state: BeaconState,
                                 attestation_data: AttestationData,
                                 aggregation_bitfield: bytes) -> List[ValidatorIndex]:
    """
    Returns the participant indices at for the ``attestation_data`` and ``aggregation_bitfield``.
    """

    # Find the committee in the list with the desired shard
    crosslink_committees = get_crosslink_committees_at_slot(state, attestation_data.slot)

    assert attestation_data.shard in [shard for _, shard in crosslink_committees]
    crosslink_committee = [committee for committee, shard in crosslink_committees if shard == attestation_data.shard][0]
    assert len(aggregation_bitfield) == (len(crosslink_committee) + 7) // 8

    # Find the participating attesters in the committee
    participants = []
    for i, validator_index in enumerate(crosslink_committee):
        aggregation_bit = (aggregation_bitfield[i // 8] >> (7 - (i % 8))) % 2
        if aggregation_bit == 1:
            participants.append(validator_index)
    return participants

int_to_bytes1, int_to_bytes2, ...

int_to_bytes1(x): return x.to_bytes(1, 'big'), int_to_bytes2(x): return x.to_bytes(2, 'big'), and so on for all integers, particularly 1, 2, 3, 4, 8, 32, 48, 96.

get_effective_balance

def get_effective_balance(state: State, index: ValidatorIndex) -> Gwei:
    """
    Returns the effective balance (also known as "balance at stake") for a ``validator`` with the given ``index``.
    """
    return min(state.validator_balances[index], MAX_DEPOSIT_AMOUNT)

get_fork_version

def get_fork_version(fork: Fork,
                     epoch: EpochNumber) -> int:
    if epoch < fork.epoch:
        return fork.previous_version
    else:
        return fork.current_version

get_domain

def get_domain(fork: Fork,
               epoch: EpochNumber,
               domain_type: int) -> int:
    return get_fork_version(
        fork,
        epoch,
    ) * 2**32 + domain_type

verify_slashable_vote_data

def verify_slashable_vote_data(state: BeaconState, vote_data: SlashableVoteData) -> bool:
    if len(vote_data.custody_bit_0_indices) + len(vote_data.custody_bit_1_indices) > MAX_CASPER_VOTES:
        return False

    return bls_verify_multiple(
        pubkeys=[
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in vote_data.custody_bit_0_indices]),
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in vote_data.custody_bit_1_indices]),
        ],
        messages=[
            hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, False)),
            hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, True)),
        ],
        signature=vote_data.aggregate_signature,
        domain=get_domain(
            state.fork,
            slot_to_epoch(vote_data.data.slot),
            DOMAIN_ATTESTATION,
        ),
    )

is_double_vote

def is_double_vote(attestation_data_1: AttestationData,
                   attestation_data_2: AttestationData) -> bool
    """
    Assumes ``attestation_data_1`` is distinct from ``attestation_data_2``.
    Returns True if the provided ``AttestationData`` are slashable
    due to a 'double vote'.
    """
    target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
    target_epoch_2 = slot_to_epoch(attestation_data_2.slot)
    return target_epoch_1 == target_epoch_2

is_surround_vote

def is_surround_vote(attestation_data_1: AttestationData,
                     attestation_data_2: AttestationData) -> bool:
    """
    Assumes ``attestation_data_1`` is distinct from ``attestation_data_2``.
    Returns True if the provided ``AttestationData`` are slashable
    due to a 'surround vote'.
    Note: parameter order matters as this function only checks
    that ``attestation_data_1`` surrounds ``attestation_data_2``.
    """
    source_epoch_1 = attestation_data_1.justified_epoch
    source_epoch_2 = attestation_data_2.justified_epoch
    target_epoch_1 = slot_to_epoch(attestation_data_1.slot)
    target_epoch_2 = slot_to_epoch(attestation_data_2.slot)
    return (
        (source_epoch_1 < source_epoch_2) and
        (source_epoch_2 + 1 == target_epoch_2) and
        (target_epoch_2 < target_epoch_1)
    )

integer_squareroot

def integer_squareroot(n: int) -> int:
    """
    The largest integer ``x`` such that ``x**2`` is less than or equal to ``n``.
    """
    assert n >= 0
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

get_entry_exit_effect_epoch

def get_entry_exit_effect_epoch(epoch: EpochNumber) -> EpochNumber:
    """
    An entry or exit triggered in the ``epoch`` given by the input takes effect at
    the epoch given by the output.
    """
    return epoch + 1 + ENTRY_EXIT_DELAY

bls_verify

bls_verify is a function for verifying a BLS signature, defined in the BLS Signature spec.

bls_verify_multiple

bls_verify_multiple is a function for verifying a BLS signature constructed from multiple messages, defined in the BLS Signature spec.

bls_aggregate_pubkeys

bls_aggregate_pubkeys is a function for aggregating multiple BLS public keys into a single aggregate key, defined in the BLS Signature spec.

On startup

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

{
    slot=GENESIS_SLOT,
    parent_root=ZERO_HASH,
    state_root=STARTUP_STATE_ROOT,
    randao_reveal=EMPTY_SIGNATURE,
    eth1_data=Eth1Data(
        deposit_root=ZERO_HASH,
        block_hash=ZERO_HASH
    ),
    signature=EMPTY_SIGNATURE,
    body=BeaconBlockBody(
        proposer_slashings=[],
        casper_slashings=[],
        attestations=[],
        custody_reseeds=[],
        custody_challenges=[],
        custody_responses=[],
        deposits=[],
        exits=[],
    ),
}

STARTUP_STATE_ROOT (in the above "genesis block") is generated from the get_initial_beacon_state function below. When enough full deposits have been made to the deposit contract and the ChainStart log has been emitted, get_initial_beacon_state will execute to compute the hash_tree_root of BeaconState.

def get_initial_beacon_state(initial_validator_deposits: List[Deposit],
                             genesis_time: int,
                             latest_eth1_data: Eth1Data) -> BeaconState:
    state = BeaconState(
        # Misc
        slot=GENESIS_SLOT,
        genesis_time=genesis_time,
        fork=Fork(
            previous_version=GENESIS_FORK_VERSION,
            current_version=GENESIS_FORK_VERSION,
            epoch=GENESIS_EPOCH,
        ),

        # Validator registry
        validator_registry=[],
        validator_balances=[],
        validator_registry_update_epoch=GENESIS_EPOCH,
        validator_registry_exit_count=0,

        # Randomness and committees
        latest_randao_mixes=[ZERO_HASH for _ in range(LATEST_RANDAO_MIXES_LENGTH)],
        latest_vdf_outputs=[ZERO_HASH for _ in range(LATEST_RANDAO_MIXES_LENGTH // EPOCH_LENGTH)],
        previous_epoch_start_shard=GENESIS_START_SHARD,
        current_epoch_start_shard=GENESIS_START_SHARD,
        previous_calculation_epoch=GENESIS_EPOCH,
        current_calculation_epoch=GENESIS_EPOCH,
        previous_epoch_seed=ZERO_HASH,
        current_epoch_seed=ZERO_HASH,

        # Custody challenges
        custody_challenges=[],

        # Finality
        previous_justified_epoch=GENESIS_EPOCH,
        justified_epoch=GENESIS_EPOCH,
        justification_bitfield=0,
        finalized_epoch=GENESIS_EPOCH,

        # Recent state
        latest_crosslinks=[Crosslink(epoch=GENESIS_EPOCH, shard_block_root=ZERO_HASH) for _ in range(SHARD_COUNT)],
        latest_block_roots=[ZERO_HASH for _ in range(LATEST_BLOCK_ROOTS_LENGTH)],
        latest_index_roots=[ZERO_HASH for _ in range(LATEST_INDEX_ROOTS_LENGTH)],
        latest_penalized_balances=[0 for _ in range(LATEST_PENALIZED_EXIT_LENGTH)],
        latest_attestations=[],
        batched_block_roots=[],

        # Ethereum 1.0 chain data
        latest_eth1_data=latest_eth1_data,
        eth1_data_votes=[],
    )

    # Process initial deposits
    for deposit in initial_validator_deposits:
        process_deposit(
            state=state,
            pubkey=deposit.deposit_data.deposit_input.pubkey,
            amount=deposit.deposit_data.amount,
            proof_of_possession=deposit.deposit_data.deposit_input.proof_of_possession,
            withdrawal_credentials=deposit.deposit_data.deposit_input.withdrawal_credentials,
        )

    # Process initial activations
    for validator_index, _ in enumerate(state.validator_registry):
        if get_effective_balance(state, validator_index) >= MAX_DEPOSIT_AMOUNT:
            activate_validator(state, validator_index, True)

    state.latest_index_roots[GENESIS_EPOCH % LATEST_INDEX_ROOTS_LENGTH] = hash_tree_root(get_active_validator_indices(state, GENESIS_EPOCH))
    state.current_epoch_seed = generate_seed(state, GENESIS_EPOCH)

    return state

Routine for processing deposits

First, a helper function:

def validate_proof_of_possession(state: BeaconState,
                                 pubkey: BLSPubkey,
                                 proof_of_possession: BLSSignature,
                                 withdrawal_credentials: Bytes32) -> bool:
    proof_of_possession_data = DepositInput(
        pubkey=pubkey,
        withdrawal_credentials=withdrawal_credentials,
        proof_of_possession=EMPTY_SIGNATURE,
    )

    return bls_verify(
        pubkey=pubkey,
        message=hash_tree_root(proof_of_possession_data),
        signature=proof_of_possession,
        domain=get_domain(
            state.fork,
            get_current_epoch(state),
            DOMAIN_DEPOSIT,
        )
    )

Now, to add a validator or top up an existing validator's balance by some deposit amount:

def process_deposit(state: BeaconState,
                    pubkey: BLSPubkey,
                    amount: Gwei,
                    proof_of_possession: BLSSignature,
                    withdrawal_credentials: Bytes32) -> None:
    """
    Process a deposit from Ethereum 1.0.
    Note that this function mutates ``state``.
    """
    # Validate the given `proof_of_possession`
    assert validate_proof_of_possession(
        state,
        pubkey,
        proof_of_possession,
        withdrawal_credentials,
    )

    validator_pubkeys = [v.pubkey for v in state.validator_registry]

    if pubkey not in validator_pubkeys:
        # Add new validator
        validator = Validator(
            pubkey=pubkey,
            withdrawal_credentials=withdrawal_credentials,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawal_epoch=FAR_FUTURE_EPOCH,
            penalized_epoch=FAR_FUTURE_EPOCH,
            exit_count=0,
            status_flags=0,
            latest_custody_reseed_slot=GENESIS_SLOT,
            penultimate_custody_reseed_slot=GENESIS_SLOT,
        )

        # Note: In phase 2 registry indices that have been withdrawn for a long time will be recycled.
        state.validator_registry.append(validator)
        state.validator_balances.append(amount)
    else:
        # Increase balance by deposit amount
        index = validator_pubkeys.index(pubkey)
        assert state.validator_registry[index].withdrawal_credentials == withdrawal_credentials

        state.validator_balances[index] += amount

Routines for updating validator status

Note: All functions in this section mutate state.

def activate_validator(state: BeaconState, index: ValidatorIndex, genesis: bool) -> None:
    validator = state.validator_registry[index]

    validator.activation_epoch = GENESIS_EPOCH if genesis else get_entry_exit_effect_epoch(get_current_epoch(state))

def initiate_validator_exit(state: BeaconState, index: ValidatorIndex) -> None:
    validator = state.validator_registry[index]
    validator.status_flags |= INITIATED_EXIT

def exit_validator(state: BeaconState, index: ValidatorIndex) -> None:
    validator = state.validator_registry[index]

    # The following updates only occur if not previous exited
    if validator.exit_epoch <= get_entry_exit_effect_epoch(get_current_epoch(state)):
        return

    validator.exit_epoch = get_entry_exit_effect_epoch(get_current_epoch(state))

    state.validator_registry_exit_count += 1
    validator.exit_count = state.validator_registry_exit_count

def penalize_validator(state: BeaconState, index: ValidatorIndex) -> None:
    exit_validator(state, index)
    validator = state.validator_registry[index]
    state.latest_penalized_balances[get_current_epoch(state) % LATEST_PENALIZED_EXIT_LENGTH] += get_effective_balance(state, index)

    whistleblower_index = get_beacon_proposer_index(state, state.slot)
    whistleblower_reward = get_effective_balance(state, index) // WHISTLEBLOWER_REWARD_QUOTIENT
    state.validator_balances[whistleblower_index] += whistleblower_reward
    state.validator_balances[index] -= whistleblower_reward
    validator.penalized_epoch = get_current_epoch(state)

def prepare_validator_for_withdrawal(state: BeaconState, index: ValidatorIndex) -> None:
    validator = state.validator_registry[index]
    validator.status_flags |= WITHDRAWABLE

Per-slot processing

Below are the processing steps that happen at every slot.

Slot

  • Set state.slot += 1.

Block roots

  • Let previous_block_root be the tree_hash_root of the previous beacon block processed in the chain.
  • Set state.latest_block_roots[(state.slot - 1) % LATEST_BLOCK_ROOTS_LENGTH] = previous_block_root.
  • If state.slot % LATEST_BLOCK_ROOTS_LENGTH == 0 append merkle_root(state.latest_block_roots) to state.batched_block_roots.

Per-block processing

Below are the processing steps that happen at every block.

Slot

  • Verify that block.slot == state.slot.

Proposer signature

  • Let block_without_signature_root be the hash_tree_root of block where block.signature is set to EMPTY_SIGNATURE.
  • Let proposal_root = hash_tree_root(ProposalSignedData(state.slot, BEACON_CHAIN_SHARD_NUMBER, block_without_signature_root)).
  • Verify that bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, message=proposal_root, signature=block.signature, domain=get_domain(state.fork, get_current_epoch(state), DOMAIN_PROPOSAL)).

RANDAO

  • Let proposer = state.validator_registry[get_beacon_proposer_index(state, state.slot)].
  • Verify that bls_verify(pubkey=proposer.pubkey, message=int_to_bytes32(get_current_epoch(state)), signature=block.randao_reveal, domain=get_domain(state.fork, get_current_epoch(state), DOMAIN_RANDAO)).
  • Set state.latest_randao_mixes[get_current_epoch(state) % LATEST_RANDAO_MIXES_LENGTH] = xor(get_randao_mix(state, get_current_epoch(state)), hash(block.randao_reveal)).

Eth1 data

  • If block.eth1_data equals eth1_data_vote.eth1_data for some eth1_data_vote in state.eth1_data_votes, set eth1_data_vote.vote_count += 1.
  • Otherwise, append to state.eth1_data_votes a new Eth1DataVote(eth1_data=block.eth1_data, vote_count=1).

Operations

Proposer slashings

Verify that len(block.body.proposer_slashings) <= MAX_PROPOSER_SLASHINGS.

For each proposer_slashing in block.body.proposer_slashings:

  • Let proposer = state.validator_registry[proposer_slashing.proposer_index].
  • Verify that proposer_slashing.proposal_data_1.slot == proposer_slashing.proposal_data_2.slot.
  • Verify that proposer_slashing.proposal_data_1.shard == proposer_slashing.proposal_data_2.shard.
  • Verify that proposer_slashing.proposal_data_1.block_root != proposer_slashing.proposal_data_2.block_root.
  • Verify that proposer.penalized_epoch > get_current_epoch(state).
  • Verify that bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=proposer_slashing.proposal_signature_1, domain=get_domain(state.fork, slot_to_epoch(proposer_slashing.proposal_data_1.slot), DOMAIN_PROPOSAL)).
  • Verify that bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=proposer_slashing.proposal_signature_2, domain=get_domain(state.fork, slot_to_epoch(proposer_slashing.proposal_data_2.slot), DOMAIN_PROPOSAL)).
  • Run penalize_validator(state, proposer_slashing.proposer_index).

Casper slashings

Verify that len(block.body.casper_slashings) <= MAX_CASPER_SLASHINGS.

For each casper_slashing in block.body.casper_slashings:

  • Let slashable_vote_data_1 = casper_slashing.slashable_vote_data_1.
  • Let slashable_vote_data_2 = casper_slashing.slashable_vote_data_2.
  • Let indices(slashable_vote_data) = slashable_vote_data.custody_bit_0_indices + slashable_vote_data.custody_bit_1_indices.
  • Let intersection = [x for x in indices(slashable_vote_data_1) if x in indices(slashable_vote_data_2)].
  • Verify that len(intersection) >= 1.
  • Verify that slashable_vote_data_1.data != slashable_vote_data_2.data.
  • Verify that is_double_vote(slashable_vote_data_1.data, slashable_vote_data_2.data) or is_surround_vote(slashable_vote_data_1.data, slashable_vote_data_2.data).
  • Verify that verify_slashable_vote_data(state, slashable_vote_data_1).
  • Verify that verify_slashable_vote_data(state, slashable_vote_data_2).
  • For each validator index i in intersection run penalize_validator(state, i) if state.validator_registry[i].penalized_epoch > get_current_epoch(state).

Attestations

Verify that len(block.body.attestations) <= MAX_ATTESTATIONS.

For each attestation in block.body.attestations:

  • Verify that attestation.data.slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot.
  • Verify that attestation.data.slot + EPOCH_LENGTH >= state.slot.
  • Verify that attestation.data.justified_epoch is equal to state.justified_epoch if attestation.data.slot >= get_epoch_start_slot(get_current_epoch(state)) else state.previous_justified_epoch.
  • Verify that attestation.data.justified_block_root is equal to get_block_root(state, get_epoch_start_slot(attestation.data.justified_epoch)).
  • Verify that either attestation.data.latest_crosslink_root or attestation.data.shard_block_root equals state.latest_crosslinks[shard].shard_block_root.
  • aggregate_signature verification:
    • Let participants = get_attestation_participants(state, attestation.data, attestation.aggregation_bitfield).
    • Let group_public_key = bls_aggregate_pubkeys([state.validator_registry[v].pubkey for v in participants]).
    • Verify that bls_verify(pubkey=group_public_key, message=hash_tree_root(AttestationDataAndCustodyBit(attestation.data, False)), signature=attestation.aggregate_signature, domain=get_domain(state.fork, slot_to_epoch(attestation.data.slot), DOMAIN_ATTESTATION)).
  • [TO BE REMOVED IN PHASE 1] Verify that attestation.data.shard_block_root == ZERO_HASH.
  • Append PendingAttestation(data=attestation.data, aggregation_bitfield=attestation.aggregation_bitfield, custody_bitfield=attestation.custody_bitfield, slot_included=state.slot) to state.latest_attestations.

Deposits

Verify that len(block.body.deposits) <= MAX_DEPOSITS.

[TODO: add logic to ensure that deposits from 1.0 chain are processed in order] [TODO: update the call to verify_merkle_branch below if it needs to change after we process deposits in order]

For each deposit in block.body.deposits:

  • Let serialized_deposit_data be the serialized form of deposit.deposit_data. It should be 8 bytes for deposit_data.amount followed by 8 bytes for deposit_data.timestamp and then the DepositInput bytes. That is, it should match deposit_data in the Ethereum 1.0 deposit contract of which the hash was placed into the Merkle tree.
  • Verify that verify_merkle_branch(hash(serialized_deposit_data), deposit.branch, DEPOSIT_CONTRACT_TREE_DEPTH, deposit.index, state.latest_eth1_data.deposit_root) is True.
def verify_merkle_branch(leaf: Bytes32, branch: List[Bytes32], depth: int, index: int, root: Bytes32) -> bool:
    value = leaf
    for i in range(depth):
        if index // (2**i) % 2:
            value = hash(branch[i] + value)
        else:
            value = hash(value + branch[i])
    return value == root
  • Run the following:
process_deposit(
    state=state,
    pubkey=deposit.deposit_data.deposit_input.pubkey,
    amount=deposit.deposit_data.amount,
    proof_of_possession=deposit.deposit_data.deposit_input.proof_of_possession,
    withdrawal_credentials=deposit.deposit_data.deposit_input.withdrawal_credentials,
)

Exits

Verify that len(block.body.exits) <= MAX_EXITS.

For each exit in block.body.exits:

  • Let validator = state.validator_registry[exit.validator_index].
  • Verify that validator.exit_epoch > get_entry_exit_effect_epoch(get_current_epoch(state)).
  • Verify that get_current_epoch(state) >= exit.epoch.
  • Let exit_message = hash_tree_root(Exit(epoch=exit.epoch, validator_index=exit.validator_index, signature=EMPTY_SIGNATURE)).
  • Verify that bls_verify(pubkey=validator.pubkey, message=exit_message, signature=exit.signature, domain=get_domain(state.fork, exit.epoch, DOMAIN_EXIT)).
  • Run initiate_validator_exit(state, exit.validator_index).

Custody

[TO BE REMOVED IN PHASE 1] Verify that len(block.body.custody_reseeds) == len(block.body.custody_challenges) == len(block.body.custody_responses) == 0.

Per-epoch processing

The steps below happen when (state.slot + 1) % EPOCH_LENGTH == 0.

Helpers

  • Let current_epoch = get_current_epoch(state).
  • Let previous_epoch = current_epoch - 1 if current_epoch > GENESIS_EPOCH else current_epoch.
  • Let next_epoch = current_epoch + 1.

Validators attesting during the current epoch:

  • Let current_total_balance = sum([get_effective_balance(state, i) for i in get_active_validator_indices(state.validator_registry, current_epoch)]).
  • Let current_epoch_attestations = [a for a in state.latest_attestations if current_epoch == slot_to_epoch(a.data.slot)]. (Note: this is the set of attestations of slots in the epoch current_epoch, not attestations that got included in the chain during the epoch current_epoch.)
  • Validators justifying the epoch boundary block at the start of the current epoch:
    • Let current_epoch_boundary_attestations = [a for a in current_epoch_attestations if a.data.epoch_boundary_root == get_block_root(state, get_epoch_start_slot(current_epoch)) and a.data.justified_epoch == state.justified_epoch].
    • Let current_epoch_boundary_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_boundary_attestations].
    • Let current_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in current_epoch_boundary_attester_indices]).

Validators attesting during the previous epoch:

  • Let previous_total_balance = sum([get_effective_balance(state, i) for i in get_active_validator_indices(state.validator_registry, previous_epoch)]).
  • Validators that made an attestation during the previous epoch:
    • Let previous_epoch_attestations = [a for a in state.latest_attestations if previous_epoch == slot_to_epoch(a.data.slot)].
    • Let previous_epoch_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_attestations].
  • Validators targeting the previous justified slot:
    • Let previous_epoch_justified_attestations = [a for a in current_epoch_attestations + previous_epoch_attestations if a.data.justified_epoch == state.previous_justified_epoch].
    • Let previous_epoch_justified_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_justified_attestations].
    • Let previous_epoch_justified_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_justified_attester_indices]).
  • Validators justifying the epoch boundary block at the start of the previous epoch:
    • Let previous_epoch_boundary_attestations = [a for a in previous_epoch_justified_attestations if a.data.epoch_boundary_root == get_block_root(state, get_epoch_start_slot(previous_epoch))].
    • Let previous_epoch_boundary_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_boundary_attestations].
    • Let previous_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_boundary_attester_indices]).
  • Validators attesting to the expected beacon chain head during the previous epoch:
    • Let previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.data.beacon_block_root == get_block_root(state, a.data.slot)].
    • Let previous_epoch_head_attester_indices be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_head_attestations].
    • Let previous_epoch_head_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_head_attester_indices]).

Note: previous_total_balance and previous_epoch_boundary_attesting_balance balance might be marginally different than the actual balances during previous epoch transition. Due to the tight bound on validator churn each epoch and small per-epoch rewards/penalties, the potential balance difference is very low and only marginally affects consensus safety.

For every slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(next_epoch)), let crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot). For every (crosslink_committee, shard) in crosslink_committees_at_slot, compute:

  • Let shard_block_root be state.latest_crosslinks[shard].shard_block_root
  • Let attesting_validator_indices(crosslink_committee, shard_block_root) be the union of the validator index sets given by [get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_attestations + previous_epoch_attestations if a.data.shard == shard and a.data.shard_block_root == shard_block_root].
  • Let winning_root(crosslink_committee) be equal to the value of shard_block_root such that sum([get_effective_balance(state, i) for i in attesting_validator_indices(crosslink_committee, shard_block_root)]) is maximized (ties broken by favoring lower shard_block_root values).
  • Let attesting_validators(crosslink_committee) be equal to attesting_validator_indices(crosslink_committee, winning_root(crosslink_committee)) for convenience.
  • Let total_attesting_balance(crosslink_committee) = sum([get_effective_balance(state, i) for i in attesting_validators(crosslink_committee)]).
  • Let total_balance(crosslink_committee) = sum([get_effective_balance(state, i) for i in crosslink_committee]).

Define the following helpers to process attestation inclusion rewards and inclusion distance reward/penalty. For every attestation a in previous_epoch_attestations:

  • Let inclusion_slot(state, index) = a.slot_included for the attestation a where index is in get_attestation_participants(state, a.data, a.aggregation_bitfield). If multiple attestations are applicable, the attestation with lowest slot_included is considered.
  • Let inclusion_distance(state, index) = a.slot_included - a.data.slot where a is the above attestation.

Eth1 data

If next_epoch % ETH1_DATA_VOTING_PERIOD == 0:

  • Set state.latest_eth1_data = eth1_data_vote.data if eth1_data_vote.vote_count * 2 > ETH1_DATA_VOTING_PERIOD * EPOCH_LENGTH for some eth1_data_vote in state.eth1_data_votes.
  • Set state.eth1_data_votes = [].

Justification

First, update the justification bitfield:

  • Let new_justified_epoch = state.justified_epoch.
  • Set state.justification_bitfield = state.justification_bitfield << 1.
  • Set state.justification_bitfield |= 2 and new_justified_epoch = previous_epoch if 3 * previous_epoch_boundary_attesting_balance >= 2 * previous_total_balance.
  • Set state.justification_bitfield |= 1 and new_justified_epoch = current_epoch if 3 * current_epoch_boundary_attesting_balance >= 2 * current_total_balance.

Next, update last finalized epoch if possible:

  • Set state.finalized_epoch = state.previous_justified_epoch if (state.justification_bitfield >> 1) % 8 == 0b111 and state.previous_justified_epoch == previous_epoch - 2.
  • Set state.finalized_epoch = state.previous_justified_epoch if (state.justification_bitfield >> 1) % 4 == 0b11 and state.previous_justified_epoch == previous_epoch - 1.
  • Set state.finalized_epoch = state.justified_epoch if (state.justification_bitfield >> 0) % 8 == 0b111 and state.justified_epoch == previous_epoch - 1.
  • Set state.finalized_epoch = state.justified_epoch if (state.justification_bitfield >> 0) % 4 == 0b11 and state.justified_epoch == previous_epoch.

Finally, update the following:

  • Set state.previous_justified_epoch = state.justified_epoch.
  • Set state.justified_epoch = new_justified_epoch.

Crosslinks

For every slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(next_epoch)), let crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot). For every (crosslink_committee, shard) in crosslink_committees_at_slot, compute:

  • Set state.latest_crosslinks[shard] = Crosslink(epoch=current_epoch, shard_block_root=winning_root(crosslink_committee)) if 3 * total_attesting_balance(crosslink_committee) >= 2 * total_balance(crosslink_committee).

Rewards and penalties

First, we define some additional helpers:

  • Let base_reward_quotient = integer_squareroot(previous_total_balance) // BASE_REWARD_QUOTIENT.
  • Let base_reward(state, index) = get_effective_balance(state, index) // base_reward_quotient // 5 for any validator with the given index.
  • Let inactivity_penalty(state, index, epochs_since_finality) = base_reward(state, index) + get_effective_balance(state, index) * epochs_since_finality // INACTIVITY_PENALTY_QUOTIENT // 2 for any validator with the given index.

Justification and finalization

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

  • Let epochs_since_finality = next_epoch - state.finalized_epoch.

Case 1: epochs_since_finality <= 4:

  • Expected FFG source:
    • Any validator index in previous_epoch_justified_attester_indices gains base_reward(state, index) * previous_epoch_justified_attesting_balance // previous_total_balance.
    • Any active validator v not in previous_epoch_justified_attester_indices loses base_reward(state, index).
  • Expected FFG target:
    • Any validator index in previous_epoch_boundary_attester_indices gains base_reward(state, index) * previous_epoch_boundary_attesting_balance // previous_total_balance.
    • Any active validator index not in previous_epoch_boundary_attester_indices loses base_reward(state, index).
  • Expected beacon chain head:
    • Any validator index in previous_epoch_head_attester_indices gains base_reward(state, index) * previous_epoch_head_attesting_balance // previous_total_balance).
    • Any active validator index not in previous_epoch_head_attester_indices loses base_reward(state, index).
  • Inclusion distance:
    • Any validator index in previous_epoch_attester_indices gains base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // inclusion_distance(state, index)

Case 2: epochs_since_finality > 4:

  • Any active validator index not in previous_epoch_justified_attester_indices, loses inactivity_penalty(state, index, epochs_since_finality).
  • Any active validator index not in previous_epoch_boundary_attester_indices, loses inactivity_penalty(state, index, epochs_since_finality).
  • Any active validator index not in previous_epoch_head_attester_indices, loses base_reward(state, index).
  • Any active_validator index with validator.penalized_epoch <= current_epoch, loses 2 * inactivity_penalty(state, index, epochs_since_finality) + base_reward(state, index).
  • Any validator index in previous_epoch_attester_indices loses base_reward(state, index) - base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // inclusion_distance(state, index)

Attestation inclusion

For each index in previous_epoch_attester_indices, we determine the proposer proposer_index = get_beacon_proposer_index(state, inclusion_slot(state, index)) and set state.validator_balances[proposer_index] += base_reward(state, index) // INCLUDER_REWARD_QUOTIENT.

Crosslinks

For every slot in range(get_epoch_start_slot(previous_epoch), get_epoch_start_slot(current_epoch)):

  • Let crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot).
  • For every (crosslink_committee, shard) in crosslink_committees_at_slot:
    • If index in attesting_validators(crosslink_committee), state.validator_balances[index] += base_reward(state, index) * total_attesting_balance(crosslink_committee) // total_balance(crosslink_committee)).
    • If index not in attesting_validators(crosslink_committee), state.validator_balances[index] -= base_reward(state, index).

Ejections

  • Run process_ejections(state).
def process_ejections(state: BeaconState) -> None:
    """
    Iterate through the validator registry
    and eject active validators with balance below ``EJECTION_BALANCE``.
    """
    for index in get_active_validator_indices(state.validator_registry, current_epoch(state)):
        if state.validator_balances[index] < EJECTION_BALANCE:
            exit_validator(state, index)

Validator registry and shuffling seed data

First, update the following:

  • Set state.previous_calculation_epoch = state.current_calculation_epoch.
  • Set state.previous_epoch_start_shard = state.current_epoch_start_shard.
  • Set state.previous_epoch_seed = state.current_epoch_seed.
  • Set state.latest_index_roots[next_epoch % LATEST_INDEX_ROOTS_LENGTH] = hash_tree_root(get_active_validator_indices(state, next_epoch)).

If the following are satisfied:

  • state.finalized_epoch > state.validator_registry_update_epoch
  • state.latest_crosslinks[shard].epoch > state.validator_registry_update_epoch for every shard number shard in [(state.current_epoch_start_shard + i) % SHARD_COUNT for i in range(get_current_epoch_committee_count(state))] (that is, for every shard in the current committees)

update the validator registry and associated fields by running

def update_validator_registry(state: BeaconState) -> None:
    """
    Update validator registry.
    Note that this function mutates ``state``.
    """
    current_epoch = get_current_epoch(state)
    # The active validators
    active_validator_indices = get_active_validator_indices(state.validator_registry, current_epoch)
    # The total effective balance of active validators
    total_balance = sum([get_effective_balance(state, i) for i in active_validator_indices])

    # The maximum balance churn in Gwei (for deposits and exits separately)
    max_balance_churn = max(
        MAX_DEPOSIT_AMOUNT,
        total_balance // (2 * MAX_BALANCE_CHURN_QUOTIENT)
    )

    # Activate validators within the allowable balance churn
    balance_churn = 0
    for index, validator in enumerate(state.validator_registry):
        if validator.activation_epoch > get_entry_exit_effect_epoch(current_epoch) and state.validator_balances[index] >= MAX_DEPOSIT_AMOUNT:
            # Check the balance churn would be within the allowance
            balance_churn += get_effective_balance(state, index)
            if balance_churn > max_balance_churn:
                break

            # Activate validator
            activate_validator(state, index, False)

    # Exit validators within the allowable balance churn
    balance_churn = 0
    for index, validator in enumerate(state.validator_registry):
        if validator.exit_epoch > get_entry_exit_effect_epoch(current_epoch) and validator.status_flags & INITIATED_EXIT:
            # Check the balance churn would be within the allowance
            balance_churn += get_effective_balance(state, index)
            if balance_churn > max_balance_churn:
                break

            # Exit validator
            exit_validator(state, index)

    state.validator_registry_update_epoch = current_epoch

and perform the following updates:

  • Set state.current_calculation_epoch = next_epoch
  • Set state.current_epoch_start_shard = (state.current_epoch_start_shard + get_current_epoch_committee_count(state)) % SHARD_COUNT
  • Set state.current_epoch_seed = generate_seed(state, state.current_calculation_epoch)

If a validator registry update does not happen do the following:

  • Let epochs_since_last_registry_change = current_epoch - state.validator_registry_update_epoch.
  • If epochs_since_last_registry_change is an exact power of 2:
    • Set state.current_calculation_epoch = next_epoch.
    • Set state.current_epoch_seed = generate_seed(state, state.current_calculation_epoch)
    • Note that state.current_epoch_start_shard is left unchanged.

Invariant: the active index root that is hashed into the shuffling seed actually is the hash_tree_root of the validator set that is used for that epoch.

Regardless of whether or not a validator set change happens, run the following:

def process_penalties_and_exits(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    # The active validators
    active_validator_indices = get_active_validator_indices(state.validator_registry, current_epoch)
    # The total effective balance of active validators
    total_balance = sum([get_effective_balance(state, i) for i in active_validator_indices])

    for index, validator in enumerate(state.validator_registry):
        if current_epoch == validator.penalized_epoch + LATEST_PENALIZED_EXIT_LENGTH // 2:
            e = current_epoch % LATEST_PENALIZED_EXIT_LENGTH
            total_at_start = state.latest_penalized_balances[(e + 1) % LATEST_PENALIZED_EXIT_LENGTH]
            total_at_end = state.latest_penalized_balances[e]
            total_penalties = total_at_end - total_at_start
            penalty = get_effective_balance(state, index) * min(total_penalties * 3, total_balance) // total_balance
            state.validator_balances[index] -= penalty

    def eligible(index):
        validator = state.validator_registry[index]
        if validator.penalized_epoch <= current_epoch:
            penalized_withdrawal_epochs = LATEST_PENALIZED_EXIT_LENGTH // 2
            return current_epoch >= validator.penalized_epoch + penalized_withdrawal_epochs
        else:
            return current_epoch >= validator.exit_epoch + MIN_VALIDATOR_WITHDRAWAL_EPOCHS

    all_indices = list(range(len(state.validator_registry)))
    eligible_indices = filter(eligible, all_indices)
    sorted_indices = sorted(eligible_indices, key=lambda index: state.validator_registry[index].exit_count)
    withdrawn_so_far = 0
    for index in sorted_indices:
        prepare_validator_for_withdrawal(state, index)
        withdrawn_so_far += 1
        if withdrawn_so_far >= MAX_WITHDRAWALS_PER_EPOCH:
            break

Final updates

  • Set state.latest_penalized_balances[(next_epoch) % LATEST_PENALIZED_EXIT_LENGTH] = state.latest_penalized_balances[current_epoch % LATEST_PENALIZED_EXIT_LENGTH].
  • Set state.latest_randao_mixes[next_epoch % LATEST_RANDAO_MIXES_LENGTH] = get_randao_mix(state, current_epoch).
  • Remove any attestation in state.latest_attestations such that slot_to_epoch(attestation.data.slot) < current_epoch.

State root processing

Verify block.state_root == hash_tree_root(state) if there exists a block for the slot being processed.

References

This section is divided into Normative and Informative references. Normative references are those that must be read in order to implement this specification, while Informative references are merely that, information. An example of the former might be the details of a required consensus algorithm, and an example of the latter might be a pointer to research that demonstrates why a particular consensus algorithm might be better suited for inclusion in the standard than another.

Normative

Informative

casper-ffg
  Casper the Friendly Finality Gadget. V. Buterin and V. Griffith. URL: https://arxiv.org/abs/1710.09437

python-poc
  Python proof-of-concept implementation. Ethereum Foundation. URL: https://github.com/ethereum/beacon_chain

Copyright

Copyright and related rights waived via CC0.