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

# Ethereum 2.0 Phase 0 -- The Beacon Chain

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

## Table of contents
<!-- TOC -->

- [Ethereum 2.0 Phase 0 -- The Beacon Chain](#ethereum-20-phase-0----the-beacon-chain)
    - [Table of contents](#table-of-contents)
    - [Introduction](#introduction)
    - [Notation](#notation)
    - [Terminology](#terminology)
    - [Constants](#constants)
        - [Misc](#misc)
        - [Deposit contract](#deposit-contract)
        - [Gwei values](#gwei-values)
        - [Initial values](#initial-values)
        - [Time parameters](#time-parameters)
        - [State list lengths](#state-list-lengths)
        - [Reward and penalty quotients](#reward-and-penalty-quotients)
        - [Max operations per block](#max-operations-per-block)
        - [Signature domains](#signature-domains)
    - [Data structures](#data-structures)
        - [Misc dependencies](#misc-dependencies)
            - [`Fork`](#fork)
            - [`Crosslink`](#crosslink)
            - [`Eth1Data`](#eth1data)
            - [`AttestationData`](#attestationdata)
            - [`AttestationDataAndCustodyBit`](#attestationdataandcustodybit)
            - [`IndexedAttestation`](#indexedattestation)
            - [`DepositData`](#depositdata)
            - [`BeaconBlockHeader`](#beaconblockheader)
            - [`Validator`](#validator)
            - [`PendingAttestation`](#pendingattestation)
            - [`HistoricalBatch`](#historicalbatch)
        - [Beacon operations](#beacon-operations)
            - [`ProposerSlashing`](#proposerslashing)
            - [`AttesterSlashing`](#attesterslashing)
            - [`Attestation`](#attestation)
            - [`Deposit`](#deposit)
            - [`VoluntaryExit`](#voluntaryexit)
            - [`Transfer`](#transfer)
        - [Beacon blocks](#beacon-blocks)
            - [`BeaconBlockBody`](#beaconblockbody)
            - [`BeaconBlock`](#beaconblock)
        - [Beacon state](#beacon-state)
            - [`BeaconState`](#beaconstate)
    - [Custom types](#custom-types)
    - [Helper functions](#helper-functions)
        - [`xor`](#xor)
        - [`hash`](#hash)
        - [`hash_tree_root`](#hash_tree_root)
        - [`signing_root`](#signing_root)
        - [`bls_domain`](#bls_domain)
        - [`slot_to_epoch`](#slot_to_epoch)
        - [`get_previous_epoch`](#get_previous_epoch)
        - [`get_current_epoch`](#get_current_epoch)
        - [`get_epoch_start_slot`](#get_epoch_start_slot)
        - [`is_active_validator`](#is_active_validator)
        - [`is_slashable_validator`](#is_slashable_validator)
        - [`get_active_validator_indices`](#get_active_validator_indices)
        - [`increase_balance`](#increase_balance)
        - [`decrease_balance`](#decrease_balance)
        - [`get_epoch_committee_count`](#get_epoch_committee_count)
        - [`get_shard_delta`](#get_shard_delta)
        - [`get_epoch_start_shard`](#get_epoch_start_shard)
        - [`get_attestation_data_slot`](#get_attestation_data_slot)
        - [`get_block_root_at_slot`](#get_block_root_at_slot)
        - [`get_block_root`](#get_block_root)
        - [`get_randao_mix`](#get_randao_mix)
        - [`get_active_index_root`](#get_active_index_root)
        - [`generate_seed`](#generate_seed)
        - [`get_beacon_proposer_index`](#get_beacon_proposer_index)
        - [`verify_merkle_branch`](#verify_merkle_branch)
        - [`get_shuffled_index`](#get_shuffled_index)
        - [`compute_committee`](#compute_committee)
        - [`get_crosslink_committee`](#get_crosslink_committee)
        - [`get_attesting_indices`](#get_attesting_indices)
        - [`int_to_bytes`](#int_to_bytes)
        - [`bytes_to_int`](#bytes_to_int)
        - [`get_total_balance`](#get_total_balance)
        - [`get_domain`](#get_domain)
        - [`get_bitfield_bit`](#get_bitfield_bit)
        - [`verify_bitfield`](#verify_bitfield)
        - [`convert_to_indexed`](#convert_to_indexed)
        - [`validate_indexed_attestation`](#validate_indexed_attestation)
        - [`is_slashable_attestation_data`](#is_slashable_attestation_data)
        - [`integer_squareroot`](#integer_squareroot)
        - [`get_delayed_activation_exit_epoch`](#get_delayed_activation_exit_epoch)
        - [`get_churn_limit`](#get_churn_limit)
        - [`bls_verify`](#bls_verify)
        - [`bls_verify_multiple`](#bls_verify_multiple)
        - [`bls_aggregate_pubkeys`](#bls_aggregate_pubkeys)
        - [Routines for updating validator status](#routines-for-updating-validator-status)
            - [`initiate_validator_exit`](#initiate_validator_exit)
            - [`slash_validator`](#slash_validator)
    - [Genesis](#genesis)
        - [`Eth2Genesis`](#eth2genesis)
        - [Genesis state](#genesis-state)
        - [Genesis block](#genesis-block)
    - [Beacon chain state transition function](#beacon-chain-state-transition-function)
        - [Epoch processing](#epoch-processing)
            - [Helper functions](#helper-functions-1)
            - [Justification and finalization](#justification-and-finalization)
            - [Crosslinks](#crosslinks)
            - [Rewards and penalties](#rewards-and-penalties)
            - [Registry updates](#registry-updates)
            - [Slashings](#slashings)
            - [Final updates](#final-updates)
        - [Block processing](#block-processing)
            - [Block header](#block-header)
            - [RANDAO](#randao)
            - [Eth1 data](#eth1-data)
            - [Operations](#operations)
                - [Proposer slashings](#proposer-slashings)
                - [Attester slashings](#attester-slashings)
                - [Attestations](#attestations)
                - [Deposits](#deposits)
                - [Voluntary exits](#voluntary-exits)
                - [Transfers](#transfers)

<!-- /TOC -->

## 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](#dfn-validator). In the initial deployment phases of Ethereum 2.0, the only mechanism to become a [validator](#dfn-validator) is to make a one-way ETH transaction to a deposit contract on Ethereum 1.0. Activation as a [validator](#dfn-validator) happens when Ethereum 1.0 deposit receipts are processed by the beacon chain, the activation balance is reached, and a queuing process is completed. 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 simultaneously availability votes for a shard block and 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.

## Terminology

* **Validator**<a id="dfn-validator"></a>—a registered participant in the beacon chain. You can become one by sending ether into the Ethereum 1.0 deposit contract.
* **Active validator**<a id="dfn-active-validator"></a>—an active participant in the Ethereum 2.0 consensus invited to, among other things, propose and attest to blocks and vote for crosslinks.
* **Committee**—a (pseudo-) randomly sampled subset of [active validators](#dfn-active-validator). 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](#dfn-validator) that the protocol recognizes it as representing the committee".
* **Proposer**—the [validator](#dfn-validator) that creates a beacon chain block.
* **Attester**—a [validator](#dfn-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 that 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 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](#dfn-validator) get exactly one chance to make an attestation.
* **Finalized**, **justified**—see the [Casper FFG paper](https://arxiv.org/abs/1710.09437).
* **Withdrawal period**—the number of slots between a [validator](#dfn-validator) exit and the [validator](#dfn-validator) balance being withdrawable.
* **Genesis time**—the Unix time of the genesis beacon chain block at slot 0.

## Constants

*Note*: The default mainnet values for the constants are included here for spec-design purposes.
The different configurations for mainnet, testnets, and YAML-based testing can be found in the `configs/constant_presets/` directory.
These configurations are updated for releases, but may be out of sync during `dev` changes.

### Misc

| Name | Value |
| - | - |
| `SHARD_COUNT` | `2**10` (= 1,024) |
| `TARGET_COMMITTEE_SIZE` | `2**7` (= 128) |
| `MAX_INDICES_PER_ATTESTATION` | `2**12` (= 4,096) |
| `MIN_PER_EPOCH_CHURN_LIMIT` | `2**2` (= 4) |
| `CHURN_LIMIT_QUOTIENT` | `2**16` (= 65,536) |
| `BASE_REWARDS_PER_EPOCH` | `5` |
| `SHUFFLE_ROUND_COUNT` | `90` |

* For the safety of crosslinks `TARGET_COMMITTEE_SIZE` exceeds [the recommended minimum committee size of 111](https://vitalik.ca/files/Ithaca201807_Sharding.pdf); with sufficient active validators (at least `SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE`), the shuffling algorithm ensures committee sizes of 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 |
| - | - |
| `DEPOSIT_CONTRACT_TREE_DEPTH` | `2**5` (= 32) |

### Gwei values

| Name | Value | Unit |
| - | - | :-: |
| `MIN_DEPOSIT_AMOUNT` | `2**0 * 10**9` (= 1,000,000,000) | Gwei |
| `MAX_EFFECTIVE_BALANCE` | `2**5 * 10**9` (= 32,000,000,000) | Gwei |
| `EJECTION_BALANCE` | `2**4 * 10**9` (= 16,000,000,000) | Gwei |
| `EFFECTIVE_BALANCE_INCREMENT` | `2**0 * 10**9` (= 1,000,000,000) | Gwei |

### Initial values

| Name | Value |
| - | - |
| `GENESIS_SLOT` | `0` |
| `GENESIS_EPOCH` | `0` |
| `FAR_FUTURE_EPOCH` | `2**64 - 1` |
| `ZERO_HASH` | `b'\x00' * 32` |
| `BLS_WITHDRAWAL_PREFIX` | `0` |

### Time parameters

| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `MIN_ATTESTATION_INCLUSION_DELAY` | `2**2` (= 4) | slots | 24 seconds |
| `SLOTS_PER_EPOCH` | `2**6` (= 64) | slots | 6.4 minutes |
| `MIN_SEED_LOOKAHEAD` | `2**0` (= 1) | epochs | 6.4 minutes |
| `ACTIVATION_EXIT_DELAY` | `2**2` (= 4) | epochs | 25.6 minutes |
| `SLOTS_PER_ETH1_VOTING_PERIOD` | `2**10` (= 1,024) | slots | ~1.7 hours |
| `SLOTS_PER_HISTORICAL_ROOT` | `2**13` (= 8,192) | slots | ~13 hours |
| `MIN_VALIDATOR_WITHDRAWABILITY_DELAY` | `2**8` (= 256) | epochs | ~27 hours |
| `PERSISTENT_COMMITTEE_PERIOD` | `2**11` (= 2,048)  | epochs | 9 days  |
| `MAX_EPOCHS_PER_CROSSLINK` | `2**6` (= 64) | epochs | ~7 hours |
| `MIN_EPOCHS_TO_INACTIVITY_PENALTY` | `2**2` (= 4) | epochs | 25.6 minutes |

* `MAX_EPOCHS_PER_CROSSLINK` should be a small constant times `SHARD_COUNT // SLOTS_PER_EPOCH`

### State list lengths

| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `LATEST_RANDAO_MIXES_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |
| `LATEST_ACTIVE_INDEX_ROOTS_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |
| `LATEST_SLASHED_EXIT_LENGTH` | `2**13` (= 8,192) | epochs | ~36 days |

### Reward and penalty quotients

| Name | Value |
| - | - |
| `BASE_REWARD_QUOTIENT` | `2**5` (= 32) |
| `WHISTLEBLOWING_REWARD_QUOTIENT` | `2**9` (= 512) |
| `PROPOSER_REWARD_QUOTIENT` | `2**3` (= 8) |
| `INACTIVITY_PENALTY_QUOTIENT` | `2**25` (= 33,554,432) |
| `MIN_SLASHING_PENALTY_QUOTIENT` | `2**5` (= 32) |

* **The `BASE_REWARD_QUOTIENT` is NOT final. Once all other protocol details are finalized, it will be adjusted to target a theoretical maximum total issuance of `2**21` ETH per year if `2**27` ETH is validating (and therefore `2**20` per year if `2**25` ETH is validating, etc.)**
* 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](#dfn-validator) to about `1/sqrt(e) ~= 60.6%`. Indeed, the balance retained by offline [validators](#dfn-validator) 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)`.

### Max operations per block

| Name | Value |
| - | - |
| `MAX_PROPOSER_SLASHINGS` | `2**4` (= 16) |
| `MAX_ATTESTER_SLASHINGS` | `2**0` (= 1) |
| `MAX_ATTESTATIONS` | `2**7` (= 128) |
| `MAX_DEPOSITS` | `2**4` (= 16) |
| `MAX_VOLUNTARY_EXITS` | `2**4` (= 16) |
| `MAX_TRANSFERS` | `0` |

### Signature domains

| Name | Value |
| - | - |
| `DOMAIN_BEACON_PROPOSER` | `0` |
| `DOMAIN_RANDAO` | `1` |
| `DOMAIN_ATTESTATION` | `2` |
| `DOMAIN_DEPOSIT` | `3` |
| `DOMAIN_VOLUNTARY_EXIT` | `4` |
| `DOMAIN_TRANSFER` | `5` |

## Data structures

The following data structures are defined as [SimpleSerialize (SSZ)](../simple-serialize.md) objects.

The types are defined topologically to aid in facilitating an executable version of the spec.

### Misc dependencies

#### `Fork`

```python
{
    # Previous fork version
    'previous_version': 'bytes4',
    # Current fork version
    'current_version': 'bytes4',
    # Fork epoch number
    'epoch': 'uint64',
}
```

#### `Crosslink`

```python
{
    # Shard number
    'shard': 'uint64',
    # Crosslinking data from epochs [start....end-1]
    'start_epoch': 'uint64',
    'end_epoch': 'uint64',
    # Root of the previous crosslink
    'parent_root': 'bytes32',
    # Root of the crosslinked shard data since the previous crosslink
    'data_root': 'bytes32',
}
```

#### `Eth1Data`

```python
{
    # Root of the deposit tree
    'deposit_root': 'bytes32',
    # Total number of deposits
    'deposit_count': 'uint64',
    # Block hash
    'block_hash': 'bytes32',
}
```

#### `AttestationData`

```python
{
    # LMD GHOST vote
    'beacon_block_root': 'bytes32',

    # FFG vote
    'source_epoch': 'uint64',
    'source_root': 'bytes32',
    'target_epoch': 'uint64',
    'target_root': 'bytes32',

    # Crosslink vote
    'crosslink': Crosslink,
}
```

#### `AttestationDataAndCustodyBit`

```python
{
    # Attestation data
    'data': AttestationData,
    # Custody bit
    'custody_bit': 'bool',
}
```

#### `IndexedAttestation`

```python
{
    # Validator indices
    'custody_bit_0_indices': ['uint64'],
    'custody_bit_1_indices': ['uint64'],
    # Attestation data
    'data': AttestationData,
    # Aggregate signature
    'signature': 'bytes96',
}
```

#### `DepositData`

```python
{
    # BLS pubkey
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Amount in Gwei
    'amount': 'uint64',
    # Container self-signature
    'signature': 'bytes96',
}
```

#### `BeaconBlockHeader`

```python
{
    'slot': 'uint64',
    'parent_root': 'bytes32',
    'state_root': 'bytes32',
    'body_root': 'bytes32',
    'signature': 'bytes96',
}
```

#### `Validator`

```python
{
    # BLS public key
    'pubkey': 'bytes48',
    # Withdrawal credentials
    'withdrawal_credentials': 'bytes32',
    # Epoch when became eligible for activation
    'activation_eligibility_epoch': 'uint64',
    # Epoch when validator activated
    'activation_epoch': 'uint64',
    # Epoch when validator exited
    'exit_epoch': 'uint64',
    # Epoch when validator is eligible to withdraw
    'withdrawable_epoch': 'uint64',
    # Was the validator slashed
    'slashed': 'bool',
    # Effective balance
    'effective_balance': 'uint64',
}
```

#### `PendingAttestation`

```python
{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Inclusion delay
    'inclusion_delay': 'uint64',
    # Proposer index
    'proposer_index': 'uint64',
}
```

#### `HistoricalBatch`

```python
{
    # Block roots
    'block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    # State roots
    'state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
}
```

### Beacon operations

#### `ProposerSlashing`

```python
{
    # Proposer index
    'proposer_index': 'uint64',
    # First block header
    'header_1': BeaconBlockHeader,
    # Second block header
    'header_2': BeaconBlockHeader,
}
```

#### `AttesterSlashing`

```python
{
    # First attestation
    'attestation_1': IndexedAttestation,
    # Second attestation
    'attestation_2': IndexedAttestation,
}
```

#### `Attestation`

```python
{
    # Attester aggregation bitfield
    'aggregation_bitfield': 'bytes',
    # Attestation data
    'data': AttestationData,
    # Custody bitfield
    'custody_bitfield': 'bytes',
    # BLS aggregate signature
    'signature': 'bytes96',
}
```

#### `Deposit`

```python
{
    # Branch in the deposit tree
    'proof': ['bytes32', DEPOSIT_CONTRACT_TREE_DEPTH],
    # Index in the deposit tree
    'index': 'uint64',
    # Data
    'data': DepositData,
}
```

#### `VoluntaryExit`

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

#### `Transfer`

```python
{
    # Sender index
    'sender': 'uint64',
    # Recipient index
    'recipient': 'uint64',
    # Amount in Gwei
    'amount': 'uint64',
    # Fee in Gwei for block proposer
    'fee': 'uint64',
    # Inclusion slot
    'slot': 'uint64',
    # Sender withdrawal pubkey
    'pubkey': 'bytes48',
    # Sender signature
    'signature': 'bytes96',
}
```

### Beacon blocks

#### `BeaconBlockBody`

```python
{
    'randao_reveal': 'bytes96',
    'eth1_data': Eth1Data,
    'graffiti': 'bytes32',
    'proposer_slashings': [ProposerSlashing],
    'attester_slashings': [AttesterSlashing],
    'attestations': [Attestation],
    'deposits': [Deposit],
    'voluntary_exits': [VoluntaryExit],
    'transfers': [Transfer],
}
```

#### `BeaconBlock`

```python
{
    # Header
    'slot': 'uint64',
    'parent_root': 'bytes32',
    'state_root': 'bytes32',
    'body': BeaconBlockBody,
    'signature': 'bytes96',
}
```

### Beacon state

#### `BeaconState`

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

    # Validator registry
    'validator_registry': [Validator],
    'balances': ['uint64'],

    # Randomness and committees
    'latest_randao_mixes': ['bytes32', LATEST_RANDAO_MIXES_LENGTH],
    'latest_start_shard': 'uint64',

    # Finality
    'previous_epoch_attestations': [PendingAttestation],
    'current_epoch_attestations': [PendingAttestation],
    'previous_justified_epoch': 'uint64',
    'current_justified_epoch': 'uint64',
    'previous_justified_root': 'bytes32',
    'current_justified_root': 'bytes32',
    'justification_bitfield': 'uint64',
    'finalized_epoch': 'uint64',
    'finalized_root': 'bytes32',

    # Recent state
    'current_crosslinks': [Crosslink, SHARD_COUNT],
    'previous_crosslinks': [Crosslink, SHARD_COUNT],
    'latest_block_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    'latest_state_roots': ['bytes32', SLOTS_PER_HISTORICAL_ROOT],
    'latest_active_index_roots': ['bytes32', LATEST_ACTIVE_INDEX_ROOTS_LENGTH],
    'latest_slashed_balances': ['uint64', LATEST_SLASHED_EXIT_LENGTH],
    'latest_block_header': BeaconBlockHeader,
    'historical_roots': ['bytes32'],

    # Ethereum 1.0 chain data
    'latest_eth1_data': Eth1Data,
    'eth1_data_votes': [Eth1Data],
    'deposit_index': 'uint64',
}
```

## Custom types

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

| Name | SSZ equivalent | Description |
| - | - | - |
| `Slot` | `uint64` | a slot number |
| `Epoch` | `uint64` | an epoch number |
| `Shard` | `uint64` | a shard number |
| `ValidatorIndex` | `uint64` | a validator registry index |
| `Gwei` | `uint64` | an amount in Gwei |
| `Bytes32` | `bytes32` | 32 bytes of binary data |
| `BLSPubkey` | `bytes48` | a BLS12-381 public key |
| `BLSSignature` | `bytes96` | a BLS12-381 signature |

## Helper functions

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

### `xor`

```python
def xor(bytes1: Bytes32, bytes2: Bytes32) -> Bytes32:
    return bytes(a ^ b for a, b in zip(bytes1, bytes2))
```

### `hash`

The `hash` function is SHA256.

*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](../simple-serialize.md#merkleization).

### `signing_root`

`def signing_root(object: SSZContainer) -> Bytes32` is a function defined in the [SimpleSerialize spec](../simple-serialize.md#self-signed-containers) to compute signing messages.

### `bls_domain`

```python
def bls_domain(domain_type: int, fork_version: bytes=b'\x00\x00\x00\x00') -> int:
    """
    Return the bls domain given by the ``domain_type`` and optional 4 byte ``fork_version`` (defaults to zero).
    """
    return bytes_to_int(int_to_bytes(domain_type, length=4) + fork_version)
```

### `slot_to_epoch`

```python
def slot_to_epoch(slot: Slot) -> Epoch:
    """
    Return the epoch number of the given ``slot``.
    """
    return slot // SLOTS_PER_EPOCH
```

### `get_previous_epoch`

```python
def get_previous_epoch(state: BeaconState) -> Epoch:
    """`
    Return the previous epoch of the given ``state``.
    Return the current epoch if it's genesis epoch.
    """
    current_epoch = get_current_epoch(state)
    return GENESIS_EPOCH if current_epoch == GENESIS_EPOCH else current_epoch - 1
```

### `get_current_epoch`

```python
def get_current_epoch(state: BeaconState) -> Epoch:
    """
    Return the current epoch of the given ``state``.
    """
    return slot_to_epoch(state.slot)
```

### `get_epoch_start_slot`

```python
def get_epoch_start_slot(epoch: Epoch) -> Slot:
    """
    Return the starting slot of the given ``epoch``.
    """
    return epoch * SLOTS_PER_EPOCH
```

### `is_active_validator`

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

### `is_slashable_validator`

```python
def is_slashable_validator(validator: Validator, epoch: Epoch) -> bool:
    """
    Check if ``validator`` is slashable.
    """
    return validator.slashed is False and (validator.activation_epoch <= epoch < validator.withdrawable_epoch)
```

### `get_active_validator_indices`

```python
def get_active_validator_indices(state: BeaconState, epoch: Epoch) -> List[ValidatorIndex]:
    """
    Get active validator indices at ``epoch``.
    """
    return [i for i, v in enumerate(state.validator_registry) if is_active_validator(v, epoch)]
```

### `increase_balance`

```python
def increase_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
    """
    Increase validator balance by ``delta``.
    """
    state.balances[index] += delta
```

### `decrease_balance`

```python
def decrease_balance(state: BeaconState, index: ValidatorIndex, delta: Gwei) -> None:
    """
    Decrease validator balance by ``delta`` with underflow protection.
    """
    state.balances[index] = 0 if delta > state.balances[index] else state.balances[index] - delta
```

### `get_epoch_committee_count`

```python
def get_epoch_committee_count(state: BeaconState, epoch: Epoch) -> int:
    """
    Return the number of committees at ``epoch``.
    """
    active_validator_indices = get_active_validator_indices(state, epoch)
    return max(
        1,
        min(
            SHARD_COUNT // SLOTS_PER_EPOCH,
            len(active_validator_indices) // SLOTS_PER_EPOCH // TARGET_COMMITTEE_SIZE,
        )
    ) * SLOTS_PER_EPOCH
```

### `get_shard_delta`

```python
def get_shard_delta(state: BeaconState, epoch: Epoch) -> int:
    """
    Return the number of shards to increment ``state.latest_start_shard`` during ``epoch``.
    """
    return min(get_epoch_committee_count(state, epoch), SHARD_COUNT - SHARD_COUNT // SLOTS_PER_EPOCH)
```

### `get_epoch_start_shard`

```python
def get_epoch_start_shard(state: BeaconState, epoch: Epoch) -> Shard:
    assert epoch <= get_current_epoch(state) + 1
    check_epoch = get_current_epoch(state) + 1
    shard = (state.latest_start_shard + get_shard_delta(state, get_current_epoch(state))) % SHARD_COUNT
    while check_epoch > epoch:
        check_epoch -= 1
        shard = (shard + SHARD_COUNT - get_shard_delta(state, check_epoch)) % SHARD_COUNT
    return shard
```

### `get_attestation_data_slot`

```python
def get_attestation_data_slot(state: BeaconState, data: AttestationData) -> Slot:
    committee_count = get_epoch_committee_count(state, data.target_epoch)
    offset = (data.crosslink.shard + SHARD_COUNT - get_epoch_start_shard(state, data.target_epoch)) % SHARD_COUNT
    return get_epoch_start_slot(data.target_epoch) + offset // (committee_count // SLOTS_PER_EPOCH)
```

### `get_block_root_at_slot`

```python
def get_block_root_at_slot(state: BeaconState,
                           slot: Slot) -> Bytes32:
    """
    Return the block root at a recent ``slot``.
    """
    assert slot < state.slot <= slot + SLOTS_PER_HISTORICAL_ROOT
    return state.latest_block_roots[slot % SLOTS_PER_HISTORICAL_ROOT]
```

### `get_block_root`

```python
def get_block_root(state: BeaconState,
                   epoch: Epoch) -> Bytes32:
    """
    Return the block root at a recent ``epoch``.
    """
    return get_block_root_at_slot(state, get_epoch_start_slot(epoch))
```

### `get_randao_mix`

```python
def get_randao_mix(state: BeaconState,
                   epoch: Epoch) -> Bytes32:
    """
    Return the randao mix at a recent ``epoch``.
    ``epoch`` expected to be between (current_epoch - LATEST_RANDAO_MIXES_LENGTH, current_epoch].
    """
    return state.latest_randao_mixes[epoch % LATEST_RANDAO_MIXES_LENGTH]
```

### `get_active_index_root`

```python
def get_active_index_root(state: BeaconState,
                          epoch: Epoch) -> Bytes32:
    """
    Return the index root at a recent ``epoch``.
    ``epoch`` expected to be between
    (current_epoch - LATEST_ACTIVE_INDEX_ROOTS_LENGTH + ACTIVATION_EXIT_DELAY, current_epoch + ACTIVATION_EXIT_DELAY].
    """
    return state.latest_active_index_roots[epoch % LATEST_ACTIVE_INDEX_ROOTS_LENGTH]
```

### `generate_seed`

```python
def generate_seed(state: BeaconState,
                  epoch: Epoch) -> Bytes32:
    """
    Generate a seed for the given ``epoch``.
    """
    return hash(
        get_randao_mix(state, epoch + LATEST_RANDAO_MIXES_LENGTH - MIN_SEED_LOOKAHEAD) +
        get_active_index_root(state, epoch) +
        int_to_bytes(epoch, length=32)
    )
```

### `get_beacon_proposer_index`

```python
def get_beacon_proposer_index(state: BeaconState) -> ValidatorIndex:
    """
    Return the current beacon proposer index.
    """
    epoch = get_current_epoch(state)
    committees_per_slot = get_epoch_committee_count(state, epoch) // SLOTS_PER_EPOCH
    offset = committees_per_slot * (state.slot % SLOTS_PER_EPOCH)
    shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
    first_committee = get_crosslink_committee(state, epoch, shard)
    MAX_RANDOM_BYTE = 2**8 - 1
    seed = generate_seed(state, epoch)
    i = 0
    while True:
        candidate_index = first_committee[(epoch + i) % len(first_committee)]
        random_byte = hash(seed + int_to_bytes(i // 32, length=8))[i % 32]
        effective_balance = state.validator_registry[candidate_index].effective_balance
        if effective_balance * MAX_RANDOM_BYTE >= MAX_EFFECTIVE_BALANCE * random_byte:
            return candidate_index
        i += 1
```

### `verify_merkle_branch`

```python
def verify_merkle_branch(leaf: Bytes32, proof: List[Bytes32], depth: int, index: int, root: Bytes32) -> bool:
    """
    Verify that the given ``leaf`` is on the merkle branch ``proof``
    starting with the given ``root``.
    """
    value = leaf
    for i in range(depth):
        if index // (2**i) % 2:
            value = hash(proof[i] + value)
        else:
            value = hash(value + proof[i])
    return value == root
```

### `get_shuffled_index`

```python
def get_shuffled_index(index: ValidatorIndex, index_count: int, seed: Bytes32) -> ValidatorIndex:
    """
    Return the shuffled validator index corresponding to ``seed`` (and ``index_count``).
    """
    assert index < index_count
    assert index_count <= 2**40

    # Swap or not (https://link.springer.com/content/pdf/10.1007%2F978-3-642-32009-5_1.pdf)
    # See the 'generalized domain' algorithm on page 3
    for round in range(SHUFFLE_ROUND_COUNT):
        pivot = bytes_to_int(hash(seed + int_to_bytes(round, length=1))[0:8]) % index_count
        flip = (pivot + index_count - index) % index_count
        position = max(index, flip)
        source = hash(seed + int_to_bytes(round, length=1) + int_to_bytes(position // 256, length=4))
        byte = source[(position % 256) // 8]
        bit = (byte >> (position % 8)) % 2
        index = flip if bit else index

    return index
```

### `compute_committee`

```python
def compute_committee(indices: List[ValidatorIndex], seed: Bytes32, index: int, count: int) -> List[ValidatorIndex]:
    start = (len(indices) * index) // count
    end = (len(indices) * (index + 1)) // count
    return [indices[get_shuffled_index(i, len(indices), seed)] for i in range(start, end)]
```

### `get_crosslink_committee`

```python
def get_crosslink_committee(state: BeaconState, epoch: Epoch, shard: Shard) -> List[ValidatorIndex]:
    return compute_committee(
        indices=get_active_validator_indices(state, epoch),
        seed=generate_seed(state, epoch),
        index=(shard + SHARD_COUNT - get_epoch_start_shard(state, epoch)) % SHARD_COUNT,
        count=get_epoch_committee_count(state, epoch),
    )
```

### `get_attesting_indices`

```python
def get_attesting_indices(state: BeaconState,
                          attestation_data: AttestationData,
                          bitfield: bytes) -> List[ValidatorIndex]:
    """
    Return the sorted attesting indices corresponding to ``attestation_data`` and ``bitfield``.
    """
    committee = get_crosslink_committee(state, attestation_data.target_epoch, attestation_data.crosslink.shard)
    assert verify_bitfield(bitfield, len(committee))
    return sorted([index for i, index in enumerate(committee) if get_bitfield_bit(bitfield, i) == 0b1])
```

### `int_to_bytes`

```python
def int_to_bytes(integer: int, length: int) -> bytes:
    return integer.to_bytes(length, 'little')
```

### `bytes_to_int`

```python
def bytes_to_int(data: bytes) -> int:
    return int.from_bytes(data, 'little')
```

### `get_total_balance`

```python
def get_total_balance(state: BeaconState, indices: List[ValidatorIndex]) -> Gwei:
    """
    Return the combined effective balance of the ``indices``. (1 Gwei minimum to avoid divisions by zero.)
    """
    return max(sum([state.validator_registry[index].effective_balance for index in indices]), 1)
```

### `get_domain`

```python
def get_domain(state: BeaconState,
               domain_type: int,
               message_epoch: int=None) -> int:
    """
    Return the signature domain (fork version concatenated with domain type) of a message.
    """
    epoch = get_current_epoch(state) if message_epoch is None else message_epoch
    fork_version = state.fork.previous_version if epoch < state.fork.epoch else state.fork.current_version
    return bls_domain(domain_type, fork_version)
```

### `get_bitfield_bit`

```python
def get_bitfield_bit(bitfield: bytes, i: int) -> int:
    """
    Extract the bit in ``bitfield`` at position ``i``.
    """
    return (bitfield[i // 8] >> (i % 8)) % 2
```

### `verify_bitfield`

```python
def verify_bitfield(bitfield: bytes, committee_size: int) -> bool:
    """
    Verify ``bitfield`` against the ``committee_size``.
    """
    if len(bitfield) != (committee_size + 7) // 8:
        return False

    # Check `bitfield` is padded with zero bits only
    for i in range(committee_size, len(bitfield) * 8):
        if get_bitfield_bit(bitfield, i) == 0b1:
            return False

    return True
```

### `convert_to_indexed`

```python
def convert_to_indexed(state: BeaconState, attestation: Attestation) -> IndexedAttestation:
    """
    Convert ``attestation`` to (almost) indexed-verifiable form.
    """
    attesting_indices = get_attesting_indices(state, attestation.data, attestation.aggregation_bitfield)
    custody_bit_1_indices = get_attesting_indices(state, attestation.data, attestation.custody_bitfield)
    custody_bit_0_indices = [index for index in attesting_indices if index not in custody_bit_1_indices]

    return IndexedAttestation(
        custody_bit_0_indices=custody_bit_0_indices,
        custody_bit_1_indices=custody_bit_1_indices,
        data=attestation.data,
        signature=attestation.signature,
    )
```

### `validate_indexed_attestation`

```python
def validate_indexed_attestation(state: BeaconState, indexed_attestation: IndexedAttestation) -> None:
    """
    Verify validity of ``indexed_attestation``.
    """
    bit_0_indices = indexed_attestation.custody_bit_0_indices
    bit_1_indices = indexed_attestation.custody_bit_1_indices

    # Verify no index has custody bit equal to 1 [to be removed in phase 1]
    assert len(bit_1_indices) == 0
    # Verify max number of indices
    assert len(bit_0_indices) + len(bit_1_indices) <= MAX_INDICES_PER_ATTESTATION
    # Verify index sets are disjoint
    assert len(set(bit_0_indices).intersection(bit_1_indices)) == 0
    # Verify indices are sorted
    assert bit_0_indices == sorted(bit_0_indices) and bit_1_indices == sorted(bit_1_indices)
    # Verify aggregate signature
    assert bls_verify_multiple(
        pubkeys=[
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in bit_0_indices]),
            bls_aggregate_pubkeys([state.validator_registry[i].pubkey for i in bit_1_indices]),
        ],
        message_hashes=[
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b0)),
            hash_tree_root(AttestationDataAndCustodyBit(data=indexed_attestation.data, custody_bit=0b1)),
        ],
        signature=indexed_attestation.signature,
        domain=get_domain(state, DOMAIN_ATTESTATION, indexed_attestation.data.target_epoch),
    )
```

### `is_slashable_attestation_data`

```python
def is_slashable_attestation_data(data_1: AttestationData, data_2: AttestationData) -> bool:
    """
    Check if ``data_1`` and ``data_2`` are slashable according to Casper FFG rules.
    """
    return (
        # Double vote
        (data_1 != data_2 and data_1.target_epoch == data_2.target_epoch) or
        # Surround vote
        (data_1.source_epoch < data_2.source_epoch and data_2.target_epoch < data_1.target_epoch)
    )
```

### `integer_squareroot`

```python
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_delayed_activation_exit_epoch`

```python
def get_delayed_activation_exit_epoch(epoch: Epoch) -> Epoch:
    """
    Return the epoch at which an activation or exit triggered in ``epoch`` takes effect.
    """
    return epoch + 1 + ACTIVATION_EXIT_DELAY
```

### `get_churn_limit`

```python
def get_churn_limit(state: BeaconState) -> int:
    """
    Return the churn limit based on the active validator count.
    """
    return max(
        MIN_PER_EPOCH_CHURN_LIMIT,
        len(get_active_validator_indices(state, get_current_epoch(state))) // CHURN_LIMIT_QUOTIENT
    )
```

### `bls_verify`

`bls_verify` is a function for verifying a BLS signature, defined in the [BLS Signature spec](../bls_signature.md#bls_verify).

### `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_signature.md#bls_verify_multiple).

### `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](../bls_signature.md#bls_aggregate_pubkeys).

### Routines for updating validator status

*Note*: All functions in this section mutate `state`.

#### `initiate_validator_exit`

```python
def initiate_validator_exit(state: BeaconState, index: ValidatorIndex) -> None:
    """
    Initiate the exit of the validator of the given ``index``.
    """
    # Return if validator already initiated exit
    validator = state.validator_registry[index]
    if validator.exit_epoch != FAR_FUTURE_EPOCH:
        return

    # Compute exit queue epoch
    exit_epochs = [v.exit_epoch for v in state.validator_registry if v.exit_epoch != FAR_FUTURE_EPOCH]
    exit_queue_epoch = max(exit_epochs + [get_delayed_activation_exit_epoch(get_current_epoch(state))])
    exit_queue_churn = len([v for v in state.validator_registry if v.exit_epoch == exit_queue_epoch])
    if exit_queue_churn >= get_churn_limit(state):
        exit_queue_epoch += 1

    # Set validator exit epoch and withdrawable epoch
    validator.exit_epoch = exit_queue_epoch
    validator.withdrawable_epoch = validator.exit_epoch + MIN_VALIDATOR_WITHDRAWABILITY_DELAY
```

#### `slash_validator`

```python
def slash_validator(state: BeaconState,
                    slashed_index: ValidatorIndex,
                    whistleblower_index: ValidatorIndex=None) -> None:
    """
    Slash the validator with index ``slashed_index``.
    """
    current_epoch = get_current_epoch(state)
    initiate_validator_exit(state, slashed_index)
    state.validator_registry[slashed_index].slashed = True
    state.validator_registry[slashed_index].withdrawable_epoch = current_epoch + LATEST_SLASHED_EXIT_LENGTH
    slashed_balance = state.validator_registry[slashed_index].effective_balance
    state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH] += slashed_balance

    proposer_index = get_beacon_proposer_index(state)
    if whistleblower_index is None:
        whistleblower_index = proposer_index
    whistleblowing_reward = slashed_balance // WHISTLEBLOWING_REWARD_QUOTIENT
    proposer_reward = whistleblowing_reward // PROPOSER_REWARD_QUOTIENT
    increase_balance(state, proposer_index, proposer_reward)
    increase_balance(state, whistleblower_index, whistleblowing_reward - proposer_reward)
    decrease_balance(state, slashed_index, whistleblowing_reward)
```

## Genesis

### `Eth2Genesis`

When enough deposits of size `MAX_EFFECTIVE_BALANCE` have been made to the deposit contract an `Eth2Genesis` log is emitted triggering the genesis of the beacon chain. Let:

* `eth2genesis` be the object corresponding to `Eth2Genesis`
* `genesis_eth1_data` be object of type `Eth1Data` where
    * `genesis_eth1_data.deposit_root = eth2genesis.deposit_root`
    * `genesis_eth1_data.deposit_count = eth2genesis.deposit_count`
    * `genesis_eth1_data.block_hash` is the hash of the Ethereum 1.0 block that emitted the `Eth2Genesis` log
* `genesis_deposits` be the object of type `List[Deposit]` with deposits ordered chronologically up to and including the deposit that triggered the `Eth2Genesis` log

### Genesis state

Let `genesis_state = get_genesis_beacon_state(genesis_deposits, eth2genesis.genesis_time, genesis_eth1_data)`.

```python
def get_genesis_beacon_state(deposits: List[Deposit], genesis_time: int, genesis_eth1_data: Eth1Data) -> BeaconState:
    state = BeaconState(
        genesis_time=genesis_time,
        latest_eth1_data=genesis_eth1_data,
        latest_block_header=BeaconBlockHeader(body_root=hash_tree_root(BeaconBlockBody())),
    )

    # Process genesis deposits
    for deposit in deposits:
        process_deposit(state, deposit)

    # Process genesis activations
    for validator in state.validator_registry:
        if validator.effective_balance >= MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = GENESIS_EPOCH
            validator.activation_epoch = GENESIS_EPOCH

    # Populate latest_active_index_roots
    genesis_active_index_root = hash_tree_root(get_active_validator_indices(state, GENESIS_EPOCH))
    for index in range(LATEST_ACTIVE_INDEX_ROOTS_LENGTH):
        state.latest_active_index_roots[index] = genesis_active_index_root

    return state
```

### Genesis block

Let `genesis_block = BeaconBlock(state_root=hash_tree_root(genesis_state))`.

## Beacon chain state transition function

The post-state corresponding to a pre-state `state` and a block `block` is defined as `state_transition(state, block)`. State transitions that trigger an unhandled excpetion (e.g. a failed `assert` or an out-of-range list access) are considered invalid.

```python
def state_transition(state: BeaconState, block: BeaconBlock, validate_state_root: bool=False) -> BeaconState:
    # Process slots (including those with no blocks) since block
    process_slots(state, block.slot)
    # Process block
    process_block(state, block)
    # Validate state root (`validate_state_root == True` in production)
    if validate_state_root:
        assert block.state_root == hash_tree_root(state)
    # Return post-state
    return state
```

```python
def process_slots(state: BeaconState, slot: Slot) -> None:
    assert state.slot < slot
    while state.slot < slot:
        process_slot(state)
        # Process epoch on the first slot of the next epoch
        if (state.slot + 1) % SLOTS_PER_EPOCH == 0:
            process_epoch(state)
        state.slot += 1
```

```python
def process_slot(state: BeaconState) -> None:
    # Cache state root
    previous_state_root = hash_tree_root(state)
    state.latest_state_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_state_root

    # Cache latest block header state root
    if state.latest_block_header.state_root == ZERO_HASH:
        state.latest_block_header.state_root = previous_state_root

    # Cache block root
    previous_block_root = signing_root(state.latest_block_header)
    state.latest_block_roots[state.slot % SLOTS_PER_HISTORICAL_ROOT] = previous_block_root
```

### Epoch processing

```python
def process_epoch(state: BeaconState) -> None:
    process_justification_and_finalization(state)
    process_crosslinks(state)
    process_rewards_and_penalties(state)
    process_registry_updates(state)
    process_slashings(state)
    process_final_updates(state)
```

#### Helper functions

```python
def get_total_active_balance(state: BeaconState) -> Gwei:
    return get_total_balance(state, get_active_validator_indices(state, get_current_epoch(state)))
```

```python
def get_matching_source_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    assert epoch in (get_current_epoch(state), get_previous_epoch(state))
    return state.current_epoch_attestations if epoch == get_current_epoch(state) else state.previous_epoch_attestations
```

```python
def get_matching_target_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    return [
        a for a in get_matching_source_attestations(state, epoch)
        if a.data.target_root == get_block_root(state, epoch)
    ]
```

```python
def get_matching_head_attestations(state: BeaconState, epoch: Epoch) -> List[PendingAttestation]:
    return [
        a for a in get_matching_source_attestations(state, epoch)
        if a.data.beacon_block_root == get_block_root_at_slot(state, get_attestation_data_slot(state, a.data))
    ]
```

```python
def get_unslashed_attesting_indices(state: BeaconState,
                                    attestations: List[PendingAttestation]) -> List[ValidatorIndex]:
    output = set()
    for a in attestations:
        output = output.union(get_attesting_indices(state, a.data, a.aggregation_bitfield))
    return sorted(filter(lambda index: not state.validator_registry[index].slashed, list(output)))
```

```python
def get_attesting_balance(state: BeaconState, attestations: List[PendingAttestation]) -> Gwei:
    return get_total_balance(state, get_unslashed_attesting_indices(state, attestations))
```

```python
def get_winning_crosslink_and_attesting_indices(state: BeaconState,
                                                epoch: Epoch,
                                                shard: Shard) -> Tuple[Crosslink, List[ValidatorIndex]]:
    attestations = [a for a in get_matching_source_attestations(state, epoch) if a.data.crosslink.shard == shard]
    crosslinks = list(filter(
        lambda c: hash_tree_root(state.current_crosslinks[shard]) in (c.parent_root, hash_tree_root(c)),
        [a.data.crosslink for a in attestations]
    ))
    # Winning crosslink has the crosslink data root with the most balance voting for it (ties broken lexicographically)
    winning_crosslink = max(crosslinks, key=lambda c: (
        get_attesting_balance(state, [a for a in attestations if a.data.crosslink == c]), c.data_root
    ), default=Crosslink())
    winning_attestations = [a for a in attestations if a.data.crosslink == winning_crosslink]
    return winning_crosslink, get_unslashed_attesting_indices(state, winning_attestations)
```

#### Justification and finalization

```python
def process_justification_and_finalization(state: BeaconState) -> None:
    if get_current_epoch(state) <= GENESIS_EPOCH + 1:
        return

    previous_epoch = get_previous_epoch(state)
    current_epoch = get_current_epoch(state)
    old_previous_justified_epoch = state.previous_justified_epoch
    old_current_justified_epoch = state.current_justified_epoch

    # Process justifications
    state.previous_justified_epoch = state.current_justified_epoch
    state.previous_justified_root = state.current_justified_root
    state.justification_bitfield = (state.justification_bitfield << 1) % 2**64
    previous_epoch_matching_target_balance = get_attesting_balance(
        state, get_matching_target_attestations(state, previous_epoch)
    )
    if previous_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
        state.current_justified_epoch = previous_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 1)
    current_epoch_matching_target_balance = get_attesting_balance(
        state, get_matching_target_attestations(state, current_epoch)
    )
    if current_epoch_matching_target_balance * 3 >= get_total_active_balance(state) * 2:
        state.current_justified_epoch = current_epoch
        state.current_justified_root = get_block_root(state, state.current_justified_epoch)
        state.justification_bitfield |= (1 << 0)

    # Process finalizations
    bitfield = state.justification_bitfield
    # The 2nd/3rd/4th most recent epochs are justified, the 2nd using the 4th as source
    if (bitfield >> 1) % 8 == 0b111 and old_previous_justified_epoch + 3 == current_epoch:
        state.finalized_epoch = old_previous_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 2nd/3rd most recent epochs are justified, the 2nd using the 3rd as source
    if (bitfield >> 1) % 4 == 0b11 and old_previous_justified_epoch + 2 == current_epoch:
        state.finalized_epoch = old_previous_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 1st/2nd/3rd most recent epochs are justified, the 1st using the 3rd as source
    if (bitfield >> 0) % 8 == 0b111 and old_current_justified_epoch + 2 == current_epoch:
        state.finalized_epoch = old_current_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
    # The 1st/2nd most recent epochs are justified, the 1st using the 2nd as source
    if (bitfield >> 0) % 4 == 0b11 and old_current_justified_epoch + 1 == current_epoch:
        state.finalized_epoch = old_current_justified_epoch
        state.finalized_root = get_block_root(state, state.finalized_epoch)
```

#### Crosslinks

```python
def process_crosslinks(state: BeaconState) -> None:
    state.previous_crosslinks = [c for c in state.current_crosslinks]
    for epoch in (get_previous_epoch(state), get_current_epoch(state)):
        for offset in range(get_epoch_committee_count(state, epoch)):
            shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
            crosslink_committee = get_crosslink_committee(state, epoch, shard)
            winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
            if 3 * get_total_balance(state, attesting_indices) >= 2 * get_total_balance(state, crosslink_committee):
                state.current_crosslinks[shard] = winning_crosslink
```

#### Rewards and penalties

```python
def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
    total_balance = get_total_active_balance(state)
    effective_balance = state.validator_registry[index].effective_balance
    return effective_balance * BASE_REWARD_QUOTIENT // integer_squareroot(total_balance) // BASE_REWARDS_PER_EPOCH
```

```python
def get_attestation_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    previous_epoch = get_previous_epoch(state)
    total_balance = get_total_active_balance(state)
    rewards = [0 for _ in range(len(state.validator_registry))]
    penalties = [0 for _ in range(len(state.validator_registry))]
    eligible_validator_indices = [
        index for index, v in enumerate(state.validator_registry)
        if is_active_validator(v, previous_epoch) or (v.slashed and previous_epoch + 1 < v.withdrawable_epoch)
    ]

    # Micro-incentives for matching FFG source, FFG target, and head
    matching_source_attestations = get_matching_source_attestations(state, previous_epoch)
    matching_target_attestations = get_matching_target_attestations(state, previous_epoch)
    matching_head_attestations = get_matching_head_attestations(state, previous_epoch)
    for attestations in (matching_source_attestations, matching_target_attestations, matching_head_attestations):
        unslashed_attesting_indices = get_unslashed_attesting_indices(state, attestations)
        attesting_balance = get_attesting_balance(state, attestations)
        for index in eligible_validator_indices:
            if index in unslashed_attesting_indices:
                rewards[index] += get_base_reward(state, index) * attesting_balance // total_balance
            else:
                penalties[index] += get_base_reward(state, index)

    # Proposer and inclusion delay micro-rewards
    for index in get_unslashed_attesting_indices(state, matching_source_attestations):
        attestation = min([
            a for a in matching_source_attestations
            if index in get_attesting_indices(state, a.data, a.aggregation_bitfield)
        ], key=lambda a: a.inclusion_delay)
        rewards[attestation.proposer_index] += get_base_reward(state, index) // PROPOSER_REWARD_QUOTIENT
        rewards[index] += get_base_reward(state, index) * MIN_ATTESTATION_INCLUSION_DELAY // attestation.inclusion_delay

    # Inactivity penalty
    finality_delay = previous_epoch - state.finalized_epoch
    if finality_delay > MIN_EPOCHS_TO_INACTIVITY_PENALTY:
        matching_target_attesting_indices = get_unslashed_attesting_indices(state, matching_target_attestations)
        for index in eligible_validator_indices:
            penalties[index] += BASE_REWARDS_PER_EPOCH * get_base_reward(state, index)
            if index not in matching_target_attesting_indices:
                penalties[index] += (
                    state.validator_registry[index].effective_balance * finality_delay // INACTIVITY_PENALTY_QUOTIENT
                )

    return rewards, penalties
```

```python
def get_crosslink_deltas(state: BeaconState) -> Tuple[List[Gwei], List[Gwei]]:
    rewards = [0 for index in range(len(state.validator_registry))]
    penalties = [0 for index in range(len(state.validator_registry))]
    epoch = get_previous_epoch(state)
    for offset in range(get_epoch_committee_count(state, epoch)):
        shard = (get_epoch_start_shard(state, epoch) + offset) % SHARD_COUNT
        crosslink_committee = get_crosslink_committee(state, epoch, shard)
        winning_crosslink, attesting_indices = get_winning_crosslink_and_attesting_indices(state, epoch, shard)
        attesting_balance = get_total_balance(state, attesting_indices)
        committee_balance = get_total_balance(state, crosslink_committee)
        for index in crosslink_committee:
            base_reward = get_base_reward(state, index)
            if index in attesting_indices:
                rewards[index] += base_reward * attesting_balance // committee_balance
            else:
                penalties[index] += base_reward
    return rewards, penalties
```

```python
def process_rewards_and_penalties(state: BeaconState) -> None:
    if get_current_epoch(state) == GENESIS_EPOCH:
        return

    rewards1, penalties1 = get_attestation_deltas(state)
    rewards2, penalties2 = get_crosslink_deltas(state)
    for i in range(len(state.validator_registry)):
        increase_balance(state, i, rewards1[i] + rewards2[i])
        decrease_balance(state, i, penalties1[i] + penalties2[i])
```

#### Registry updates

```python
def process_registry_updates(state: BeaconState) -> None:
    # Process activation eligibility and ejections
    for index, validator in enumerate(state.validator_registry):
        if (
            validator.activation_eligibility_epoch == FAR_FUTURE_EPOCH and
            validator.effective_balance >= MAX_EFFECTIVE_BALANCE
        ):
            validator.activation_eligibility_epoch = get_current_epoch(state)

        if is_active_validator(validator, get_current_epoch(state)) and validator.effective_balance <= EJECTION_BALANCE:
            initiate_validator_exit(state, index)

    # Queue validators eligible for activation and not dequeued for activation prior to finalized epoch
    activation_queue = sorted([
        index for index, validator in enumerate(state.validator_registry) if
        validator.activation_eligibility_epoch != FAR_FUTURE_EPOCH and
        validator.activation_epoch >= get_delayed_activation_exit_epoch(state.finalized_epoch)
    ], key=lambda index: state.validator_registry[index].activation_eligibility_epoch)
    # Dequeued validators for activation up to churn limit (without resetting activation epoch)
    for index in activation_queue[:get_churn_limit(state)]:
        validator = state.validator_registry[index]
        if validator.activation_epoch == FAR_FUTURE_EPOCH:
            validator.activation_epoch = get_delayed_activation_exit_epoch(get_current_epoch(state))
```

#### Slashings

```python
def process_slashings(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    total_balance = get_total_active_balance(state)

    # Compute slashed balances in the current epoch
    total_at_start = state.latest_slashed_balances[(current_epoch + 1) % LATEST_SLASHED_EXIT_LENGTH]
    total_at_end = state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
    total_penalties = total_at_end - total_at_start

    for index, validator in enumerate(state.validator_registry):
        if validator.slashed and current_epoch == validator.withdrawable_epoch - LATEST_SLASHED_EXIT_LENGTH // 2:
            penalty = max(
                validator.effective_balance * min(total_penalties * 3, total_balance) // total_balance,
                validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT
            )
            decrease_balance(state, index, penalty)
```

#### Final updates

```python
def process_final_updates(state: BeaconState) -> None:
    current_epoch = get_current_epoch(state)
    next_epoch = current_epoch + 1
    # Reset eth1 data votes
    if (state.slot + 1) % SLOTS_PER_ETH1_VOTING_PERIOD == 0:
        state.eth1_data_votes = []
    # Update effective balances with hysteresis
    for index, validator in enumerate(state.validator_registry):
        balance = state.balances[index]
        HALF_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // 2
        if balance < validator.effective_balance or validator.effective_balance + 3 * HALF_INCREMENT < balance:
            validator.effective_balance = min(balance - balance % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
    # Update start shard
    state.latest_start_shard = (state.latest_start_shard + get_shard_delta(state, current_epoch)) % SHARD_COUNT
    # Set active index root
    index_root_position = (next_epoch + ACTIVATION_EXIT_DELAY) % LATEST_ACTIVE_INDEX_ROOTS_LENGTH
    state.latest_active_index_roots[index_root_position] = hash_tree_root(
        get_active_validator_indices(state, next_epoch + ACTIVATION_EXIT_DELAY)
    )
    # Set total slashed balances
    state.latest_slashed_balances[next_epoch % LATEST_SLASHED_EXIT_LENGTH] = (
        state.latest_slashed_balances[current_epoch % LATEST_SLASHED_EXIT_LENGTH]
    )
    # Set randao mix
    state.latest_randao_mixes[next_epoch % LATEST_RANDAO_MIXES_LENGTH] = get_randao_mix(state, current_epoch)
    # Set historical root accumulator
    if next_epoch % (SLOTS_PER_HISTORICAL_ROOT // SLOTS_PER_EPOCH) == 0:
        historical_batch = HistoricalBatch(
            block_roots=state.latest_block_roots,
            state_roots=state.latest_state_roots,
        )
        state.historical_roots.append(hash_tree_root(historical_batch))
    # Rotate current/previous epoch attestations
    state.previous_epoch_attestations = state.current_epoch_attestations
    state.current_epoch_attestations = []
```

### Block processing

```python
def process_block(state: BeaconState, block: BeaconBlock) -> None:
    process_block_header(state, block)
    process_randao(state, block.body)
    process_eth1_data(state, block.body)
    process_operations(state, block.body)
```

#### Block header

```python
def process_block_header(state: BeaconState, block: BeaconBlock) -> None:
    # Verify that the slots match
    assert block.slot == state.slot
    # Verify that the parent matches
    assert block.parent_root == signing_root(state.latest_block_header)
    # Save current block as the new latest block
    state.latest_block_header = BeaconBlockHeader(
        slot=block.slot,
        parent_root=block.parent_root,
        body_root=hash_tree_root(block.body),
    )
    # Verify proposer is not slashed
    proposer = state.validator_registry[get_beacon_proposer_index(state)]
    assert not proposer.slashed
    # Verify proposer signature
    assert bls_verify(proposer.pubkey, signing_root(block), block.signature, get_domain(state, DOMAIN_BEACON_PROPOSER))
```

#### RANDAO

```python
def process_randao(state: BeaconState, body: BeaconBlockBody) -> None:
    proposer = state.validator_registry[get_beacon_proposer_index(state)]
    # Verify that the provided randao value is valid
    assert bls_verify(
        proposer.pubkey,
        hash_tree_root(get_current_epoch(state)),
        body.randao_reveal,
        get_domain(state, DOMAIN_RANDAO),
    )
    # Mix it in
    state.latest_randao_mixes[get_current_epoch(state) % LATEST_RANDAO_MIXES_LENGTH] = (
        xor(get_randao_mix(state, get_current_epoch(state)),
            hash(body.randao_reveal))
    )
```

#### Eth1 data

```python
def process_eth1_data(state: BeaconState, body: BeaconBlockBody) -> None:
    state.eth1_data_votes.append(body.eth1_data)
    if state.eth1_data_votes.count(body.eth1_data) * 2 > SLOTS_PER_ETH1_VOTING_PERIOD:
        state.latest_eth1_data = body.eth1_data
```

#### Operations

```python
def process_operations(state: BeaconState, body: BeaconBlockBody) -> None:
    # Verify that outstanding deposits are processed up to the maximum number of deposits
    assert len(body.deposits) == min(MAX_DEPOSITS, state.latest_eth1_data.deposit_count - state.deposit_index)
    # Verify that there are no duplicate transfers
    assert len(body.transfers) == len(set(body.transfers))

    for operations, max_operations, function in (
        (body.proposer_slashings, MAX_PROPOSER_SLASHINGS, process_proposer_slashing),
        (body.attester_slashings, MAX_ATTESTER_SLASHINGS, process_attester_slashing),
        (body.attestations, MAX_ATTESTATIONS, process_attestation),
        (body.deposits, MAX_DEPOSITS, process_deposit),
        (body.voluntary_exits, MAX_VOLUNTARY_EXITS, process_voluntary_exit),
        (body.transfers, MAX_TRANSFERS, process_transfer),
    ):
        assert len(operations) <= max_operations
        for operation in operations:
            function(state, operation)
```

##### Proposer slashings

```python
def process_proposer_slashing(state: BeaconState, proposer_slashing: ProposerSlashing) -> None:
    """
    Process ``ProposerSlashing`` operation.
    """
    proposer = state.validator_registry[proposer_slashing.proposer_index]
    # Verify that the epoch is the same
    assert slot_to_epoch(proposer_slashing.header_1.slot) == slot_to_epoch(proposer_slashing.header_2.slot)
    # But the headers are different
    assert proposer_slashing.header_1 != proposer_slashing.header_2
    # Check proposer is slashable
    assert is_slashable_validator(proposer, get_current_epoch(state))
    # Signatures are valid
    for header in (proposer_slashing.header_1, proposer_slashing.header_2):
        domain = get_domain(state, DOMAIN_BEACON_PROPOSER, slot_to_epoch(header.slot))
        assert bls_verify(proposer.pubkey, signing_root(header), header.signature, domain)

    slash_validator(state, proposer_slashing.proposer_index)
```

##### Attester slashings

```python
def process_attester_slashing(state: BeaconState, attester_slashing: AttesterSlashing) -> None:
    """
    Process ``AttesterSlashing`` operation.
    """
    attestation_1 = attester_slashing.attestation_1
    attestation_2 = attester_slashing.attestation_2
    assert is_slashable_attestation_data(attestation_1.data, attestation_2.data)
    validate_indexed_attestation(state, attestation_1)
    validate_indexed_attestation(state, attestation_2)

    slashed_any = False
    attesting_indices_1 = attestation_1.custody_bit_0_indices + attestation_1.custody_bit_1_indices
    attesting_indices_2 = attestation_2.custody_bit_0_indices + attestation_2.custody_bit_1_indices
    for index in set(attesting_indices_1).intersection(attesting_indices_2):
        if is_slashable_validator(state.validator_registry[index], get_current_epoch(state)):
            slash_validator(state, index)
            slashed_any = True
    assert slashed_any
```

##### Attestations

```python
def process_attestation(state: BeaconState, attestation: Attestation) -> None:
    """
    Process ``Attestation`` operation.
    """
    data = attestation.data
    attestation_slot = get_attestation_data_slot(state, data)
    assert attestation_slot + MIN_ATTESTATION_INCLUSION_DELAY <= state.slot <= attestation_slot + SLOTS_PER_EPOCH

    pending_attestation = PendingAttestation(
        data=data,
        aggregation_bitfield=attestation.aggregation_bitfield,
        inclusion_delay=state.slot - attestation_slot,
        proposer_index=get_beacon_proposer_index(state),
    )

    assert data.target_epoch in (get_previous_epoch(state), get_current_epoch(state))
    if data.target_epoch == get_current_epoch(state):
        ffg_data = (state.current_justified_epoch, state.current_justified_root, get_current_epoch(state))
        parent_crosslink = state.current_crosslinks[data.crosslink.shard]
        state.current_epoch_attestations.append(pending_attestation)
    else:
        ffg_data = (state.previous_justified_epoch, state.previous_justified_root, get_previous_epoch(state))
        parent_crosslink = state.previous_crosslinks[data.crosslink.shard]
        state.previous_epoch_attestations.append(pending_attestation)

    # Check FFG data, crosslink data, and signature
    assert ffg_data == (data.source_epoch, data.source_root, data.target_epoch)
    assert data.crosslink.start_epoch == parent_crosslink.end_epoch
    assert data.crosslink.end_epoch == min(data.target_epoch, parent_crosslink.end_epoch + MAX_EPOCHS_PER_CROSSLINK)
    assert data.crosslink.parent_root == hash_tree_root(parent_crosslink)
    assert data.crosslink.data_root == ZERO_HASH  # [to be removed in phase 1]
    validate_indexed_attestation(state, convert_to_indexed(state, attestation))
```

##### Deposits

```python
def process_deposit(state: BeaconState, deposit: Deposit) -> None:
    """
    Process an Eth1 deposit, registering a validator or increasing its balance.
    """
    # Verify the Merkle branch
    assert verify_merkle_branch(
        leaf=hash_tree_root(deposit.data),
        proof=deposit.proof,
        depth=DEPOSIT_CONTRACT_TREE_DEPTH,
        index=deposit.index,
        root=state.latest_eth1_data.deposit_root,
    )

    # Deposits must be processed in order
    assert deposit.index == state.deposit_index
    state.deposit_index += 1

    pubkey = deposit.data.pubkey
    amount = deposit.data.amount
    validator_pubkeys = [v.pubkey for v in state.validator_registry]
    if pubkey not in validator_pubkeys:
        # Verify the deposit signature (proof of possession)
        # Note: deposits are valid across forks, hence the deposit domain is retrieved directly from `bls_domain`
        if not bls_verify(
            pubkey, signing_root(deposit.data), deposit.data.signature, bls_domain(DOMAIN_DEPOSIT)
        ):
            return

        # Add validator and balance entries
        state.validator_registry.append(Validator(
            pubkey=pubkey,
            withdrawal_credentials=deposit.data.withdrawal_credentials,
            activation_eligibility_epoch=FAR_FUTURE_EPOCH,
            activation_epoch=FAR_FUTURE_EPOCH,
            exit_epoch=FAR_FUTURE_EPOCH,
            withdrawable_epoch=FAR_FUTURE_EPOCH,
            effective_balance=min(amount - amount % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
        ))
        state.balances.append(amount)
    else:
        # Increase balance by deposit amount
        index = validator_pubkeys.index(pubkey)
        increase_balance(state, index, amount)
```

##### Voluntary exits

```python
def process_voluntary_exit(state: BeaconState, exit: VoluntaryExit) -> None:
    """
    Process ``VoluntaryExit`` operation.
    """
    validator = state.validator_registry[exit.validator_index]
    # Verify the validator is active
    assert is_active_validator(validator, get_current_epoch(state))
    # Verify the validator has not yet exited
    assert validator.exit_epoch == FAR_FUTURE_EPOCH
    # Exits must specify an epoch when they become valid; they are not valid before then
    assert get_current_epoch(state) >= exit.epoch
    # Verify the validator has been active long enough
    assert get_current_epoch(state) >= validator.activation_epoch + PERSISTENT_COMMITTEE_PERIOD
    # Verify signature
    domain = get_domain(state, DOMAIN_VOLUNTARY_EXIT, exit.epoch)
    assert bls_verify(validator.pubkey, signing_root(exit), exit.signature, domain)
    # Initiate exit
    initiate_validator_exit(state, exit.validator_index)
```

##### Transfers

```python
def process_transfer(state: BeaconState, transfer: Transfer) -> None:
    """
    Process ``Transfer`` operation.
    """
    # Verify the amount and fee are not individually too big (for anti-overflow purposes)
    assert state.balances[transfer.sender] >= max(transfer.amount, transfer.fee)
    # A transfer is valid in only one slot
    assert state.slot == transfer.slot
    # Sender must be not yet eligible for activation, withdrawn, or transfer balance over MAX_EFFECTIVE_BALANCE
    assert (
        state.validator_registry[transfer.sender].activation_eligibility_epoch == FAR_FUTURE_EPOCH or
        get_current_epoch(state) >= state.validator_registry[transfer.sender].withdrawable_epoch or
        transfer.amount + transfer.fee + MAX_EFFECTIVE_BALANCE <= state.balances[transfer.sender]
    )
    # Verify that the pubkey is valid
    assert (
        state.validator_registry[transfer.sender].withdrawal_credentials ==
        int_to_bytes(BLS_WITHDRAWAL_PREFIX, length=1) + hash(transfer.pubkey)[1:]
    )
    # Verify that the signature is valid
    assert bls_verify(transfer.pubkey, signing_root(transfer), transfer.signature, get_domain(state, DOMAIN_TRANSFER))
    # Process the transfer
    decrease_balance(state, transfer.sender, transfer.amount + transfer.fee)
    increase_balance(state, transfer.recipient, transfer.amount)
    increase_balance(state, get_beacon_proposer_index(state), transfer.fee)
    # Verify balances are not dust
    assert not (0 < state.balances[transfer.sender] < MIN_DEPOSIT_AMOUNT)
    assert not (0 < state.balances[transfer.recipient] < MIN_DEPOSIT_AMOUNT)
```