eth2.0-specs/specs/beacon-chain.md

# Ethereum 2.0 spec—Casper and sharding

###### tags: `spec`, `eth2.0`, `casper`, `sharding`
###### spec version: 2.2 (October 2018)

**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](https://github.com/ethereum/beacon_chain).

### Introduction

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

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

### Terminology

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

### Constants

| Constant | Value | Unit | Approximation |
| --- | --- | :---: | - |
| `SHARD_COUNT` | 2**10 (= 1,024)| shards |
| `DEPOSIT_SIZE` | 2**5 (= 32) | ETH |
| `MIN_COMMITTEE_SIZE` | 2**7 (= 128) | validators |
| `GENESIS_TIME` | **TBD** | seconds |
| `SLOT_DURATION` | 2**4 (= 16) | seconds |
| `CYCLE_LENGTH` | 2**6 (= 64) | slots | ~17 minutes |
| `MIN_DYNASTY_LENGTH` | 2**8 (= 256) | slots | ~1.1 hours |
| `SQRT_E_DROP_TIME` | 2**16 (= 65,536) | slots | ~12 days |
| `WITHDRAWAL_PERIOD` | 2**19 (= 524,288) | slots | ~97 days |
| `BASE_REWARD_QUOTIENT` | 2**15 (= 32,768) | — |
| `MAX_VALIDATOR_CHURN_QUOTIENT` | 2**5 (= 32) | — | 

**Notes**

* The `SQRT_E_DROP_TIME` constant is the amount of time it takes for the quadratic leak to cut deposits of non-participating validators by ~39.4%. 
* The `BASE_REWARD_QUOTIENT` constant is the per-slot interest rate assuming all validators are participating, assuming total deposits of 1 ETH. It corresponds to ~3.88% annual interest assuming 10 million participating ETH.
* At most `1/MAX_VALIDATOR_CHURN_QUOTIENT` of the validators can change during each dynasty.

**Status codes**

| Status code | Value |
| - | :-: |
| `PENDING_LOG_IN` | `0` |
| `LOGGED_IN` | `1` |
| `PENDING_EXIT` | `2` |
| `PENDING_WITHDRAW` | `3` |
| `WITHDRAWN` | `4` |
| `PENALIZED` | `128` |
| `ENTRY` | `1` |
| `EXIT` | `2` |

### PoW chain registration contract

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

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

## Data structures
### Beacon chain blocks

A `BeaconBlock` has the following fields:

```python
{
    # Slot number
    'slot': 'int64',
    # Proposer RANDAO reveal
    'randao_reveal': 'hash32',
    # Recent PoW chain reference (block hash)
    'pow_chain_reference': 'hash32',
    # Skip list of ancestor block hashes (i'th item is 2**i'th ancestor (or zero) for i = 0, ..., 31)
    'ancestor_hashes': ['hash32'],
    # Active state root
    'active_state_root': 'hash32',
    # Crystallized state root
    'crystallized_state_root': 'hash32',
    # Attestations
    'attestations': [AttestationRecord],
    # Specials (e.g. logouts, penalties)
    'specials': [SpecialRecord]
}
```

An `AttestationRecord` has the following fields:

```python
{
    # Slot number
    'slot': 'int64',
    # Shard number
    'shard': 'int16',
    # Block hashes not part of the current chain, oldest to newest
    'oblique_parent_hashes': ['hash32'],
    # Shard block hash being attested to
    'shard_block_hash': 'hash32',
    # Attester participation bitfield (1 bit per attester)
    'attester_bitfield': 'bytes',
    # Slot of last justified block
    'justified_slot': 'int64',
    # Hash of last justified block
    'justified_block_hash': 'hash32',
    # BLS aggregate signature
    'aggregate_sig': ['int256']
}
```

An `AttestationSignedData` has the following fields:

```python
{
    # Chain version
    'version': 'int64',
    # Slot number
    'slot': 'int64',
    # Shard number
    'shard': 'int16',
    # 31 parent hashes
    'parent_hashes': ['hash32'],
    # Shard block hash
    'shard_block_hash': 'hash32',
    # Slot of last justified block
    'justified_slot': 'int64'
}
```

A `SpecialRecord` has the following fields:

```python
{
    # Kind
    'kind': 'int8',
    # Data
    'data': ['bytes']
}
```

### Beacon chain state

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

The `ActiveState` has the following fields:

```python
{
    # Most recent 2 * CYCLE_LENGTH block hashes, oldest to newest
    'recent_block_hashes': ['hash32'],
    # Attestations not yet processed
    'pending_attestations': [AttestationRecord],
    # Specials not yet been processed
    'pending_specials': [SpecialRecord]
    # Most recent 2 * CYCLE_LENGTH block hashes, older to newer
    'recent_block_hashes': ['hash32'],
    # RANDAO state
    'randao_mix': 'hash32'
}
```

The `CrystallizedState` has the following fields:

```python
{
    # Dynasty number
    'dynasty': 'int64',
    # Dynasty seed (from randomness beacon)
    'dynasty_seed': 'hash32',
    # Dynasty start
    'dynasty_start_slot': 'int64',
    # List of validators
    'validators': [ValidatorRecord],
    # Most recent crosslink for each shard
    'crosslinks': [CrosslinkRecord],
    # Last crystallized state recalculation
    'last_state_recalculation_slot': 'int64',
    # Last finalized slot
    'last_finalized_slot': 'int64',
    # Last justified slot
    'last_justified_slot': 'int64',
    # Number of consecutive justified slots
    'justified_streak': 'int64',
    # Committee members and their assigned shard, per slot
    'shard_and_committee_for_slots': [[ShardAndCommittee]],
    # Total deposits penalized in the given withdrawal period
    'deposits_penalized_in_period': ['int32'],
    # Hash chain of validator set changes (for light clients to easily track deltas)
    'validator_set_delta_hash_chain': 'hash32'
    # Parameters relevant to hard forks / versioning.
    # Should be updated only by hard forks.
    'pre_fork_version': 'int32',
    'post_fork_version': 'int32',
    'fork_slot_number': 'int64',
}
```

A `ValidatorRecord` has the following fields:

```python
{
    # BLS public key
    'pubkey': 'int256',
    # Withdrawal shard number
    'withdrawal_shard': 'int16',
    # Withdrawal address
    'withdrawal_address': 'address',
    # RANDAO commitment
    'randao_commitment': 'hash32',
    # Balance
    'balance': 'int64',
    # Status code
    'status': 'int8',
    # Slot when validator exited (or 0)
    'exit_slot': 'int64'
}
```

A `CrosslinkRecord` has the following fields:

```python
{
    # Dynasty number
    'dynasty': 'int64',
    # Slot number
    'slot': 'int64',
    # Beacon chain block hash
    'shard_block_hash': 'hash32'
}
```

A `ShardAndCommittee` object has the following fields:

```python
{
    # Shard number
    'shard': 'int16',
    # Validator indices
    'committee': ['int24']
}
```

## Beacon chain processing

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

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

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

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

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

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

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

### Beacon chain fork choice rule

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

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

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

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

![](https://vitalik.ca/files/RPJ.png)

## Beacon chain state transition function

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

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


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

### Helper functions

We start off by defining some helper algorithms. First, the function that selects the active validators:

```python
def get_active_validator_indices(validators):
    return [i for i, v in enumerate(validators) if v.status == LOGGED_IN]
```

Now, a function that shuffles this list:

```python
def shuffle(lst, seed):
    # entropy is consumed in 3 byte chunks
    # rand_max is defined to remove the modulo bias from this entropy source
    rand_max = 2**24
    assert len(lst) <= rand_max

    o = [x for x in lst]
    source = seed
    i = 0
    while i < len(lst):
        source = hash(source)
        for pos in range(0, 30, 3):
            m = int.from_bytes(source[pos:pos+3], 'big')
            remaining = len(lst) - i
            if remaining == 0:
                break
            rand_max = rand_max - rand_max % remaining
            if m < rand_max:
                replacement_pos = (m % remaining) + i
                o[i], o[replacement_pos] = o[replacement_pos], o[i]
                i += 1
    return o
```

Here's a function that splits a list into `N` pieces:

```python
def split(lst, N):
    return [lst[len(lst)*i//N: len(lst)*(i+1)//N] for i in range(N)]
```

Now, our combined helper method:

```python
def get_new_shuffling(seed, validators, crosslinking_start_shard):
    active_validators = get_active_validator_indices(validators)
    if len(active_validators) >= CYCLE_LENGTH * MIN_COMMITTEE_SIZE:
        committees_per_slot = min(len(active_validators) // CYCLE_LENGTH // (MIN_COMMITTEE_SIZE * 2) + 1, SHARD_COUNT // CYCLE_LENGTH)
        slots_per_committee = 1
    else:
        committees_per_slot = 1
        slots_per_committee = 1
        while len(active_validators) * slots_per_committee < CYCLE_LENGTH * MIN_COMMITTEE_SIZE \
                and slots_per_committee < CYCLE_LENGTH:
            slots_per_committee *= 2
    o = []
    for i, slot_indices in enumerate(split(shuffle(active_validators, seed), CYCLE_LENGTH)):
        shard_indices = split(slot_indices, committees_per_slot)
        shard_start = crosslinking_start_shard + \
            i * committees_per_slot // slots_per_committee
        o.append([ShardAndCommittee(
            shard = (shard_start + j) % SHARD_COUNT,
            committee = indices
        ) for j, indices in enumerate(shard_indices)])
    return o
```

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

![](http://vitalik.ca/files/ShuffleAndAssign.png?1)

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

```python
def get_shards_and_committees_for_slot(crystallized_state, slot):
    earliest_slot_in_array = crystallized_state.last_state_recalculation_slot - CYCLE_LENGTH
    assert earliest_slot_in_array <= slot < earliest_slot_in_array + CYCLE_LENGTH * 2
    return crystallized_state.shard_and_committee_for_slots[slot - earliest_slot_in_array]

def get_block_hash(active_state, curblock, slot):
    earliest_slot_in_array = curblock.slot - CYCLE_LENGTH * 2
    assert earliest_slot_in_array <= slot < earliest_slot_in_array + CYCLE_LENGTH * 2
    return active_state.recent_block_hashes[slot - earliest_slot_in_array]
```

`get_block_hash(_, _, s)` should always return the block in the chain at slot `s`, and `get_shards_and_committees_for_slot(_, s)` should not change unless the dynasty changes.

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

```python
def add_validator_set_change_record(crystallized_state, index, pubkey, flag):
    crystallized_state.validator_set_delta_hash_chain = \
        hash(crystallized_state.validator_set_delta_hash_chain +
             bytes1(flag) + bytes3(index) + bytes32(pubkey))
```

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

```python
def int_sqrt(n):
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x
```


### On startup

Run the following code:

```python
def on_startup(initial_validator_entries):
    # Induct validators
    validators = []
    for pubkey, proof_of_possession, withdrawal_shard, withdrawal_address, \
            randao_commitment in initial_validator_entries:
        add_validator(validators, pubkey, proof_of_possession,
                      withdrawal_shard, withdrawal_address, randao_commitment)
    # Setup crystallized state
    cs = CrystallizedState()
    x = get_new_shuffling(bytes([0] * 32), validators, 0)
    cs.shard_and_committee_for_slots = x + x
    cs.dynasty = 1
    cs.crosslinks = [CrosslinkRecord(dynasty=0, slot=0, hash=bytes([0] * 32))
                            for i in range(SHARD_COUNT)]
    # Setup active state
    as = ActiveState()
    as.recent_block_hashes = [bytes([0] * 32) for _ in range(CYCLE_LENGTH * 2)]
```

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

### Routine for adding a validator

This routine should be run for every validator that is inducted as part of a log created on the PoW chain [TODO: explain where to check for these logs]. These logs should be processed in the order in which they are emitted by the PoW chain. Define `min_empty_validator(validators)` as a function that returns the lowest validator index `i` such that `validators[i].status == WITHDRAWN`, otherwise `None`.

```python
def add_validator(validators, pubkey, proof_of_possession, withdrawal_shard,
                  withdrawal_address, randao_commitment):
    # if following assert fails, validator induction failed
    # move on to next validator registration log
    assert BLSVerify(pub=pubkey,
                     msg=hash(pubkey),
                     sig=proof_of_possession)
    rec = ValidatorRecord(
        pubkey=pubkey,
        withdrawal_shard=withdrawal_shard,
        withdrawal_address=withdrawal_address,
        randao_commitment=randao_commitment,
        balance=DEPOSIT_SIZE * 10**9, # in Gwei
        status=PENDING_LOG_IN,
        exit_slot=0
    )
    index = min_empty_validator(validators)
    if index is None:
        validators.append(rec)
        return len(validators) - 1
    else:
        validators[index] = rec
        return index
```

### Per-block processing

This procedure should be carried out every block.

First, set `recent_block_hashes` to the output of the following, where `parent_hash` is the hash of the immediate previous block (ie. must be equal to `ancestor_hashes[0]`):

```python
def get_new_recent_block_hashes(old_block_hashes, parent_slot,
                                current_slot, parent_hash):
    d = current_slot - parent_slot
    return old_block_hashes[d:] + [parent_hash] * min(d, len(old_block_hashes))
```

The output of `get_block_hash` should not change, except that it will no longer throw for `current_slot - 1`, and will now throw for `current_slot - CYCLE_LENGTH * 2 - 1`. Also, check that the block's `ancestor_hashes` array was correctly updated, using the following algorithm:

```python
def update_ancestor_hashes(parent_ancestor_hashes, parent_slot_number, parent_hash):
    new_ancestor_hashes = copy.copy(parent_ancestor_hashes)
    for i in range(32):
        if parent_slot_number % 2**i == 0:
            new_ancestor_hashes[i] = parent_hash
    return new_ancestor_hashes
```

A block can have 0 or more `AttestationRecord` objects

For each one of these attestations:

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

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

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

Additionally, verify that `hash(block.randao_reveal) == crystallized_state.validators[proposer_index].randao_commitment`, and set `active_state.randao_mix = xor(active_state.randao_mix, block.randao_reveal)` and `crystallized_state.validators[proposer_index].randao_commitment = block.randao_reveal`.

### State recalculations (every `CYCLE_LENGTH` slots)

Repeat while `slot - last_state_recalculation_slot >= CYCLE_LENGTH`:

#### Adjust justified slots and crosslink status

For all slots `s` in `last_state_recalculation_slot - CYCLE_LENGTH ... last_state_recalculation_slot - 1`:

* Determine the total set of validators that attested to that block at least once
* Determine the total balance of these validators. If this value times three equals or exceeds the total balance of all active validators times two, set `last_justified_slot = max(last_justified_slot, s)` and `justified_streak += 1`. Otherwise, set `justified_streak = 0`
* If `justified_streak >= CYCLE_LENGTH + 1`, set `last_finalized_slot = max(last_finalized_slot, s - CYCLE_LENGTH - 1)`

For all (`shard`, `shard_block_hash`) tuples, compute the total deposit size of validators that attested to that block hash for that shard. If this value times three equals or exceeds the total balance of all validators in the committee times two, and the current dynasty exceeds `crosslinks[shard].dynasty`, set `crosslinks[shard] = CrosslinkRecord(dynasty=dynasty, slot=block.last_state_recalculation_slot + CYCLE_LENGTH, hash=shard_block_hash)`.

#### Balance recalculations related to FFG rewards

Let `time_since_finality = block.slot - last_finalized_slot`, and let `B` be the balance of any given validator whose balance we are adjusting, not including any balance changes from this round of state recalculation. Let:

* `total_deposits = sum([v.balance for i, v in enumerate(validators) if i in get_active_validator_indices(validators, dynasty)])` and `total_deposits_in_gwei = total_deposits // 10**9`
* `reward_quotient = BASE_REWARD_QUOTIENT * int_sqrt(total_deposits_in_gwei)` (`1/reward_quotient` is the per-slot max interest rate)
* `quadratic_penalty_quotient = SQRT_E_DROP_TIME**2` (after `D` slots about `D*D/2/quadratic_penalty_quotient` is the portion lost by offline validators)

For each slot `S` in the range `last_state_recalculation_slot - CYCLE_LENGTH ... last_state_recalculation_slot - 1`:

* Let `total_participated_deposits` be the total balance of validators that voted for the correct hash in slot `S` (ie. the hash that actually is the hash of the block at that slot in the current chain); note that in the normal case, every validator will be in one of the `CYCLE_LENGTH` slots following the slot and so can vote for a hash in slot `S`. If `time_since_finality <= 3 * CYCLE_LENGTH`, then adjust participating and non-participating validators' balances as follows:
    * Participating validators gain `B // reward_quotient * (2 * total_participated_deposits - total_deposits) // total_deposits` (note: this may be negative)
    * Nonparticipating validators lose `B // reward_quotient`
* Otherwise, adjust as follows:
    * Participating validators' balances are unchanged
    * Nonparticipating validators lose `B // reward_quotient + B * time_since_finality // quadratic_penalty_quotient`

Validators with `status == PENALIZED` also lose `B // reward_quotient + B * time_since_finality // quadratic_penalty_quotient`.

#### Balance recalculations related to crosslink rewards

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

* Let `total_v_deposits` be the total balance of `V`
* Let `total_participated_v_deposits` be the total balance of the subset of `V` that participated (note that `total_participated_v_deposits <= total_v_deposits`)
* Let `time_since_last_confirmation` be `block.slot - crosslinks[S].slot`
* Adjust balances as follows:
    * If `crosslinks[S].dynasty == dynasty`, no reward adjustments
    * Otherwise, participating validators' balances are increased by `B // reward_quotient * (2 * total_participated_v_deposits - total_v_deposits) // total_v_deposits`, and the balances of non-participating validators are decreased by `B // reward_quotient + B * time_since_last_confirmation // quadratic_penalty_quotient`

Let `committees` be the set of committees processed and `time_since_last_confirmation(c)` be the value of `time_since_last_confirmation` in that committee. Validators with `status == PENALIZED` lose `B // reward_quotient + B * sum([time_since_last_confirmation(c) for c in committees]) // len(committees) // quadratic_penalty_quotient`.

#### Process penalties, logouts and other special objects

For each `SpecialRecord` `obj` in `active_state.pending_specials`:

* **[covers logouts]**: If `obj.kind == 0`, interpret `data[0]` as a validator index as an `int32` and `data[1]` as a signature. If `BLSVerify(pubkey=validators[data[0]].pubkey, msg=hash("bye bye"), sig=data[1])`, and `validators[i].status == LOGGED_IN`, set `validators[i].status = PENDING_EXIT` and `validators[i].exit_slot = current_slot`
* **[covers `NO_DBL_VOTE`, `NO_SURROUND`, `NO_DBL_PROPOSE` slashing conditions]:** If `obj.kind == 1`, interpret `data[0]` as a list of concatenated `int32` values where each value represents an index into `validators`, `data[1]` as the data being signed and `data[2]` as an aggregate signature. Interpret `data[3:6]` similarly. Verify that both signatures are valid, that the two signatures are signing distinct data, and that they are either signing the same slot number, or that one surrounds the other (ie. `source1 < source2 < target2 < target1`). Let `inds` be the list of indices in both signatures; verify that its length is at least 1. For each validator index `v` in `inds`, set their end dynasty to equal the current dynasty plus 1, and if its `status` does not equal `PENALIZED`, then:

1. Set its `exit_slot` to equal the current `slot`
2. Set its `status` to `PENALIZED`
3. Set `crystallized_state.deposits_penalized_in_period[slot // WITHDRAWAL_PERIOD] += validators[v].balance`, extending the array if needed
4. Run `add_validator_set_change_record(crystallized_state, v, validators[v].pubkey, EXIT)`

#### Finally...

* Set `crystallized_state.last_state_recalculation_slot += CYCLE_LENGTH`
* Remove all attestation records older than slot `crystallized_state.last_state_recalculation_slot`
* Empty the `active_state.pending_specials` list
* Set `shard_and_committee_for_slots[:CYCLE_LENGTH] = shard_and_committee_for_slots[CYCLE_LENGTH:]`

### Dynasty transition

A dynasty transition can happen after a state recalculation if all of the following criteria are satisfied:

* `block.slot - crystallized_state.dynasty_start_slot >= MIN_DYNASTY_LENGTH`
* `last_finalized_slot > dynasty_start_slot`
* For every shard `S` in `shard_and_committee_for_slots`, `crosslinks[S].slot > dynasty_start_slot`

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

```python
def change_validators(validators):
    # The active validator set
    active_validators = get_active_validator_indices(validators, dynasty)
    # The total size of active deposits
    total_deposits = sum([v.balance for i, v in enumerate(validators) if i in active_validators])
    # The maximum total Gwei that can be deposited and withdrawn
    max_allowable_change = max(
        2 * DEPOSIT_SIZE * 10**9,
        total_deposits // MAX_VALIDATOR_CHURN_QUOTIENT
    )
    # Go through the list start to end depositing+withdrawing as many as possible
    total_changed = 0
    for i in range(len(validators)):
        if validators[i].status == PENDING_LOG_IN:
            validators[i].status = LOGGED_IN
            total_changed += DEPOSIT_SIZE
            add_validator_set_change_record(crystallized_state, i, validators[i].pubkey, ENTRY)
        if validators[i].status == PENDING_EXIT:
            validators[i].status = PENDING_WITHDRAW
            validators[i].exit_slot = current_slot
            total_changed += validators[i].balance
            add_validator_set_change_record(crystallized_state, i, validators[i].pubkey, EXIT)
        if total_changed >= max_allowable_change:
            break

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

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

Finally:

* Set `last_dynasty_start_slot = crystallized_state.last_state_recalculation_slot`
* Set `crystallized_state.dynasty += 1`
* Let `next_start_shard = (shard_and_committee_for_slots[-1][-1].shard + 1) % SHARD_COUNT`
* Set `shard_and_committee_for_slots[CYCLE_LENGTH:] = get_new_shuffling(active_state.randao_mix, validators, next_start_shard)`

### TODO

Note: This spec is ~60% complete.

**Missing**

* [ ] Specify how `crystallized_state_root` and `active_state_root` are constructed, including Merklelisation logic for light clients
* [ ] Specify the rules around acceptable values for `pow_chain_reference`
* [ ] Specify the shard chain blocks, blobs, proposers, etc.
* [ ] Specify the rules for forced deregistrations
* [ ] Specify the various assumptions (global clock, networking latency, validator honesty, validator liveness, etc.)
* [ ] Specify (in a separate Vyper file) the registration contract on the PoW chain
* [ ] Specify the bootstrapping logic for the beacon chain genesis (e.g. specify a minimum number validators before the genesis block)
* [ ] Specify the logic for proofs of custody, including slashing conditions
* [ ] Add an appendix about the BLS12-381 curve
* [ ] Add an appendix on gossip networks and the offchain signature aggregation logic
* [ ] Add a glossary (in a separate `glossary.md`) to comprehensively and precisely define all the terms
* [ ] Undergo peer review, security audits and formal verification

**Possible rework/additions**

* [ ] Replace the IMD fork choice rule with LMD
* [ ] Merklelise `crystallized_state_root` and `active_state_root` into a single root
* [ ] Replace Blake with a STARK-friendly hash function
* [ ] Get rid of dynasties
* [ ] Reduce the slot duration to 8 seconds
* [ ] Allow for the delayed inclusion of aggregated signatures
* [ ] Use a separate networking-optimised serialisation format for networking
* [ ] Harden RANDAO against orphaned reveals
* [ ] Introduce a RANDAO slashing condition for early leakage
* [ ] Use a separate hash function for the proof of possession
* [ ] Rework the `ShardAndCommittee` data structures
* [ ] Add a double-batched Merkle accumulator for historical beacon chain blocks
* [ ] Allow for deposits larger than 32 ETH, as well as deposit top-ups
* [ ] Add penalties for a deposit below 32 ETH (or some other threshold)
* [ ] Add a `SpecialRecord` to (re)register
* [ ] Rework the document for readability
* [ ] Clearly document the various edge cases, e.g. with committee sizing

# Appendix
## Appendix A - Hash function

We aim to have a STARK-friendly hash function `hash(x)` for the production launch of the beacon chain. While the standardisation process for a STARK-friendly hash function takes place—led by STARKware, who will produce a detailed report with recommendations—we use `BLAKE2b-512` as a placeholder. Specifically, we set `hash(x) := BLAKE2b-512(x)[0:32]` where the `BLAKE2b-512` algorithm is defined in [RFC 7693](https://tools.ietf.org/html/rfc7693) and the input `x` is of type `bytes`.

## Copyright
Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).