# Casper+Sharding chain v2.1

## WORK IN PROGRESS!!!!!!!

This is the work-in-progress document describing the specification for the Casper+Sharding (shasper) chain, version 2.1.

In this protocol, there is a central PoS chain which stores and manages the current set of active PoS validators. The only mechanism available to become a validator initially is to send a transaction on the existing PoW main chain containing 32 ETH. When you do so, as soon as the PoS chain processes that block, you will be queued, and eventually inducted as an active validator until you either voluntarily deregister or you are forcibly deregistered as a penalty for misbehavior.

The primary source of load on the PoS chain is **attestations**. An attestation has a double role:

1. It attests to some parent block in the beacon chain
2. It attests to a block hash in a shard (a sufficient number of such attestations create a "crosslink", confirming that shard block into the beacon chain).

Every shard (e.g. there might be 1024 shards in total) is itself a PoS chain, and the shard chains are where the transactions and accounts will be stored. The crosslinks serve to "confirm" segments of the shard chains into the beacon chain, and are also the primary way through which the different shards will be able to talk to each other.

Note that one can also consider a simpler "minimal sharding algorithm" where crosslinks are simply hashes of proposed blocks of data that are not themselves chained to each other in any way.

Note: the python code at https://github.com/ethereum/beacon_chain and [an ethresear.ch post](https://ethresear.ch/t/convenience-link-to-full-casper-chain-v2-spec/2332) do not reflect all of the latest changes. If there is a discrepancy, this document is likely to reflect the more recent changes.

### 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 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 8 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

### Constants

* **SHARD_COUNT** - a constant referring to the number of shards. Currently set to 1024.
* **DEPOSIT_SIZE** - 32 ETH
* **MAX_VALIDATOR_COUNT** - 2<sup>22</sup> = 4194304 # Note: this means that up to ~134 million ETH can stake at the same time
* **GENESIS_TIME** - time of beacon chain startup (slot 0) in seconds since the Unix epoch
* **SLOT_DURATION** - 8 seconds
* **CYCLE_LENGTH** - 64 slots
* **MIN_DYNASTY_LENGTH** - 256 slots
* **MIN_COMMITTEE_SIZE** - 128 (rationale: see recommended minimum 111 here https://vitalik.ca/files/Ithaca201807_Sharding.pdf)
* **SQRT_E_DROP_TIME** - a constant set to reflect the amount of time it will take for the quadratic leak to cut nonparticipating validators' deposits by ~39.4%. Currently set to 2**20 seconds (~12 days).
* **BASE_REWARD_QUOTIENT** - 1/this is the per-slot interest rate assuming all validators are participating, assuming total deposits of 1 ETH. Currently set to `2**15 = 32768`, corresponding to ~3.88% annual interest assuming 10 million participating ETH.

### PoW main chain changes

This PoS/sharding proposal can be implemented separately from the existing PoW main chain. Only two changes to the PoW main chain are required (and the second one is technically not strictly necessary).

* On the PoW main chain a contract is added; this contract allows you to deposit `DEPOSIT_SIZE` ETH; the `deposit` function also takes as arguments (i) `pubkey` (bytes), (ii) `withdrawal_shard_id` (int), (iii)  `withdrawal_addr` (address), (iv) `randao_commitment` (bytes32), (v) `bls_proof_of_possession`
* PoW Main chain clients will implement a method, `prioritize(block_hash, value)`. If the block is available and has been verified, this method sets its score to the given value, and recursively adjusts the scores of all descendants. This allows the PoS beacon chain's finality gadget to also implicitly finalize PoW main chain blocks. Note that implementing this into the PoW client *is* a change to the PoW fork choice rule so is a sort of fork.

### Beacon chain

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

Here are the fields that go into every beacon chain block:

```python
fields = {
    # Hash of the parent block
    'parent_hash': 'hash32',
    # Slot number (for the PoS mechanism)
    'slot_number': 'int64',
    # Randao commitment reveal
    'randao_reveal': 'hash32',
    # Attestations
    'attestations': [AttestationRecord],
    # Reference to PoW chain block
    'pow_chain_ref': 'hash32',
    # Hash of the active state
    'active_state_root': 'hash32',
    # Hash of the crystallized state
    'crystallized_state_root': 'hash32',
}
```

The beacon chain state is split into two parts, _active state_ and _crystallized state_.

Here's the `ActiveState`:

```python
fields = {
    # Attestations that have not yet been processed
    'pending_attestations': [AttestationRecord],
    # Most recent 2 * CYCLE_LENGTH block hashes, older to newer
    'recent_block_hashes': ['hash32']
}
```

Here's the `CrystallizedState`:

```python
fields = {
    # List of validators
    'validators': [ValidatorRecord],
    # Last CrystallizedState recalculation
    'last_state_recalc': 'int64',
    # What active validators are part of the attester set
    # at what slot, and in what shard. Starts at slot
    # last_state_recalc - CYCLE_LENGTH
    'shard_and_committee_for_slots': [[ShardAndCommittee]],
    # The last justified slot
    'last_justified_slot': 'int64',
    # Number of consecutive justified slots ending at this one
    'justified_streak': 'int64',
    # The last finalized slot
    'last_finalized_slot': 'int64',
    # The current dynasty
    'current_dynasty': 'int64',
    # Records about the most recent crosslink `for each shard
    'crosslink_records': [CrosslinkRecord],
    # Used to select the committees for each shard
    'dynasty_seed': 'hash32',
    # Start of the current dynasty
    'dynasty_start': 'int64'
}
```

A `ShardAndCommittee` object is of the form:

```python
fields = {
    # The shard ID
    'shard_id': 'int16',
    # Validator indices
    'committee': ['int24']
}
```

Each `ValidatorRecord` is an object containing information about a validator:

```python
fields = {
    # The validator's public key
    'pubkey': 'int256',
    # What shard the validator's balance will be sent to 
    # after withdrawal
    'withdrawal_shard': 'int16',
    # And what address
    'withdrawal_address': 'address',
    # The validator's current RANDAO beacon commitment
    'randao_commitment': 'hash32',
    # Current balance
    'balance': 'int128',
    # Dynasty where the validator  is inducted
    'start_dynasty': 'int64',
    # Dynasty where the validator leaves
    'end_dynasty': 'int64'
}
```

And a `CrosslinkRecord` contains information about the last fully formed crosslink to be submitted into the chain:

```python
fields = {
    # What dynasty the crosslink was submitted in
    'dynasty': 'int64',
    # What slot
    'slot': 'int64',
    # The block hash
    'hash': 'hash32'
}
```

### Beacon chain processing

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 `parent_hash` 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_ref` 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 + slot_number * 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.

### 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 crystallized state recalculation, which happens only if `block.slot_number >= last_state_recalc + CYCLE_LENGTH`, and affects the `CrystallizedState` and `ActiveState`
2. The per-block processing, which happens every block (if during a crystallized state recalculation block, it happens after the crystallized state recalculation), and affects the `ActiveState` only

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, dynasty):
    o = []
    for i in range(len(validators)):
        if validators[i].start_dynasty <= dynasty < \
                validators[i].end_dynasty:
            o.append(i)
    return o
```

Now, a function that shuffles this list:

```python
def shuffle(lst, seed):
    assert len(lst) <= 16777216
    o = [x for x in lst]
    source = seed
    i = 0
    while i < len(lst):
        source = blake(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 = 16777216 - 16777216 % 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, dynasty, crosslinking_start_shard):
    avs = get_active_validator_indices(validators, dynasty)
    if len(avs) >= CYCLE_LENGTH * MIN_COMMITTEE_SIZE:
        committees_per_slot = len(avs) // CYCLE_LENGTH // (MIN_COMMITTEE_SIZE * 2) + 1 
        slots_per_committee = 1
    else:
        committees_per_slot = 1
        slots_per_committee = 1
        while len(avs) * 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(avs, seed), CYCLE_LENGTH)):
        shard_indices = split(slot_indices, committees_per_slot)
        shard_id_start = crosslinking_start_shard + \
            i * committees_per_slot // slots_per_committee
        o.append([ShardAndCommittee(
            shard_id = (shard_id_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:

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

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

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

### On startup

* Let `x = get_new_shuffling(bytes([0] * 32), validators, 1, 0)` and set `crystallized_state.shard_and_committee_for_slots` to `x + x`
* Set `crystallized_state.dynasty = 1`
* Set `crystallized_state.crosslink_records` to `[CrosslinkRecord(dynasty=0, slot=0, hash=bytes([0] * 32)) for i in range(SHARD_COUNT)]`
* Set `active_state.recent_block_hashes` to `[bytes([0] * 32) for _ in range(CYCLE_LENGTH * 2)]`

Any value not explicitly set above in the active and crystallized state should be set to zero or an empty array depending on context.

### Per-block processing

First, set `recent_block_hashes` to the output of the following:

```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`

A block can have 0 or more `AttestationRecord` objects, where each `AttestationRecord` object has the following fields:

```python
fields = {
    # Slot number
    'slot': 'int64',
    # Shard ID
    'shard_id': 'int16',
    # List of block hashes that this signature is signing over that
    # are NOT part of the current chain, in order of oldest to newest
    'oblique_parent_hashes': ['hash32'],
    # Block hash in the shard that we are attesting to
    'shard_block_hash': 'hash32',
    # Who is participating
    'attester_bitfield': 'bytes',
    # Last justified block
    'justified_slot': 'int64',
    'justified_block_hash': 'hash32',
    # The actual signature
    'aggregate_sig': ['int256']
}
```

For each one of these attestations [TODO]:

* Verify that `slot <= parent.slot_number` and `slot >= max(parent.slot_number - 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_id` equals the `shard_id` 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
* Verify that `aggregate_sig` verifies using the group pubkey generated and `hash(slot.to_bytes(8, 'big') + parent_hashes + shard_id + shard_block_hash + justified_slot.to_bytes(8, 'big'))` as the message.

Extend the list of `AttestationRecord` objects in the `active_state`, ordering the new additions in the same order as they came in the block.

Verify that the `parent.slot_number % len(get_shards_and_committees_for_slot(crystallized_state, parent.slot_number)[0].committee)`'th attester in `get_shards_and_committees_for_slot(crystallized_state, parent.slot_number)[0]`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.

### State recalculations

Repeat while `slot - last_state_recalc >= CYCLE_LENGTH`:

For all slots `s` in `last_state_recalc - CYCLE_LENGTH ... last_state_recalc - 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_id`, `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 `crosslink_records[shard_id].dynasty`, set `crosslink_records[shard_id] = CrosslinkRecord(dynasty=current_dynasty, slot=block.last_state_recalc + CYCLE_LENGTH, hash=shard_block_hash)`.

#### Balance recalculations related to FFG rewards


Let `time_since_finality = block.slot_number - 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, current_dynasty)])` and `total_deposits_in_ETH = total_deposits // 10**18`
* `reward_quotient = BASE_REWARD_QUOTIENT * int(sqrt(total_deposits_in_ETH))` (1/this is the per-slot max interest rate)
* `quadratic_penalty_quotient = int(sqrt(SQRT_E_DROP_TIME / SLOT_DURATION))` (after D slots, ~D<sup>2</sup>/2 divided by this is the portion lost by offline validators)

For each slot `S` in the range `last_state_recalc - CYCLE_LENGTH ... last_state_recalc - 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 <= 2 * 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`

#### Balance recalculations related to crosslink rewards

For each shard S for which a crosslink committee exists in this epoch, 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 do the following:

* Let `total_v_deposits` be the total balance of V, and `total_participated_v_deposits` be the total balance of the subset of V that participated (note: it's always true that `total_participated_v_deposits <= total_v_deposits`)
* Let `time_since_last_confirmation` be `block.slot_number - crosslink_records[S].slot`
* Adjust balances as follows:
    * If `crosslink_records[S].dynasty == current_dynasty`, no reward adjustments
    * Otherwise, participating validators' balances are increased by `B // reward_quotient * (2 * total_participated_v_deposits // total_v_deposits - 1)`, and non-participating validators' balances are decreased by `B // reward_quotient + B // reward_quotient // 2 + time_since_finality // quadratic_penalty_quotient`

Finally:

* Set `crystallized_state.last_state_recalc += CYCLE_LENGTH`
* Remove all attestation records older than slot `crystallized_state.last_state_recalc`
* 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_number - crystallized_state.dynasty_start >= MIN_DYNASTY_LENGTH`
* `last_finalized_slot > dynasty_start`
* For every shard S in `shard_and_committee_for_slots`, `crosslink_records[S].slot > dynasty_start`

Then:

* Set `last_dynasty_start = crystallized_state.last_state_recalc`
* Set `crystallized_state.current_dynasty += 1`
* Let `next_start_shard = (shard_and_committee_for_slots[-1][-1].shard_id + 1) % SHARD_COUNT`
* Set `shard_and_committee_for_slots[CYCLE_LENGTH:] = get_new_shuffling(block.parent_hash, validators, crystallized_state.current_dynasty, next_start_shard)`

-------

Note: this is ~70% complete. The main sections that are missing are:

* Validator login/logout logic
* Logic for the formats of shard chains, who proposes shard blocks, etc. (in an initial release, if desired we could make crosslinks just be Merkle roots of blobs of data; in any case, one can philosophically view the whole point of the shard chains as being a coordination device for choosing what blobs of data to propose as crosslinks)
* Logic for inducting queued validators from the PoW main chain
* Penalties for signing or attesting to non-canonical-chain blocks (update: may not be necessary, see https://ethresear.ch/t/attestation-committee-based-full-pos-chains/2259)
* Slashing conditions
* Logic for withdrawing deposits to shards
* Per-validator proofs of custody
* Versioning and upgrades

Slashing conditions may include:


    Casper FFG slot equivocation
    Casper FFG surround
    Beacon chain proposal equivocation
    Shard chain proposal equivocation
    Proof of custody secret leak
    Proof of custody wrong custody bit
    Proof of custody no secret reveal
    RANDAO leak

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