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# Ethereum 2.0 Phase 1 -- Custody Game
**Notice**: This document is a work-in-progress for researchers and implementers.
## Table of contents
<!-- START doctoc generated TOC please keep comment here to allow auto update -->
<!-- DON'T EDIT THIS SECTION, INSTEAD RE-RUN doctoc TO UPDATE -->
- [Introduction](#introduction)
- [Constants](#constants)
- [Misc](#misc)
- [Configuration](#configuration)
- [Time parameters](#time-parameters)
- [Max operations per block](#max-operations-per-block)
- [Reward and penalty quotients](#reward-and-penalty-quotients)
- [Data structures](#data-structures)
- [New Beacon Chain operations](#new-beacon-chain-operations)
- [`CustodyChunkChallenge`](#custodychunkchallenge)
- [`CustodyChunkChallengeRecord`](#custodychunkchallengerecord)
- [`CustodyChunkResponse`](#custodychunkresponse)
- [`CustodySlashing`](#custodyslashing)
- [`SignedCustodySlashing`](#signedcustodyslashing)
- [`CustodyKeyReveal`](#custodykeyreveal)
- [`EarlyDerivedSecretReveal`](#earlyderivedsecretreveal)
- [Helpers](#helpers)
- [`replace_empty_or_append`](#replace_empty_or_append)
- [`legendre_bit`](#legendre_bit)
- [`get_custody_atoms`](#get_custody_atoms)
- [`get_custody_secrets`](#get_custody_secrets)
- [`universal_hash_function`](#universal_hash_function)
- [`compute_custody_bit`](#compute_custody_bit)
- [`get_randao_epoch_for_custody_period`](#get_randao_epoch_for_custody_period)
- [`get_custody_period_for_validator`](#get_custody_period_for_validator)
- [Per-block processing](#per-block-processing)
- [Custody Game Operations](#custody-game-operations)
- [Chunk challenges](#chunk-challenges)
- [Custody chunk response](#custody-chunk-response)
- [Custody key reveals](#custody-key-reveals)
- [Early derived secret reveals](#early-derived-secret-reveals)
- [Custody Slashings](#custody-slashings)
- [Per-epoch processing](#per-epoch-processing)
- [Handling of reveal deadlines](#handling-of-reveal-deadlines)
- [Final updates](#final-updates)
<!-- END doctoc generated TOC please keep comment here to allow auto update -->
## Introduction
This document details the beacon chain additions and changes in Phase 1 of Ethereum 2.0 to support the shard data custody game, building upon the [Phase 0](../phase0/beacon-chain.md) specification.
## Constants
### Misc
| Name | Value | Unit |
| - | - | - |
| `CUSTODY_PRIME` | `2 ** 256 - 189` | - |
| `CUSTODY_SECRETS` | `3` | - |
| `BYTES_PER_CUSTODY_ATOM` | `32` | bytes |
| `CUSTODY_PROBABILITY_EXPONENT` | `10` | - |
## Configuration
### Time parameters
| Name | Value | Unit | Duration |
| - | - | :-: | :-: |
| `RANDAO_PENALTY_EPOCHS` | `2**1` (= 2) | epochs | 12.8 minutes |
| `EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS` | `2**15` (= 32,768) | epochs | ~146 days |
| `EPOCHS_PER_CUSTODY_PERIOD` | `2**14` (= 16,384) | epochs | ~73 days |
| `CUSTODY_PERIOD_TO_RANDAO_PADDING` | `2**11` (= 2,048) | epochs | ~9 days |
| `MAX_CHUNK_CHALLENGE_DELAY` | `2**15` (= 32,768) | epochs | ~146 days |
| `CHUNK_RESPONSE_DEADLINE` | `2**14` (= 16,384) | epochs | ~73 days |
### Max operations per block
| Name | Value |
| - | - |
| `MAX_CUSTODY_CHUNK_CHALLENGE_RECORDS` | `2**20` (= 1,048,576) |
| `MAX_CUSTODY_KEY_REVEALS` | `2**8` (= 256) |
| `MAX_EARLY_DERIVED_SECRET_REVEALS` | `2**0` (= 1) |
| `MAX_CUSTODY_CHUNK_CHALLENGES` | `2**2` (= 4) |
| `MAX_CUSTODY_CHUNK_CHALLENGE_RESPONSES` | `2**4` (= 16) |
| `MAX_CUSTODY_SLASHINGS` | `2**0` (= 1) |
### Reward and penalty quotients
| Name | Value |
| - | - |
| `EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE` | `2**1` (= 2) |
| `MINOR_REWARD_QUOTIENT` | `2**8` (= 256) |
## Data structures
### New Beacon Chain operations
#### `CustodyChunkChallenge`
```python
class CustodyChunkChallenge(Container):
responder_index: ValidatorIndex
shard_transition: ShardTransition
attestation: Attestation
data_index: uint64
chunk_index: uint64
```
#### `CustodyChunkChallengeRecord`
```python
class CustodyChunkChallengeRecord(Container):
challenge_index: uint64
challenger_index: ValidatorIndex
responder_index: ValidatorIndex
inclusion_epoch: Epoch
data_root: Root
chunk_index: uint64
```
#### `CustodyChunkResponse`
```python
class CustodyChunkResponse(Container):
challenge_index: uint64
chunk_index: uint64
chunk: ByteVector[BYTES_PER_CUSTODY_CHUNK]
branch: Vector[Root, CUSTODY_RESPONSE_DEPTH]
```
#### `CustodySlashing`
```python
class CustodySlashing(Container):
# (Attestation.data.shard_transition_root as ShardTransition).shard_data_roots[data_index] is the root of the data.
data_index: uint64
malefactor_index: ValidatorIndex
malefactor_secret: BLSSignature
whistleblower_index: ValidatorIndex
shard_transition: ShardTransition
attestation: Attestation
data: ByteList[MAX_SHARD_BLOCK_SIZE]
```
#### `SignedCustodySlashing`
```python
class SignedCustodySlashing(Container):
message: CustodySlashing
signature: BLSSignature
```
#### `CustodyKeyReveal`
```python
class CustodyKeyReveal(Container):
# Index of the validator whose key is being revealed
revealer_index: ValidatorIndex
# Reveal (masked signature)
reveal: BLSSignature
```
#### `EarlyDerivedSecretReveal`
Represents an early (punishable) reveal of one of the derived secrets, where derived secrets are RANDAO reveals and custody reveals (both are part of the same domain).
```python
class EarlyDerivedSecretReveal(Container):
# Index of the validator whose key is being revealed
revealed_index: ValidatorIndex
# RANDAO epoch of the key that is being revealed
epoch: Epoch
# Reveal (masked signature)
reveal: BLSSignature
# Index of the validator who revealed (whistleblower)
masker_index: ValidatorIndex
# Mask used to hide the actual reveal signature (prevent reveal from being stolen)
mask: Bytes32
```
## Helpers
### `replace_empty_or_append`
```python
def replace_empty_or_append(l: List, new_element: Any) -> int:
for i in range(len(l)):
if l[i] == type(new_element)():
l[i] = new_element
return i
l.append(new_element)
return len(l) - 1
```
### `legendre_bit`
Returns the Legendre symbol `(a/q)` normalizes as a bit (i.e. `((a/q) + 1) // 2`). In a production implementation, a well-optimized library (e.g. GMP) should be used for this.
```python
def legendre_bit(a: int, q: int) -> int:
if a >= q:
return legendre_bit(a % q, q)
if a == 0:
return 0
assert(q > a > 0 and q % 2 == 1)
t = 1
n = q
while a != 0:
while a % 2 == 0:
a //= 2
r = n % 8
if r == 3 or r == 5:
t = -t
a, n = n, a
if a % 4 == n % 4 == 3:
t = -t
a %= n
if n == 1:
return (t + 1) // 2
else:
return 0
```
### `get_custody_atoms`
Given one set of data, return the custody atoms: each atom will be combined with one legendre bit.
```python
def get_custody_atoms(bytez: bytes) -> Sequence[bytes]:
bytez += b'\x00' * (-len(bytez) % BYTES_PER_CUSTODY_ATOM) # right-padding
return [bytez[i:i + BYTES_PER_CUSTODY_ATOM]
for i in range(0, len(bytez), BYTES_PER_CUSTODY_ATOM)]
```
### `get_custody_secrets`
Extract the custody secrets from the signature
```python
def get_custody_secrets(key: BLSSignature) -> Sequence[int]:
full_G2_element = bls.signature_to_G2(key)
signature = full_G2_element[0].coeffs
signature_bytes = b"".join(x.to_bytes(48, "little") for x in signature)
secrets = [int.from_bytes(signature_bytes[i:i + BYTES_PER_CUSTODY_ATOM], "little")
for i in range(0, len(signature_bytes), 32)]
return secrets
```
### `universal_hash_function`
```python
def universal_hash_function(data_chunks: Sequence[bytes], secrets: Sequence[int]) -> int:
n = len(data_chunks)
return (
sum(
secrets[i % CUSTODY_SECRETS]**i * int.from_bytes(atom, "little") % CUSTODY_PRIME
for i, atom in enumerate(data_chunks)
) + secrets[n % CUSTODY_SECRETS]**n
) % CUSTODY_PRIME
```
### `compute_custody_bit`
```python
def compute_custody_bit(key: BLSSignature, data: ByteList[MAX_SHARD_BLOCK_SIZE]) -> bit:
custody_atoms = get_custody_atoms(data)
secrets = get_custody_secrets(key)
uhf = universal_hash_function(custody_atoms, secrets)
legendre_bits = [legendre_bit(uhf + secrets[0] + i, CUSTODY_PRIME) for i in range(CUSTODY_PROBABILITY_EXPONENT)]
return all(legendre_bits)
```
### `get_randao_epoch_for_custody_period`
```python
def get_randao_epoch_for_custody_period(period: uint64, validator_index: ValidatorIndex) -> Epoch:
next_period_start = (period + 1) * EPOCHS_PER_CUSTODY_PERIOD - validator_index % EPOCHS_PER_CUSTODY_PERIOD
return Epoch(next_period_start + CUSTODY_PERIOD_TO_RANDAO_PADDING)
```
### `get_custody_period_for_validator`
```python
def get_custody_period_for_validator(validator_index: ValidatorIndex, epoch: Epoch) -> int:
'''
Return the reveal period for a given validator.
'''
return (epoch + validator_index % EPOCHS_PER_CUSTODY_PERIOD) // EPOCHS_PER_CUSTODY_PERIOD
```
## Per-block processing
### Custody Game Operations
```python
def process_custody_game_operations(state: BeaconState, body: BeaconBlockBody) -> None:
def for_ops(operations: Sequence[Any], fn: Callable[[BeaconState, Any], None]) -> None:
for operation in operations:
fn(state, operation)
for_ops(body.chunk_challenges, process_chunk_challenge)
for_ops(body.chunk_challenge_responses, process_chunk_challenge)
for_ops(body.custody_key_reveals, process_custody_key_reveal)
for_ops(body.early_derived_secret_reveals, process_early_derived_secret_reveal)
for_ops(body.custody_slashings, process_custody_slashing)
```
#### Chunk challenges
```python
def process_chunk_challenge(state: BeaconState, challenge: CustodyChunkChallenge) -> None:
# Verify the attestation
assert is_valid_indexed_attestation(state, get_indexed_attestation(state, challenge.attestation))
# Verify it is not too late to challenge the attestation
max_attestation_challenge_epoch = challenge.attestation.data.target.epoch + MAX_CHUNK_CHALLENGE_DELAY
assert get_current_epoch(state) <= max_attestation_challenge_epoch
# Verify it is not too late to challenge the responder
responder = state.validators[challenge.responder_index]
if responder.exit_epoch < FAR_FUTURE_EPOCH:
assert get_current_epoch(state) <= responder.exit_epoch + MAX_CHUNK_CHALLENGE_DELAY
# Verify responder is slashable
assert is_slashable_validator(responder, get_current_epoch(state))
# Verify the responder participated in the attestation
attesters = get_attesting_indices(state, challenge.attestation.data, challenge.attestation.aggregation_bits)
assert challenge.responder_index in attesters
# Verify shard transition is correctly given
assert hash_tree_root(challenge.shard_transition) == challenge.attestation.data.shard_transition_root
data_root = challenge.shard_transition.shard_data_roots[challenge.data_index]
# Verify the challenge is not a duplicate
for record in state.custody_chunk_challenge_records:
assert (
record.data_root != data_root or
record.chunk_index != challenge.chunk_index
)
# Verify depth
shard_block_length = challenge.shard_transition.shard_block_lengths[challenge.data_index]
transition_chunks = (shard_block_length + BYTES_PER_CUSTODY_CHUNK - 1) // BYTES_PER_CUSTODY_CHUNK
assert challenge.chunk_index < transition_chunks
# Add new chunk challenge record
new_record = CustodyChunkChallengeRecord(
challenge_index=state.custody_chunk_challenge_index,
challenger_index=get_beacon_proposer_index(state),
responder_index=challenge.responder_index,
inclusion_epoch=get_current_epoch(state),
data_root=challenge.shard_transition.shard_data_roots[challenge.data_index],
chunk_index=challenge.chunk_index,
)
replace_empty_or_append(state.custody_chunk_challenge_records, new_record)
state.custody_chunk_challenge_index += 1
# Postpone responder withdrawability
responder.withdrawable_epoch = FAR_FUTURE_EPOCH
```
#### Custody chunk response
```python
def process_chunk_challenge_response(state: BeaconState,
response: CustodyChunkResponse) -> None:
# Get matching challenge (if any) from records
matching_challenges = [
record for record in state.custody_chunk_challenge_records
if record.challenge_index == response.challenge_index
]
assert len(matching_challenges) == 1
challenge = matching_challenges[0]
# Verify chunk index
assert response.chunk_index == challenge.chunk_index
# Verify the chunk matches the crosslink data root
assert is_valid_merkle_branch(
leaf=hash_tree_root(response.chunk),
branch=response.branch,
depth=CUSTODY_RESPONSE_DEPTH,
index=response.chunk_index,
root=challenge.data_root,
)
# Clear the challenge
index_in_records = state.custody_chunk_challenge_records.index(challenge)
state.custody_chunk_challenge_records[index_in_records] = CustodyChunkChallengeRecord()
# Reward the proposer
proposer_index = get_beacon_proposer_index(state)
increase_balance(state, proposer_index, Gwei(get_base_reward(state, proposer_index) // MINOR_REWARD_QUOTIENT))
```
#### Custody key reveals
```python
def process_custody_key_reveal(state: BeaconState, reveal: CustodyKeyReveal) -> None:
"""
Process ``CustodyKeyReveal`` operation.
Note that this function mutates ``state``.
"""
revealer = state.validators[reveal.revealer_index]
epoch_to_sign = get_randao_epoch_for_custody_period(revealer.next_custody_secret_to_reveal, reveal.revealer_index)
custody_reveal_period = get_custody_period_for_validator(reveal.revealer_index, get_current_epoch(state))
# Only past custody periods can be revealed, except after exiting the exit period can be revealed
is_past_reveal = revealer.next_custody_secret_to_reveal < custody_reveal_period
is_exited = revealer.exit_epoch <= get_current_epoch(state)
is_exit_period_reveal = (
revealer.next_custody_secret_to_reveal
== get_custody_period_for_validator(reveal.revealer_index, revealer.exit_epoch - 1)
)
assert is_past_reveal or (is_exited and is_exit_period_reveal)
# Revealed validator is active or exited, but not withdrawn
assert is_slashable_validator(revealer, get_current_epoch(state))
# Verify signature
domain = get_domain(state, DOMAIN_RANDAO, epoch_to_sign)
signing_root = compute_signing_root(epoch_to_sign, domain)
assert bls.Verify(revealer.pubkey, signing_root, reveal.reveal)
# Process reveal
if is_exited and is_exit_period_reveal:
revealer.all_custody_secrets_revealed_epoch = get_current_epoch(state)
revealer.next_custody_secret_to_reveal += 1
# Reward Block Proposer
proposer_index = get_beacon_proposer_index(state)
increase_balance(
state,
proposer_index,
Gwei(get_base_reward(state, reveal.revealer_index) // MINOR_REWARD_QUOTIENT)
)
```
#### Early derived secret reveals
```python
def process_early_derived_secret_reveal(state: BeaconState, reveal: EarlyDerivedSecretReveal) -> None:
"""
Process ``EarlyDerivedSecretReveal`` operation.
Note that this function mutates ``state``.
"""
revealed_validator = state.validators[reveal.revealed_index]
derived_secret_location = reveal.epoch % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
assert reveal.epoch >= get_current_epoch(state) + RANDAO_PENALTY_EPOCHS
assert reveal.epoch < get_current_epoch(state) + EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
assert not revealed_validator.slashed
assert reveal.revealed_index not in state.exposed_derived_secrets[derived_secret_location]
# Verify signature correctness
masker = state.validators[reveal.masker_index]
pubkeys = [revealed_validator.pubkey, masker.pubkey]
domain = get_domain(state, DOMAIN_RANDAO, reveal.epoch)
signing_roots = [compute_signing_root(root, domain) for root in [hash_tree_root(reveal.epoch), reveal.mask]]
assert bls.AggregateVerify(pubkeys, signing_roots, reveal.reveal)
if reveal.epoch >= get_current_epoch(state) + CUSTODY_PERIOD_TO_RANDAO_PADDING:
# Full slashing when the secret was revealed so early it may be a valid custody
# round key
slash_validator(state, reveal.revealed_index, reveal.masker_index)
else:
# Only a small penalty proportional to proposer slot reward for RANDAO reveal
# that does not interfere with the custody period
# The penalty is proportional to the max proposer reward
# Calculate penalty
max_proposer_slot_reward = (
get_base_reward(state, reveal.revealed_index)
* SLOTS_PER_EPOCH
// len(get_active_validator_indices(state, get_current_epoch(state)))
// PROPOSER_REWARD_QUOTIENT
)
penalty = Gwei(
max_proposer_slot_reward
* EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE
* (len(state.exposed_derived_secrets[derived_secret_location]) + 1)
)
# Apply penalty
proposer_index = get_beacon_proposer_index(state)
whistleblower_index = reveal.masker_index
whistleblowing_reward = Gwei(penalty // WHISTLEBLOWER_REWARD_QUOTIENT)
proposer_reward = Gwei(whistleblowing_reward // PROPOSER_REWARD_QUOTIENT)
increase_balance(state, proposer_index, proposer_reward)
increase_balance(state, whistleblower_index, whistleblowing_reward - proposer_reward)
decrease_balance(state, reveal.revealed_index, penalty)
# Mark this derived secret as exposed so validator cannot be punished repeatedly
state.exposed_derived_secrets[derived_secret_location].append(reveal.revealed_index)
```
#### Custody Slashings
```python
def process_custody_slashing(state: BeaconState, signed_custody_slashing: SignedCustodySlashing) -> None:
custody_slashing = signed_custody_slashing.message
attestation = custody_slashing.attestation
# Any signed custody-slashing should result in at least one slashing.
# If the custody bits are valid, then the claim itself is slashed.
malefactor = state.validators[custody_slashing.malefactor_index]
whistleblower = state.validators[custody_slashing.whistleblower_index]
domain = get_domain(state, DOMAIN_CUSTODY_BIT_SLASHING, get_current_epoch(state))
signing_root = compute_signing_root(custody_slashing, domain)
assert bls.Verify(whistleblower.pubkey, signing_root, signed_custody_slashing.signature)
# Verify that the whistleblower is slashable
assert is_slashable_validator(whistleblower, get_current_epoch(state))
# Verify that the claimed malefactor is slashable
assert is_slashable_validator(malefactor, get_current_epoch(state))
# Verify the attestation
assert is_valid_indexed_attestation(state, get_indexed_attestation(state, attestation))
# TODO: can do a single combined merkle proof of data being attested.
# Verify the shard transition is indeed attested by the attestation
shard_transition = custody_slashing.shard_transition
assert hash_tree_root(shard_transition) == attestation.data.shard_transition_root
# Verify that the provided data matches the shard-transition
assert (
custody_slashing.data.get_backing().get_left().merkle_root()
== shard_transition.shard_data_roots[custody_slashing.data_index]
)
assert len(custody_slashing.data) == shard_transition.shard_block_lengths[custody_slashing.data_index]
# Verify existence and participation of claimed malefactor
attesters = get_attesting_indices(state, attestation.data, attestation.aggregation_bits)
assert custody_slashing.malefactor_index in attesters
# Verify the malefactor custody key
epoch_to_sign = get_randao_epoch_for_custody_period(
get_custody_period_for_validator(custody_slashing.malefactor_index, attestation.data.target.epoch),
custody_slashing.malefactor_index,
)
domain = get_domain(state, DOMAIN_RANDAO, epoch_to_sign)
signing_root = compute_signing_root(epoch_to_sign, domain)
assert bls.Verify(malefactor.pubkey, signing_root, custody_slashing.malefactor_secret)
# Compute the custody bit
computed_custody_bit = compute_custody_bit(custody_slashing.malefactor_secret, custody_slashing.data)
# Verify the claim
if computed_custody_bit == 1:
# Slash the malefactor, reward the other committee members
slash_validator(state, custody_slashing.malefactor_index)
committee = get_beacon_committee(state, attestation.data.slot, attestation.data.index)
others_count = len(committee) - 1
whistleblower_reward = Gwei(malefactor.effective_balance // WHISTLEBLOWER_REWARD_QUOTIENT // others_count)
for attester_index in attesters:
if attester_index != custody_slashing.malefactor_index:
increase_balance(state, attester_index, whistleblower_reward)
# No special whisteblower reward: it is expected to be an attester. Others are free to slash too however.
else:
# The claim was false, the custody bit was correct. Slash the whistleblower that induced this work.
slash_validator(state, custody_slashing.whistleblower_index)
```
## Per-epoch processing
### Handling of reveal deadlines
```python
def process_reveal_deadlines(state: BeaconState) -> None:
epoch = get_current_epoch(state)
for index, validator in enumerate(state.validators):
deadline = validator.next_custody_secret_to_reveal + 1
if get_custody_period_for_validator(ValidatorIndex(index), epoch) > deadline:
slash_validator(state, ValidatorIndex(index))
```
```python
def process_challenge_deadlines(state: BeaconState) -> None:
for custody_chunk_challenge in state.custody_chunk_challenge_records:
if get_current_epoch(state) > custody_chunk_challenge.inclusion_epoch + EPOCHS_PER_CUSTODY_PERIOD:
slash_validator(state, custody_chunk_challenge.responder_index, custody_chunk_challenge.challenger_index)
index_in_records = state.custody_chunk_challenge_records.index(custody_chunk_challenge)
state.custody_chunk_challenge_records[index_in_records] = CustodyChunkChallengeRecord()
```
### Final updates
```python
def process_custody_final_updates(state: BeaconState) -> None:
# Clean up exposed RANDAO key reveals
state.exposed_derived_secrets[get_current_epoch(state) % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS] = []
# Reset withdrawable epochs if challenge records are empty
records = state.custody_chunk_challenge_records
validator_indices_in_records = set([record.responder_index for record in records])
for index, validator in enumerate(state.validators):
if validator.exit_epoch != FAR_FUTURE_EPOCH:
not_all_secrets_are_revealed = validator.all_custody_secrets_revealed_epoch == FAR_FUTURE_EPOCH
if index in validator_indices_in_records or not_all_secrets_are_revealed:
# Delay withdrawable epochs if challenge records are not empty or not all
# custody secrets revealed
validator.withdrawable_epoch = FAR_FUTURE_EPOCH
else:
# Reset withdrawable epochs if challenge records are empty and all secrets are revealed
if validator.withdrawable_epoch == FAR_FUTURE_EPOCH:
validator.withdrawable_epoch = Epoch(validator.all_custody_secrets_revealed_epoch
+ MIN_VALIDATOR_WITHDRAWABILITY_DELAY)
```
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