Add untested attestation validation logic

This commit is contained in:
Paul Hauner 2018-12-05 16:08:03 +11:00
parent d4b6d81c9d
commit 3533b8b892
No known key found for this signature in database
GPG Key ID: 303E4494BB28068C
14 changed files with 704 additions and 18 deletions

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@ -32,6 +32,8 @@ name = "lighthouse"
[workspace]
members = [
"beacon_chain/attestation_validation",
"beacon_chain/block_validation",
"beacon_chain/chain",
"beacon_chain/naive_fork_choice",
"beacon_chain/state-transition",
@ -45,7 +47,6 @@ members = [
"beacon_chain/utils/ssz",
"beacon_chain/utils/ssz_helpers",
"beacon_chain/utils/vec_shuffle",
"beacon_chain/validation",
"beacon_chain/validator_change",
"beacon_chain/validator_induction",
"beacon_chain/validator_shuffling",

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@ -0,0 +1,12 @@
[package]
name = "attestation_validation"
version = "0.1.0"
authors = ["Paul Hauner <paul@paulhauner.com>"]
[dependencies]
bls = { path = "../utils/bls" }
db = { path = "../../lighthouse/db" }
hashing = { path = "../utils/hashing" }
ssz = { path = "../utils/ssz" }
ssz_helpers = { path = "../utils/ssz_helpers" }
types = { path = "../types" }

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@ -0,0 +1,34 @@
/// Reasons why an `AttestationRecord` can be invalid.
pub enum Invalid {
AttestationTooRecent,
AttestationTooOld,
JustifiedSlotImpermissable,
JustifiedBlockNotInChain,
JustifiedBlockHashMismatch,
UnknownShard,
ShardBlockHashMismatch,
SignatureInvalid,
}
/// The outcome of validating the `AttestationRecord`.
///
/// Distinct from the `Error` enum as an `Outcome` indicates that validation executed sucessfully
/// and determined the validity `AttestationRecord`.
pub enum Outcome {
Valid,
Invalid(Invalid),
}
/// Errors that prevent this function from correctly validating the `AttestationRecord`.
///
/// Distinct from the `Outcome` enum as `Errors` indicate that validation encountered an unexpected
/// condition and was unable to perform its duty.
pub enum Error {
BlockHasNoParent,
BadValidatorIndex,
UnableToLookupBlockAtSlot,
OutOfBoundsBitfieldIndex,
PublicKeyCorrupt,
NoPublicKeyForValidator,
DBError(String),
}

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@ -0,0 +1,18 @@
extern crate bls;
extern crate db;
extern crate hashing;
extern crate ssz;
extern crate ssz_helpers;
extern crate types;
#[macro_use]
mod macros;
mod enums;
mod validate_for_block;
mod validate_for_state;
mod validate_signature;
pub use enums::{Invalid, Outcome, Error};
pub use validate_for_block::validate_attestation_for_block;
pub use validate_for_state::validate_attestation_data_for_state;
pub use validate_signature::validate_attestation_signature;

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@ -0,0 +1,19 @@
macro_rules! verify_or {
($condition: expr, $result: expr) => {
if !$condition {
$result
}
};
}
macro_rules! reject {
($result: expr) => {
return Ok(Outcome::Invalid($result));
};
}
macro_rules! accept {
() => {
Ok(Outcome::Valid)
};
}

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@ -0,0 +1,33 @@
use super::{Error, Invalid, Outcome};
/// Check that an attestation is valid to be included in some block.
pub fn validate_attestation_for_block<T>(
attestation_slot: u64,
block_slot: u64,
parent_block_slot: u64,
min_attestation_inclusion_delay: u64,
epoch_length: u64,
) -> Result<Outcome, Error> {
/*
* There is a delay before an attestation may be included in a block, quantified by
* `slots` and defined as `min_attestation_inclusion_delay`.
*
* So, an attestation must be at least `min_attestation_inclusion_delay` slots "older" than the
* block it is contained in.
*/
verify_or!(
attestation_slot <= block_slot.saturating_sub(min_attestation_inclusion_delay),
reject!(Invalid::AttestationTooRecent)
);
/*
* A block may not include attestations reference slots more than an epoch length + 1 prior to
* the block slot.
*/
verify_or!(
attestation_slot >= parent_block_slot.saturating_sub(epoch_length + 1),
reject!(Invalid::AttestationTooOld)
);
accept!()
}

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@ -0,0 +1,107 @@
use super::db::stores::{BeaconBlockAtSlotError, BeaconBlockStore};
use super::db::ClientDB;
use super::types::Hash256;
use super::types::{AttestationData, BeaconState};
use super::{Error, Invalid, Outcome};
use std::sync::Arc;
/// Check that an attestation is valid with reference to some state.
pub fn validate_attestation_data_for_state<T>(
data: &AttestationData,
chain_tip_block_hash: &Hash256,
state: &BeaconState,
block_store: &Arc<BeaconBlockStore<T>>,
) -> Result<Outcome, Error>
where
T: ClientDB + Sized,
{
/*
* The attestation's `justified_slot` must be the same as the last justified slot known to this
* client.
*
* In the case that an attestation references a slot _before_ the latest state transition, it
* is acceptable for the attestation to reference the previous known `justified_slot`. If this
* were not the case, all attestations created _prior_ to the last state recalculation would be
* rejected if a block was justified in that state recalculation. It is both ideal and likely
* that blocks will be justified during a state recalcuation.
*/
{
let permissable_justified_slot = if data.slot >= state.latest_state_recalculation_slot {
state.justified_slot
} else {
state.previous_justified_slot
};
verify_or!(
data.justified_slot == permissable_justified_slot,
reject!(Invalid::JustifiedSlotImpermissable)
);
}
/*
* The `justified_block_hash` in the attestation must match exactly the hash of the block at
* that slot in the local chain.
*
* This condition also infers that the `justified_slot` specified in attestation must exist
* locally.
*/
match block_hash_at_slot(chain_tip_block_hash, data.justified_slot, block_store)? {
None => reject!(Invalid::JustifiedBlockNotInChain),
Some(local_justified_block_hash) => {
verify_or!(
data.justified_block_hash == local_justified_block_hash,
reject!(Invalid::JustifiedBlockHashMismatch)
);
}
};
/*
* The `shard_block_hash` in the state's `latest_crosslinks` must match either the
* `latest_crosslink_hash` or the `shard_block_hash` on the attestation.
*
* TODO: figure out the reasoning behind this.
*/
match state.latest_crosslinks.get(data.shard as usize) {
None => reject!(Invalid::UnknownShard),
Some(crosslink) => {
let local_shard_block_hash = crosslink.shard_block_hash;
let shard_block_hash_is_permissable = {
(local_shard_block_hash == data.latest_crosslink_hash)
|| (local_shard_block_hash == data.shard_block_hash)
};
verify_or!(
shard_block_hash_is_permissable,
reject!(Invalid::ShardBlockHashMismatch)
);
}
};
accept!()
}
/// Returns the hash (or None) of a block at a slot in the chain that is specified by
/// `chain_tip_hash`.
///
/// Given that the database stores multiple chains, it is possible for there to be multiple blocks
/// at the given slot. `chain_tip_hash` specifies exactly which chain should be used.
fn block_hash_at_slot<T>(
chain_tip_hash: &Hash256,
slot: u64,
block_store: &Arc<BeaconBlockStore<T>>,
) -> Result<Option<Hash256>, Error>
where
T: ClientDB + Sized,
{
match block_store.block_at_slot(&chain_tip_hash, slot)? {
None => Ok(None),
Some((hash_bytes, _)) => Ok(Some(Hash256::from(&hash_bytes[..]))),
}
}
impl From<BeaconBlockAtSlotError> for Error {
fn from(e: BeaconBlockAtSlotError) -> Self {
match e {
BeaconBlockAtSlotError::DBError(s) => Error::DBError(s),
_ => Error::UnableToLookupBlockAtSlot,
}
}
}

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@ -0,0 +1,64 @@
use super::bls::{AggregatePublicKey, AggregateSignature};
use super::db::stores::{ValidatorStore, ValidatorStoreError};
use super::db::ClientDB;
use super::types::{AttestationData, Bitfield, BitfieldError};
use super::{Error, Invalid, Outcome};
/// Validate that some signature is correct for some attestation data and known validator set.
pub fn validate_attestation_signature<T>(
attestation_data: &AttestationData,
participation_bitfield: &Bitfield,
aggregate_signature: &AggregateSignature,
attestation_indices: &[usize],
validator_store: &ValidatorStore<T>,
) -> Result<Outcome, Error>
where
T: ClientDB + Sized,
{
let mut agg_pub_key = AggregatePublicKey::new();
for i in 0..attestation_indices.len() {
let voted = participation_bitfield.get(i)?;
if voted {
// De-reference the attestation index into a canonical ValidatorRecord index.
let validator = *attestation_indices.get(i).ok_or(Error::BadValidatorIndex)?;
// Load the public key.
let pub_key = validator_store
.get_public_key_by_index(validator)?
.ok_or(Error::NoPublicKeyForValidator)?;
// Aggregate the public key.
agg_pub_key.add(&pub_key);
}
}
let signed_message = attestation_data_signing_message(attestation_data);
verify_or!(
// TODO: ensure "domain" for aggregate signatures is included.
// https://github.com/sigp/lighthouse/issues/91
aggregate_signature.verify(&signed_message, &agg_pub_key),
reject!(Invalid::SignatureInvalid)
);
accept!()
}
fn attestation_data_signing_message(attestation_data: &AttestationData) -> Vec<u8> {
let mut signed_message = attestation_data.canonical_root().to_vec();
signed_message.append(&mut vec![0]);
signed_message
}
impl From<ValidatorStoreError> for Error {
fn from(error: ValidatorStoreError) -> Self {
match error {
ValidatorStoreError::DBError(s) => Error::DBError(s),
ValidatorStoreError::DecodeError => Error::PublicKeyCorrupt,
}
}
}
impl From<BitfieldError> for Error {
fn from(_error: BitfieldError) -> Self {
Error::OutOfBoundsBitfieldIndex
}
}

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@ -0,0 +1,14 @@
[package]
name = "block_validation"
version = "0.1.0"
authors = ["Paul Hauner <paul@paulhauner.com>"]
[dependencies]
attestation_validation = { path = "../attestation_validation" }
bls = { path = "../utils/bls" }
db = { path = "../../lighthouse/db" }
hashing = { path = "../utils/hashing" }
rayon = "1.0.2"
ssz = { path = "../utils/ssz" }
ssz_helpers = { path = "../utils/ssz_helpers" }
types = { path = "../types" }

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@ -0,0 +1,371 @@
extern crate rayon;
use self::rayon::prelude::*;
use super::attestation_validation::{AttestationValidationContext, AttestationValidationError};
use super::db::stores::{BeaconBlockStore, PoWChainStore, ValidatorStore};
use super::db::{ClientDB, DBError};
use super::ssz::{Decodable, DecodeError};
use super::ssz_helpers::attestation_ssz_splitter::{
split_all_attestations, split_one_attestation, AttestationSplitError,
};
use super::ssz_helpers::ssz_beacon_block::{SszBeaconBlock, SszBeaconBlockError};
use super::types::Hash256;
use super::types::{AttestationRecord, AttesterMap, BeaconBlock, ProposerMap};
use std::sync::{Arc, RwLock};
#[derive(Debug, PartialEq)]
pub enum SszBeaconBlockValidationError {
FutureSlot,
SlotAlreadyFinalized,
UnknownPoWChainRef,
UnknownParentHash,
BadAttestationSsz,
BadAncestorHashesSsz,
BadSpecialsSsz,
ParentSlotHigherThanBlockSlot,
AttestationValidationError(AttestationValidationError),
AttestationSignatureFailed,
ProposerAttestationHasObliqueHashes,
NoProposerSignature,
BadProposerMap,
RwLockPoisoned,
DBError(String),
}
/// The context against which a block should be validated.
pub struct BeaconBlockValidationContext<T>
where
T: ClientDB + Sized,
{
/// The slot as determined by the system time.
pub present_slot: u64,
/// The cycle_length as determined by the chain configuration.
pub cycle_length: u8,
/// The last justified slot as per the client's view of the canonical chain.
pub last_justified_slot: u64,
/// The last justified block hash as per the client's view of the canonical chain.
pub last_justified_block_hash: Hash256,
/// The last finalized slot as per the client's view of the canonical chain.
pub last_finalized_slot: u64,
/// A vec of the hashes of the blocks preceeding the present slot.
pub recent_block_hashes: Arc<Vec<Hash256>>,
/// A map of slots to a block proposer validation index.
pub proposer_map: Arc<ProposerMap>,
/// A map of (slot, shard_id) to the attestation set of validation indices.
pub attester_map: Arc<AttesterMap>,
/// The store containing block information.
pub block_store: Arc<BeaconBlockStore<T>>,
/// The store containing validator information.
pub validator_store: Arc<ValidatorStore<T>>,
/// The store containing information about the proof-of-work chain.
pub pow_store: Arc<PoWChainStore<T>>,
}
impl<T> BeaconBlockValidationContext<T>
where
T: ClientDB,
{
/// Validate some SszBeaconBlock against a block validation context. An SszBeaconBlock varies from a BeaconBlock in
/// that is a read-only structure that reads directly from encoded SSZ.
///
/// The reason to validate an SzzBeaconBlock is to avoid decoding it in its entirety if there is
/// a suspicion that the block might be invalid. Such a suspicion should be applied to
/// all blocks coming from the network.
///
/// This function will determine if the block is new, already known or invalid (either
/// intrinsically or due to some application error.)
///
/// Note: this function does not implement randao_reveal checking as it is not in the
/// specification.
#[allow(dead_code)]
pub fn validate_ssz_block(
&self,
b: &SszBeaconBlock,
) -> Result<BeaconBlock, SszBeaconBlockValidationError>
where
T: ClientDB + Sized,
{
/*
* If the block slot corresponds to a slot in the future, return immediately with an error.
*
* It is up to the calling fn to determine what should be done with "future" blocks (e.g.,
* cache or discard).
*/
let block_slot = b.slot();
if block_slot > self.present_slot {
return Err(SszBeaconBlockValidationError::FutureSlot);
}
/*
* If the block is unknown (assumed unknown because we checked the db earlier in this
* function) and it comes from a slot that is already finalized, drop the block.
*
* If a slot is finalized, there's no point in considering any other blocks for that slot.
*
* TODO: We can more strongly throw away blocks based on the `last_finalized_block` related
* to this `last_finalized_slot`. Namely, any block in a future slot must include the
* `last_finalized_block` in it's chain.
*/
if block_slot <= self.last_finalized_slot {
return Err(SszBeaconBlockValidationError::SlotAlreadyFinalized);
}
/*
* If the PoW chain hash is not known to us, drop it.
*
* We only accept blocks that reference a known PoW hash.
*
* Note: it is not clear what a "known" PoW chain ref is. Likely it means the block hash is
* "sufficienty deep in the canonical PoW chain". This should be clarified as the spec
* crystallizes.
*/
let pow_chain_reference = b.pow_chain_reference();
if !self.pow_store.block_hash_exists(b.pow_chain_reference())? {
return Err(SszBeaconBlockValidationError::UnknownPoWChainRef);
}
/*
* Store a slice of the serialized attestations from the block SSZ.
*/
let attestations_ssz = &b.attestations_without_length();
/*
* Get a slice of the first serialized attestation (the 0'th) and decode it into
* a full AttestationRecord object.
*
* The first attestation must be validated separately as it must contain a signature of the
* proposer of the previous block (this is checked later in this function).
*/
let (first_attestation_ssz, next_index) = split_one_attestation(&attestations_ssz, 0)?;
let (first_attestation, _) = AttestationRecord::ssz_decode(&first_attestation_ssz, 0)?;
/*
* The first attestation may not have oblique hashes.
*
* The presence of oblique hashes in the first attestation would indicate that the proposer
* of the previous block is attesting to some other block than the one they produced.
*/
if !first_attestation.oblique_parent_hashes.is_empty() {
return Err(SszBeaconBlockValidationError::ProposerAttestationHasObliqueHashes);
}
/*
* Read the parent hash from the block we are validating then attempt to load
* that parent block ssz from the database.
*
* If that parent doesn't exist in the database or is invalid, reject the block.
*
* Also, read the slot from the parent block for later use.
*/
let parent_hash = b
.parent_hash()
.ok_or(SszBeaconBlockValidationError::BadAncestorHashesSsz)?;
let parent_block_slot = match self.block_store.get_serialized_block(&parent_hash)? {
None => return Err(SszBeaconBlockValidationError::UnknownParentHash),
Some(ssz) => {
let parent_block = SszBeaconBlock::from_slice(&ssz[..])?;
parent_block.slot()
}
};
/*
* The parent block slot must be less than the block slot.
*
* In other words, the parent must come before the child.
*/
if parent_block_slot >= block_slot {
return Err(SszBeaconBlockValidationError::ParentSlotHigherThanBlockSlot);
}
/*
* Generate the context in which attestations will be validated.
*/
let attestation_validation_context = Arc::new(AttestationValidationContext {
block_slot,
parent_block_slot,
cycle_length: self.cycle_length,
last_justified_slot: self.last_justified_slot,
recent_block_hashes: self.recent_block_hashes.clone(),
block_store: self.block_store.clone(),
validator_store: self.validator_store.clone(),
attester_map: self.attester_map.clone(),
});
/*
* Validate this first attestation.
*/
let attestation_voters =
attestation_validation_context.validate_attestation(&first_attestation)?;
/*
* Attempt to read load the parent block proposer from the proposer map. Return with an
* error if it fails.
*
* If the signature of proposer for the parent slot was not present in the first (0'th)
* attestation of this block, reject the block.
*/
let parent_block_proposer = self
.proposer_map
.get(&parent_block_slot)
.ok_or(SszBeaconBlockValidationError::BadProposerMap)?;
if !attestation_voters.contains(&parent_block_proposer) {
return Err(SszBeaconBlockValidationError::NoProposerSignature);
}
/*
* Split the remaining attestations into a vector of slices, each containing
* a single serialized attestation record.
*/
let other_attestations = split_all_attestations(attestations_ssz, next_index)?;
/*
* Verify each other AttestationRecord.
*
* This uses the `rayon` library to do "sometimes" parallelization. Put simply,
* if there are some spare threads, the verification of attestation records will happen
* concurrently.
*
* There is a thread-safe `failure` variable which is set whenever an attestation fails
* validation. This is so all attestation validation is halted if a single bad attestation
* is found.
*/
let failure: RwLock<Option<SszBeaconBlockValidationError>> = RwLock::new(None);
let mut deserialized_attestations: Vec<AttestationRecord> = other_attestations
.par_iter()
.filter_map(|attestation_ssz| {
/*
* If some thread has set the `failure` variable to `Some(error)` the abandon
* attestation serialization and validation. Also, fail early if the lock has been
* poisoned.
*/
match failure.read() {
Ok(ref option) if option.is_none() => (),
_ => return None,
}
/*
* If there has not been a failure yet, attempt to serialize and validate the
* attestation.
*/
match AttestationRecord::ssz_decode(&attestation_ssz, 0) {
/*
* Deserialization failed, therefore the block is invalid.
*/
Err(e) => {
/*
* If the failure lock isn't poisoned, set it to some error.
*/
if let Ok(mut f) = failure.write() {
*f = Some(SszBeaconBlockValidationError::from(e));
}
None
}
/*
* Deserialization succeeded and the attestation should be validated.
*/
Ok((attestation, _)) => {
match attestation_validation_context.validate_attestation(&attestation) {
/*
* Attestation validation failed with some error.
*/
Err(e) => {
/*
* If the failure lock isn't poisoned, set it to some error.
*/
if let Ok(mut f) = failure.write() {
*f = Some(SszBeaconBlockValidationError::from(e));
}
None
}
/*
* Attestation validation succeded.
*/
Ok(_) => Some(attestation),
}
}
}
}).collect();
match failure.into_inner() {
Err(_) => return Err(SszBeaconBlockValidationError::RwLockPoisoned),
Ok(failure) => match failure {
Some(error) => return Err(error),
_ => (),
},
}
/*
* Add the first attestation to the vec of deserialized attestations at
* index 0.
*/
deserialized_attestations.insert(0, first_attestation);
let (ancestor_hashes, _) = Decodable::ssz_decode(&b.ancestor_hashes(), 0)
.map_err(|_| SszBeaconBlockValidationError::BadAncestorHashesSsz)?;
let (specials, _) = Decodable::ssz_decode(&b.specials(), 0)
.map_err(|_| SszBeaconBlockValidationError::BadSpecialsSsz)?;
/*
* If we have reached this point, the block is a new valid block that is worthy of
* processing.
*/
let block = BeaconBlock {
slot: block_slot,
randao_reveal: Hash256::from(b.randao_reveal()),
pow_chain_reference: Hash256::from(pow_chain_reference),
ancestor_hashes,
active_state_root: Hash256::from(b.act_state_root()),
crystallized_state_root: Hash256::from(b.cry_state_root()),
attestations: deserialized_attestations,
specials,
};
Ok(block)
}
}
impl From<DBError> for SszBeaconBlockValidationError {
fn from(e: DBError) -> Self {
SszBeaconBlockValidationError::DBError(e.message)
}
}
impl From<AttestationSplitError> for SszBeaconBlockValidationError {
fn from(e: AttestationSplitError) -> Self {
match e {
AttestationSplitError::TooShort => SszBeaconBlockValidationError::BadAttestationSsz,
}
}
}
impl From<SszBeaconBlockError> for SszBeaconBlockValidationError {
fn from(e: SszBeaconBlockError) -> Self {
match e {
SszBeaconBlockError::TooShort => {
SszBeaconBlockValidationError::DBError("Bad parent block in db.".to_string())
}
SszBeaconBlockError::TooLong => {
SszBeaconBlockValidationError::DBError("Bad parent block in db.".to_string())
}
}
}
}
impl From<DecodeError> for SszBeaconBlockValidationError {
fn from(e: DecodeError) -> Self {
match e {
DecodeError::TooShort => SszBeaconBlockValidationError::BadAttestationSsz,
DecodeError::TooLong => SszBeaconBlockValidationError::BadAttestationSsz,
}
}
}
impl From<AttestationValidationError> for SszBeaconBlockValidationError {
fn from(e: AttestationValidationError) -> Self {
SszBeaconBlockValidationError::AttestationValidationError(e)
}
}
/*
* Tests for block validation are contained in the root directory "tests" directory (AKA
* "integration tests directory").
*/

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@ -0,0 +1,9 @@
extern crate attestation_validation;
extern crate bls;
extern crate db;
extern crate hashing;
extern crate ssz;
extern crate ssz_helpers;
extern crate types;
pub mod block_validation;

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@ -37,6 +37,12 @@ impl AttestationData {
justified_block_hash: Hash256::zero(),
}
}
// TODO: Implement this as a merkle root, once tree_ssz is implemented.
// https://github.com/sigp/lighthouse/issues/92
pub fn canonical_root(&self) -> Hash256 {
Hash256::zero()
}
}
impl Encodable for AttestationData {

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@ -2,6 +2,7 @@ use super::ValidatorRegistration;
#[derive(Debug, Clone, PartialEq)]
pub struct ChainConfig {
// Old, potentially outdated constants
pub cycle_length: u8,
pub deposit_size_gwei: u64,
pub shard_count: u16,
@ -10,6 +11,10 @@ pub struct ChainConfig {
pub genesis_time: u64,
pub slot_duration_millis: u64,
pub initial_validators: Vec<ValidatorRegistration>,
// New constants
pub epoch_length: u64,
pub min_attestation_inclusion_delay: u64,
}
/*
@ -28,6 +33,10 @@ impl ChainConfig {
genesis_time: TEST_GENESIS_TIME,
slot_duration_millis: 16 * 1000,
initial_validators: vec![],
// New
epoch_length: 64,
min_attestation_inclusion_delay: 4,
}
}
@ -54,6 +63,10 @@ impl ChainConfig {
genesis_time: TEST_GENESIS_TIME, // arbitrary
slot_duration_millis: 16 * 1000,
initial_validators: vec![],
// New constants
epoch_length: 64,
min_attestation_inclusion_delay: 4,
}
}
}

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@ -2,31 +2,16 @@ use super::Hash256;
#[derive(Clone, Debug, PartialEq)]
pub struct CrosslinkRecord {
pub recently_changed: bool,
pub slot: u64,
pub hash: Hash256,
pub shard_block_hash: Hash256,
}
impl CrosslinkRecord {
/// Generates a new instance where `dynasty` and `hash` are both zero.
pub fn zero() -> Self {
Self {
recently_changed: false,
slot: 0,
hash: Hash256::zero(),
shard_block_hash: Hash256::zero(),
}
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_crosslink_record_zero() {
let c = CrosslinkRecord::zero();
assert_eq!(c.recently_changed, false);
assert_eq!(c.slot, 0);
assert!(c.hash.is_zero());
}
}