From 8ebc4dece847bb87daf3aa4ce6974ff37b60b2c2 Mon Sep 17 00:00:00 2001
From: Anton Nashatyrev <anton.nashatyrev@gmail.com>
Date: Fri, 11 Jan 2019 15:07:18 +0300
Subject: [PATCH 01/29] Local var should stay uint

---
 specs/core/0_beacon-chain.md | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 22e8c6dfe..c1668bd98 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -921,9 +921,10 @@ def get_shard_committees_at_slot(state: BeaconState,
     """
     Returns the ``ShardCommittee`` for the ``slot``.
     """
-    earliest_slot_in_array = state.slot - (state.slot % EPOCH_LENGTH) - EPOCH_LENGTH
-    assert earliest_slot_in_array <= slot < earliest_slot_in_array + EPOCH_LENGTH * 2
-    return state.shard_committees_at_slots[slot - earliest_slot_in_array]
+    state_epoch_slot = state.slot - (state.slot % EPOCH_LENGTH) 
+    assert state_epoch_slot <= slot + EPOCH_LENGTH
+    assert slot < state_epoch_slot + EPOCH_LENGTH
+    return state.shard_committees_at_slots[slot - state_epoch_slot + EPOCH_LENGTH]
 ```
 
 #### `get_block_root`

From 23e3f4defe5cd10aa239094768fc9489a126a9ea Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:30:04 +0800
Subject: [PATCH 02/29] Rename `bytes1(x)`, `bytes2(x)`... function to
 `int_to_bytes1(x)`, `int_to_bytes2(x)`...

---
 specs/core/0_beacon-chain.md      | 14 +++++++-------
 specs/core/1_shard-data-chains.md |  2 +-
 2 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 2757d89a7..92c300b0d 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -75,7 +75,7 @@
             - [`get_beacon_proposer_index`](#get_beacon_proposer_index)
             - [`merkle_root`](#merkle_root)
             - [`get_attestation_participants`](#get_attestation_participants)
-            - [`bytes1`, `bytes2`, ...](#bytes1-bytes2-)
+            - [`int_to_bytes1`, `int_to_bytes2`, ...](#int_to_bytes1-int_to_bytes2-)
             - [`get_effective_balance`](#get_effective_balance)
             - [`get_fork_version`](#get_fork_version)
             - [`get_domain`](#get_domain)
@@ -190,9 +190,9 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 | `GENESIS_SLOT` | `0` |
 | `GENESIS_START_SHARD` | `0` |
 | `FAR_FUTURE_SLOT` | `2**64 - 1` |
-| `ZERO_HASH` | `bytes32(0)` |
-| `EMPTY_SIGNATURE` | `[bytes48(0), bytes48(0)]` |
-| `BLS_WITHDRAWAL_PREFIX_BYTE` | `bytes1(0)` |
+| `ZERO_HASH` | `int_to_bytes32(0)` |
+| `EMPTY_SIGNATURE` | `[int_to_bytes48(0), int_to_bytes48(0)]` |
+| `BLS_WITHDRAWAL_PREFIX_BYTE` | `int_to_bytes1(0)` |
 
 ### Time parameters
 
@@ -894,7 +894,7 @@ def get_shuffling(seed: Hash32,
     committees_per_slot = get_committee_count_per_slot(len(active_validator_indices))
 
     # Shuffle
-    seed = xor(seed, bytes32(slot))
+    seed = xor(seed, int_to_bytes32(slot))
     shuffled_active_validator_indices = shuffle(active_validator_indices, seed)
 
     # Split the shuffled list into epoch_length * committees_per_slot pieces
@@ -1046,9 +1046,9 @@ def get_attestation_participants(state: BeaconState,
     return participants
 ```
 
-#### `bytes1`, `bytes2`, ...
+#### `int_to_bytes1`, `int_to_bytes2`, ...
 
-`bytes1(x): return x.to_bytes(1, 'big')`, `bytes2(x): return x.to_bytes(2, 'big')`, and so on for all integers, particularly 1, 2, 3, 4, 8, 32.
+`int_to_bytes1(x): return x.to_bytes(1, 'big')`, `int_to_bytes2(x): return x.to_bytes(2, 'big')`, and so on for all integers, particularly 1, 2, 3, 4, 8, 32.
 
 #### `get_effective_balance`
 
diff --git a/specs/core/1_shard-data-chains.md b/specs/core/1_shard-data-chains.md
index 973a0d332..cabe2934e 100644
--- a/specs/core/1_shard-data-chains.md
+++ b/specs/core/1_shard-data-chains.md
@@ -70,7 +70,7 @@ To validate a block header on shard `shard_id`, compute as follows:
 * Verify that `beacon_chain_ref` is the hash of a block in the beacon chain with slot less than or equal to `slot`. Verify that `beacon_chain_ref` is equal to or a descendant of the `beacon_chain_ref` specified in the `ShardBlock` pointed to by `parent_root`.
 * Let `state` be the state of the beacon chain block referred to by `beacon_chain_ref`. Let `validators` be `[validators[i] for i in state.current_persistent_committees[shard_id]]`.
 * Assert `len(participation_bitfield) == ceil_div8(len(validators))`
-* Let `proposer_index = hash(state.randao_mix + bytes8(shard_id) + bytes8(slot)) % len(validators)`. Let `msg` be the block but with the `block.signature` set to `[0, 0]`. Verify that `BLSVerify(pub=validators[proposer_index].pubkey, msg=hash(msg), sig=block.signature, domain=get_domain(state, slot, SHARD_PROPOSER_DOMAIN))` passes.
+* Let `proposer_index = hash(state.randao_mix + int_to_bytes8(shard_id) + int_to_bytes8(slot)) % len(validators)`. Let `msg` be the block but with the `block.signature` set to `[0, 0]`. Verify that `BLSVerify(pub=validators[proposer_index].pubkey, msg=hash(msg), sig=block.signature, domain=get_domain(state, slot, SHARD_PROPOSER_DOMAIN))` passes.
 * Generate the `group_public_key` by adding the public keys of all the validators for whom the corresponding position in the bitfield is set to 1. Verify that `BLSVerify(pub=group_public_key, msg=parent_root, sig=block.aggregate_signature, domain=get_domain(state, slot, SHARD_ATTESTER_DOMAIN))` passes.
 
 ### Block Merklization helper

From accd3cfd01a5ebe55440490a45fc46b826fb8050 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:33:59 +0800
Subject: [PATCH 03/29] Replace `hash32` with `bytes32`

---
 specs/core/0_beacon-chain.md | 56 ++++++++++++++++++------------------
 1 file changed, 28 insertions(+), 28 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 92c300b0d..bc612473b 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -328,17 +328,17 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Shard number
     'shard': 'uint64',
     # Hash of root of the signed beacon block
-    'beacon_block_root': 'hash32',
+    'beacon_block_root': 'bytes32',
     # Hash of root of the ancestor at the epoch boundary
-    'epoch_boundary_root': 'hash32',
+    'epoch_boundary_root': 'bytes32',
     # Shard block's hash of root
-    'shard_block_root': 'hash32',
+    'shard_block_root': 'bytes32',
     # Last crosslink's hash of root
-    'latest_crosslink_root': 'hash32',
+    'latest_crosslink_root': 'bytes32',
     # Slot of the last justified beacon block
     'justified_slot': 'uint64',
     # Hash of the last justified beacon block
-    'justified_block_root': 'hash32',
+    'justified_block_root': 'bytes32',
 }
 ```
 
@@ -360,7 +360,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 ```python
 {
     # Branch in the deposit tree
-    'branch': '[hash32]',
+    'branch': '[bytes32]',
     # Index in the deposit tree
     'index': 'uint64',
     # Data
@@ -388,11 +388,11 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # BLS pubkey
     'pubkey': 'uint384',
     # Withdrawal credentials
-    'withdrawal_credentials': 'hash32',
+    'withdrawal_credentials': 'bytes32',
     # Initial RANDAO commitment
-    'randao_commitment': 'hash32',
+    'randao_commitment': 'bytes32',
     # Initial custody commitment
-    'custody_commitment': 'hash32',
+    'custody_commitment': 'bytes32',
     # A BLS signature of this `DepositInput`
     'proof_of_possession': ['uint384'],
 }
@@ -421,10 +421,10 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 {
     ## Header ##
     'slot': 'uint64',
-    'parent_root': 'hash32',
-    'state_root': 'hash32',
-    'randao_reveal': 'hash32',
-    'deposit_root': 'hash32',
+    'parent_root': 'bytes32',
+    'state_root': 'bytes32',
+    'randao_reveal': 'bytes32',
+    'deposit_root': 'bytes32',
     'signature': ['uint384'],
 
     ## Body ##
@@ -458,7 +458,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Shard number (`BEACON_CHAIN_SHARD_NUMBER` for beacon chain)
     'shard': 'uint64',
     # Block's hash of root
-    'block_root': 'hash32',
+    'block_root': 'bytes32',
 }
 ```
 
@@ -478,17 +478,17 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     'validator_balances': ['uint64'],
     'validator_registry_latest_change_slot': 'uint64',
     'validator_registry_exit_count': 'uint64',
-    'validator_registry_delta_chain_tip': 'hash32',  # For light clients to track deltas
+    'validator_registry_delta_chain_tip': 'bytes32',  # For light clients to track deltas
 
     # Randomness and committees
-    'latest_randao_mixes': ['hash32'],
-    'latest_vdf_outputs': ['hash32'],
+    'latest_randao_mixes': ['bytes32'],
+    'latest_vdf_outputs': ['bytes32'],
     'previous_epoch_start_shard': 'uint64',
     'current_epoch_start_shard': 'uint64',
     'previous_epoch_calculation_slot': 'uint64',
     'current_epoch_calculation_slot': 'uint64',
-    'previous_epoch_randao_mix': 'hash32',
-    'current_epoch_randao_mix': 'hash32',
+    'previous_epoch_randao_mix': 'bytes32',
+    'current_epoch_randao_mix': 'bytes32',
 
     # Custody challenges
     'custody_challenges': [CustodyChallenge],
@@ -501,13 +501,13 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 
     # Recent state
     'latest_crosslinks': [CrosslinkRecord],
-    'latest_block_roots': ['hash32'],  # Needed to process attestations, older to newer
+    'latest_block_roots': ['bytes32'],  # Needed to process attestations, older to newer
     'latest_penalized_exit_balances': ['uint64'],  # Balances penalized at every withdrawal period
     'latest_attestations': [PendingAttestationRecord],
-    'batched_block_roots': ['hash32'],
+    'batched_block_roots': ['bytes32'],
 
     # Ethereum 1.0 deposit root
-    'latest_deposit_root': 'hash32',
+    'latest_deposit_root': 'bytes32',
     'deposit_root_votes': [DepositRootVote],
 }
 ```
@@ -519,9 +519,9 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # BLS public key
     'pubkey': 'uint384',
     # Withdrawal credentials
-    'withdrawal_credentials': 'hash32',
+    'withdrawal_credentials': 'bytes32',
     # RANDAO commitment
-    'randao_commitment': 'hash32',
+    'randao_commitment': 'bytes32',
     # Slots the proposer has skipped (i.e. layers of RANDAO expected)
     'randao_layers': 'uint64',
     # Slot when validator activated
@@ -537,7 +537,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Status flags
     'status_flags': 'uint64',
     # Custody commitment
-    'custody_commitment': 'hash32',
+    'custody_commitment': 'bytes32',
     # Slot of latest custody reseed
     'latest_custody_reseed_slot': 'uint64',
     # Slot of second-latest custody reseed
@@ -552,7 +552,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Slot number
     'slot': 'uint64',
     # Shard block root
-    'shard_block_root': 'hash32',
+    'shard_block_root': 'bytes32',
 }
 ```
 
@@ -561,7 +561,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 ```python
 {
     # Deposit root
-    'deposit_root': 'hash32',
+    'deposit_root': 'bytes32',
     # Vote count
     'vote_count': 'uint64',
 }
@@ -599,7 +599,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 
 ```python
 {
-    latest_registry_delta_root: 'hash32',
+    latest_registry_delta_root: 'bytes32',
     validator_index: 'uint24',
     pubkey: 'uint384',
     slot: 'uint64',

From 8d5fd404a76d0d879b07582bcd1459183d6a3fa6 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:42:46 +0800
Subject: [PATCH 04/29] Add `Signature` SSZ obj

---
 specs/core/0_beacon-chain.md | 13 +++++++++++++
 1 file changed, 13 insertions(+)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index bc612473b..cbca62b97 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -21,6 +21,8 @@
         - [Validator registry delta flags](#validator-registry-delta-flags)
         - [Signature domains](#signature-domains)
     - [Data structures](#data-structures)
+        - [Signatures](#signatures)
+            - [`Signature`](#signature)
         - [Beacon chain operations](#beacon-chain-operations)
             - [Proposer slashings](#proposer-slashings)
                 - [`ProposerSlashing`](#proposerslashing)
@@ -253,6 +255,17 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 
 ## Data structures
 
+### Signatures
+
+#### `Signature`
+
+```python
+{
+    'z_1': 'bytes48',
+    'z_2': 'bytes48',
+}
+```
+
 ### Beacon chain operations
 
 #### Proposer slashings

From b515683e391197becf4cd788bf48de0a73147b6b Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:44:23 +0800
Subject: [PATCH 05/29] Replace `'[uint384]'` with Signature

---
 specs/core/0_beacon-chain.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index cbca62b97..cf284b169 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -279,11 +279,11 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # First proposal data
     'proposal_data_1': ProposalSignedData,
     # First proposal signature
-    'proposal_signature_1': '[uint384]',
+    'proposal_signature_1': Signature,
     # Second proposal data
     'proposal_data_2': ProposalSignedData,
     # Second proposal signature
-    'proposal_signature_2': '[uint384]',
+    'proposal_signature_2': Signature,
 }
 ```
 
@@ -422,7 +422,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Index of the exiting validator
     'validator_index': 'uint24',
     # Validator signature
-    'signature': '[uint384]',
+    'signature': Signature,
 }
 ```
 

From 7005e5fd55f007ff0c15a518036def669716f77b Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:49:08 +0800
Subject: [PATCH 06/29] Replace `['uint384']` with `Signature`

---
 specs/core/0_beacon-chain.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index cf284b169..7873a5aa4 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -311,7 +311,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Attestation data
     'data': AttestationData,
     # Aggregate signature
-    'aggregate_signature': ['uint384'],
+    'aggregate_signature': Signature,
 }
 ```
 
@@ -328,7 +328,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Custody bitfield
     'custody_bitfield': 'bytes',
     # BLS aggregate signature
-    'aggregate_signature': ['uint384'],
+    'aggregate_signature': Signature,
 }
 ```
 
@@ -407,7 +407,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Initial custody commitment
     'custody_commitment': 'bytes32',
     # A BLS signature of this `DepositInput`
-    'proof_of_possession': ['uint384'],
+    'proof_of_possession': Signature,
 }
 ```
 
@@ -438,7 +438,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     'state_root': 'bytes32',
     'randao_reveal': 'bytes32',
     'deposit_root': 'bytes32',
-    'signature': ['uint384'],
+    'signature': Signature,
 
     ## Body ##
     'body': BeaconBlockBody,

From a588643661563d5419bdbf450819e3aeeb1c2a07 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:51:50 +0800
Subject: [PATCH 07/29] Replace `uint384` with `bytes48` for pubkey

---
 specs/core/0_beacon-chain.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 7873a5aa4..bc8fcfa7c 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -399,7 +399,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 ```python
 {
     # BLS pubkey
-    'pubkey': 'uint384',
+    'pubkey': 'bytes48',
     # Withdrawal credentials
     'withdrawal_credentials': 'bytes32',
     # Initial RANDAO commitment
@@ -530,7 +530,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 ```python
 {
     # BLS public key
-    'pubkey': 'uint384',
+    'pubkey': 'bytes48',
     # Withdrawal credentials
     'withdrawal_credentials': 'bytes32',
     # RANDAO commitment
@@ -614,7 +614,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 {
     latest_registry_delta_root: 'bytes32',
     validator_index: 'uint24',
-    pubkey: 'uint384',
+    pubkey: 'bytes48',
     slot: 'uint64',
     flag: 'uint64',
 }

From 14891047d71263e34f82ee5a1e1841b1a511d5b9 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:52:56 +0800
Subject: [PATCH 08/29] Update `EMPTY_SIGNATURE`

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index bc8fcfa7c..89036acb9 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -193,7 +193,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 | `GENESIS_START_SHARD` | `0` |
 | `FAR_FUTURE_SLOT` | `2**64 - 1` |
 | `ZERO_HASH` | `int_to_bytes32(0)` |
-| `EMPTY_SIGNATURE` | `[int_to_bytes48(0), int_to_bytes48(0)]` |
+| `EMPTY_SIGNATURE` | `Signature(int_to_bytes48(0), int_to_bytes48(0))` |
 | `BLS_WITHDRAWAL_PREFIX_BYTE` | `int_to_bytes1(0)` |
 
 ### Time parameters

From 61f5d99dee28b9e974f8d66e8bd9387bb20dca03 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:54:38 +0800
Subject: [PATCH 09/29] Fix `Deposit.branch` type notation

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 89036acb9..b584bd4be 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -373,7 +373,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 ```python
 {
     # Branch in the deposit tree
-    'branch': '[bytes32]',
+    'branch': ['bytes32'],
     # Index in the deposit tree
     'index': 'uint64',
     # Data

From 0b10b0444a4e6c4f5bf548344796df76dd6d5657 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 16:56:33 +0800
Subject: [PATCH 10/29] Replace `Hash32` with `Bytes32`

---
 specs/core/0_beacon-chain.md | 26 +++++++++++++-------------
 1 file changed, 13 insertions(+), 13 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index b584bd4be..d3ffa7632 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -810,7 +810,7 @@ def get_active_validator_indices(validators: [ValidatorRecord], slot: int) -> Li
 #### `shuffle`
 
 ```python
-def shuffle(values: List[Any], seed: Hash32) -> List[Any]:
+def shuffle(values: List[Any], seed: Bytes32) -> List[Any]:
     """
     Returns the shuffled ``values`` with ``seed`` as entropy.
     """
@@ -890,7 +890,7 @@ def get_committee_count_per_slot(active_validator_count: int) -> int:
 #### `get_shuffling`
 
 ```python
-def get_shuffling(seed: Hash32,
+def get_shuffling(seed: Bytes32,
                   validators: List[ValidatorRecord],
                   slot: int) -> List[List[int]]
     """
@@ -984,7 +984,7 @@ def get_crosslink_committees_at_slot(state: BeaconState,
 
 ```python
 def get_block_root(state: BeaconState,
-                   slot: int) -> Hash32:
+                   slot: int) -> Bytes32:
     """
     Returns the block root at a recent ``slot``.
     """
@@ -999,7 +999,7 @@ def get_block_root(state: BeaconState,
 
 ```python
 def get_randao_mix(state: BeaconState,
-                   slot: int) -> Hash32:
+                   slot: int) -> Bytes32:
     """
     Returns the randao mix at a recent ``slot``.
     """
@@ -1023,7 +1023,7 @@ def get_beacon_proposer_index(state: BeaconState,
 #### `merkle_root`
 
 ```python
-def merkle_root(values: List[Hash32]) -> Hash32:
+def merkle_root(values: List[Bytes32]) -> Bytes32:
     """
     Merkleize ``values`` (where ``len(values)`` is a power of two) and return the Merkle root.
     """
@@ -1217,7 +1217,7 @@ A valid block with slot `GENESIS_SLOT` (a "genesis block") has the following val
 ```python
 def get_initial_beacon_state(initial_validator_deposits: List[Deposit],
                              genesis_time: int,
-                             latest_deposit_root: Hash32) -> BeaconState:
+                             latest_deposit_root: Bytes32) -> BeaconState:
     state = BeaconState(
         # Misc
         slot=GENESIS_SLOT,
@@ -1294,9 +1294,9 @@ First, a helper function:
 def validate_proof_of_possession(state: BeaconState,
                                  pubkey: int,
                                  proof_of_possession: bytes,
-                                 withdrawal_credentials: Hash32,
-                                 randao_commitment: Hash32,
-                                 custody_commitment: Hash32) -> bool:
+                                 withdrawal_credentials: Bytes32,
+                                 randao_commitment: Bytes32,
+                                 custody_commitment: Bytes32) -> bool:
     proof_of_possession_data = DepositInput(
         pubkey=pubkey,
         withdrawal_credentials=withdrawal_credentials,
@@ -1324,9 +1324,9 @@ def process_deposit(state: BeaconState,
                     pubkey: int,
                     amount: int,
                     proof_of_possession: bytes,
-                    withdrawal_credentials: Hash32,
-                    randao_commitment: Hash32,
-                    custody_commitment: Hash32) -> None:
+                    withdrawal_credentials: Bytes32,
+                    randao_commitment: Bytes32,
+                    custody_commitment: Bytes32) -> None:
     """
     Process a deposit from Ethereum 1.0.
     Note that this function mutates ``state``.
@@ -1550,7 +1550,7 @@ For each `deposit` in `block.body.deposits`:
 * Verify that `verify_merkle_branch(hash(serialized_deposit_data), deposit.branch, DEPOSIT_CONTRACT_TREE_DEPTH, deposit.index, state.latest_deposit_root)` is `True`.
 
 ```python
-def verify_merkle_branch(leaf: Hash32, branch: [Hash32], depth: int, index: int, root: Hash32) -> bool:
+def verify_merkle_branch(leaf: Bytes32, branch: [Bytes32], depth: int, index: int, root: Bytes32) -> bool:
     value = leaf
     for i in range(depth):
         if index // (2**i) % 2:

From 725199593bdee4389a9ded72b1c25054830485d2 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 17:00:39 +0800
Subject: [PATCH 11/29] For pubkey type hinting, use `bytes` for now.

---
 specs/core/0_beacon-chain.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index d3ffa7632..7a12c6a2a 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1292,7 +1292,7 @@ First, a helper function:
 
 ```python
 def validate_proof_of_possession(state: BeaconState,
-                                 pubkey: int,
+                                 pubkey: bytes,
                                  proof_of_possession: bytes,
                                  withdrawal_credentials: Bytes32,
                                  randao_commitment: Bytes32,
@@ -1321,7 +1321,7 @@ Now, to add a [validator](#dfn-validator) or top up an existing [validator](#dfn
 
 ```python
 def process_deposit(state: BeaconState,
-                    pubkey: int,
+                    pubkey: bytes,
                     amount: int,
                     proof_of_possession: bytes,
                     withdrawal_credentials: Bytes32,

From 713af88c4311edb1b3ce23c231ddc8aac70f7b58 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 17:03:20 +0800
Subject: [PATCH 12/29] `proof_of_possession: Signature`

---
 specs/core/0_beacon-chain.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 7a12c6a2a..56207a4ca 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1293,7 +1293,7 @@ First, a helper function:
 ```python
 def validate_proof_of_possession(state: BeaconState,
                                  pubkey: bytes,
-                                 proof_of_possession: bytes,
+                                 proof_of_possession: Signature,
                                  withdrawal_credentials: Bytes32,
                                  randao_commitment: Bytes32,
                                  custody_commitment: Bytes32) -> bool:
@@ -1323,7 +1323,7 @@ Now, to add a [validator](#dfn-validator) or top up an existing [validator](#dfn
 def process_deposit(state: BeaconState,
                     pubkey: bytes,
                     amount: int,
-                    proof_of_possession: bytes,
+                    proof_of_possession: Signature,
                     withdrawal_credentials: Bytes32,
                     randao_commitment: Bytes32,
                     custody_commitment: Bytes32) -> None:

From 209220787b4fbfb8300b685dfd37bfa681fe7a76 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 17:29:28 +0800
Subject: [PATCH 13/29] Deal with BLS APIs, will define `Bytes48` in the type
 hinting cleanup

---
 specs/bls_signature.md       |  8 ++++----
 specs/core/0_beacon-chain.md | 25 ++++++++++++++++++-------
 2 files changed, 22 insertions(+), 11 deletions(-)

diff --git a/specs/bls_signature.md b/specs/bls_signature.md
index 8e0b76c6b..75827982b 100644
--- a/specs/bls_signature.md
+++ b/specs/bls_signature.md
@@ -106,11 +106,11 @@ def modular_squareroot(value: int) -> int:
 
 ### `bls_aggregate_pubkeys`
 
-Let `bls_aggregate_pubkeys(pubkeys: [uint384]) -> uint384` return `pubkeys[0] + .... + pubkeys[len(pubkeys)-1]`, where `+` is the elliptic curve addition operation over the G1 curve.
+Let `bls_aggregate_pubkeys(pubkeys: List[Bytes48]) -> Bytes48` return `pubkeys[0] + .... + pubkeys[len(pubkeys)-1]`, where `+` is the elliptic curve addition operation over the G1 curve.
 
 ### `bls_aggregate_signatures`
 
-Let `bls_aggregate_signatures(signatures: [[uint384]]) -> [uint384]` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
+Let `bls_aggregate_signatures(signatures: Tuple[Bytes48, Bytes48]) -> Tuple[Bytes48, Bytes48]` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
 
 ## Signature verification
 
@@ -124,7 +124,7 @@ g = Fq2(g_x, g_y)
 
 ### `bls_verify`
 
-Let `bls_verify(pubkey: uint384, message: bytes32, signature: [uint384], domain: uint64) -> bool`:
+Let `bls_verify(pubkey: Bytes48, message: Bytes32, signature: Tuple[Bytes48, Bytes48], domain: uint64) -> bool`:
 
 * Verify that `pubkey` is a valid G1 point.
 * Verify that `signature` is a valid G2 point.
@@ -132,7 +132,7 @@ Let `bls_verify(pubkey: uint384, message: bytes32, signature: [uint384], domain:
 
 ### `bls_verify_multiple`
 
-Let `bls_verify_multiple(pubkeys: [uint384], messages: [bytes32], signature: [uint384], domain: uint64) -> bool`:
+Let `bls_verify_multiple(pubkeys: List[Bytes48], messages: List[Bytes32], signature: Tuple[Bytes48, Bytes48], domain: uint64) -> bool`:
 
 * Verify that each `pubkey` in `pubkeys` is a valid G1 point.
 * Verify that `signature` is a valid G2 point.
diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 56207a4ca..17e9ca24f 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -85,6 +85,7 @@
             - [`is_double_vote`](#is_double_vote)
             - [`is_surround_vote`](#is_surround_vote)
             - [`integer_squareroot`](#integer_squareroot)
+            - [`signature_to_tuple`](#signature_to_tuple)
             - [`bls_verify`](#bls_verify)
             - [`bls_verify_multiple`](#bls_verify_multiple)
             - [`bls_aggregate_pubkeys`](#bls_aggregate_pubkeys)
@@ -1112,7 +1113,7 @@ def verify_slashable_vote_data(state: BeaconState, vote_data: SlashableVoteData)
             hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, False)),
             hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, True)),
         ],
-        signature=vote_data.aggregate_signature,
+        signature=signature_to_tuple(vote_data.aggregate_signature),
         domain=get_domain(
             state.fork_data,
             state.slot,
@@ -1175,6 +1176,16 @@ def integer_squareroot(n: int) -> int:
     return x
 ```
 
+#### `signature_to_tuple`
+
+```python
+def signature_to_tuple(signature: Signature) -> Tuple[bytes, bytes]:
+    """
+    Convert SSZ object ``signature`` to ``tuple`` type for BLS APIs.
+    """
+    return (signature.z_1, signature.z_2)
+```
+
 #### `bls_verify`
 
 `bls_verify` is a function for verifying a BLS signature, defined in the [BLS Signature spec](https://github.com/ethereum/eth2.0-specs/blob/master/specs/bls_signature.md#bls_verify).
@@ -1308,7 +1319,7 @@ def validate_proof_of_possession(state: BeaconState,
     return bls_verify(
         pubkey=pubkey,
         message=hash_tree_root(proof_of_possession_data),
-        signature=proof_of_possession,
+        signature=signature_to_tuple(proof_of_possession),
         domain=get_domain(
             state.fork_data,
             state.slot,
@@ -1469,7 +1480,7 @@ Below are the processing steps that happen at every `block`.
 
 * Let `block_without_signature_root` be the `hash_tree_root` of `block` where `block.signature` is set to `EMPTY_SIGNATURE`.
 * Let `proposal_root = hash_tree_root(ProposalSignedData(state.slot, BEACON_CHAIN_SHARD_NUMBER, block_without_signature_root))`.
-* Verify that `bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, message=proposal_root, signature=block.signature, domain=get_domain(state.fork_data, state.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, message=proposal_root, signature=signature_to_tuple(block.signature), domain=get_domain(state.fork_data, state.slot, DOMAIN_PROPOSAL))`.
 
 ### RANDAO
 
@@ -1498,8 +1509,8 @@ For each `proposer_slashing` in `block.body.proposer_slashings`:
 * Verify that `proposer_slashing.proposal_data_1.shard == proposer_slashing.proposal_data_2.shard`.
 * Verify that `proposer_slashing.proposal_data_1.block_root != proposer_slashing.proposal_data_2.block_root`.
 * Verify that `proposer.penalized_slot > state.slot`.
-* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=proposer_slashing.proposal_signature_1, domain=get_domain(state.fork_data, proposer_slashing.proposal_data_1.slot, DOMAIN_PROPOSAL))`.
-* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=proposer_slashing.proposal_signature_2, domain=get_domain(state.fork_data, proposer_slashing.proposal_data_2.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=signature_to_tuple(proposer_slashing.proposal_signature_1), domain=get_domain(state.fork_data, proposer_slashing.proposal_data_1.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=signature_to_tuple(proposer_slashing.proposal_signature_2), domain=get_domain(state.fork_data, proposer_slashing.proposal_data_2.slot, DOMAIN_PROPOSAL))`.
 * Run `penalize_validator(state, proposer_slashing.proposer_index)`.
 
 #### Casper slashings
@@ -1533,7 +1544,7 @@ For each `attestation` in `block.body.attestations`:
 * `aggregate_signature` verification:
     * Let `participants = get_attestation_participants(state, attestation.data, attestation.participation_bitfield)`.
     * Let `group_public_key = bls_aggregate_pubkeys([state.validator_registry[v].pubkey for v in participants])`.
-    * Verify that `bls_verify(pubkey=group_public_key, message=hash_tree_root(AttestationDataAndCustodyBit(attestation.data, False)), signature=attestation.aggregate_signature, domain=get_domain(state.fork_data, attestation.data.slot, DOMAIN_ATTESTATION))`.
+    * Verify that `bls_verify(pubkey=group_public_key, message=hash_tree_root(AttestationDataAndCustodyBit(attestation.data, False)), signature=signature_to_tuple(attestation.aggregate_signature), domain=get_domain(state.fork_data, attestation.data.slot, DOMAIN_ATTESTATION))`.
 * [TO BE REMOVED IN PHASE 1] Verify that `attestation.data.shard_block_root == ZERO_HASH`.
 * Append `PendingAttestationRecord(data=attestation.data, participation_bitfield=attestation.participation_bitfield, custody_bitfield=attestation.custody_bitfield, slot_included=state.slot)` to `state.latest_attestations`.
 
@@ -1583,7 +1594,7 @@ For each `exit` in `block.body.exits`:
 * Let `validator = state.validator_registry[exit.validator_index]`.
 * Verify that `validator.exit_slot > state.slot + ENTRY_EXIT_DELAY`.
 * Verify that `state.slot >= exit.slot`.
-* Verify that `bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=exit.signature, domain=get_domain(state.fork_data, exit.slot, DOMAIN_EXIT))`.
+* Verify that `bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=signature_to_tuple(exit.signature), domain=get_domain(state.fork_data, exit.slot, DOMAIN_EXIT))`.
 * Run `initiate_validator_exit(state, exit.validator_index)`.
 
 #### Custody

From ba7bc29ce9308d8f0407dc34d128fa63fc53eeb4 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 17:36:31 +0800
Subject: [PATCH 14/29] Fix `bls_aggregate_signatures`

---
 specs/bls_signature.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/bls_signature.md b/specs/bls_signature.md
index 75827982b..bbb40e7c8 100644
--- a/specs/bls_signature.md
+++ b/specs/bls_signature.md
@@ -110,7 +110,7 @@ Let `bls_aggregate_pubkeys(pubkeys: List[Bytes48]) -> Bytes48` return `pubkeys[0
 
 ### `bls_aggregate_signatures`
 
-Let `bls_aggregate_signatures(signatures: Tuple[Bytes48, Bytes48]) -> Tuple[Bytes48, Bytes48]` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
+Let `bls_aggregate_signatures(signatures: List[Tuple[Bytes48, Bytes48]]) -> Tuple[Bytes48, Bytes48]` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
 
 ## Signature verification
 

From 18b97cf922b9c0d2035c8e3e014e4087e9dd3f6f Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Thu, 17 Jan 2019 19:40:26 +0800
Subject: [PATCH 15/29] pubkey: Bytes48

---
 specs/core/0_beacon-chain.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 17e9ca24f..981344116 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1303,7 +1303,7 @@ First, a helper function:
 
 ```python
 def validate_proof_of_possession(state: BeaconState,
-                                 pubkey: bytes,
+                                 pubkey: Bytes48,
                                  proof_of_possession: Signature,
                                  withdrawal_credentials: Bytes32,
                                  randao_commitment: Bytes32,
@@ -1332,7 +1332,7 @@ Now, to add a [validator](#dfn-validator) or top up an existing [validator](#dfn
 
 ```python
 def process_deposit(state: BeaconState,
-                    pubkey: bytes,
+                    pubkey: Bytes48,
                     amount: int,
                     proof_of_possession: Signature,
                     withdrawal_credentials: Bytes32,

From 4d05d3385856c73eaca5b3d6f5c894236ea1b9a0 Mon Sep 17 00:00:00 2001
From: jannikluhn <jannik@brainbot.com>
Date: Thu, 17 Jan 2019 18:35:12 +0100
Subject: [PATCH 16/29] Add links to accompanying documents to readme

---
 README.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/README.md b/README.md
index 9563c97ed..169e00cee 100644
--- a/README.md
+++ b/README.md
@@ -12,6 +12,11 @@ Core specifications for eth2.0 client validation can be found in [specs/core](sp
 * [Phase 0 -- The Beacon Chain](specs/core/0_beacon-chain.md)
 * [Phase 1 -- Shard Data Chains](specs/core/1_shard-data-chains.md)
 
+Accompanying documents can be found in [specs](specs) and include
+* [SimpleSerialize (SSZ) spec](specs/simple_serialize.md)
+* [BLS signature verification](specs/bls_signature.md)
+* [General test format](specs/test-format.md)
+
 ## Design goals
 The following are the broad design goals for Ethereum 2.0:
 * to minimize complexity, even at the cost of some losses in efficiency

From 3ada70c2f00c78135c06b5f0877ec7259408fdc1 Mon Sep 17 00:00:00 2001
From: jannikluhn <jannik@brainbot.com>
Date: Thu, 17 Jan 2019 18:39:58 +0100
Subject: [PATCH 17/29] Fix link

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index 169e00cee..f3aba3f64 100644
--- a/README.md
+++ b/README.md
@@ -13,7 +13,7 @@ Core specifications for eth2.0 client validation can be found in [specs/core](sp
 * [Phase 1 -- Shard Data Chains](specs/core/1_shard-data-chains.md)
 
 Accompanying documents can be found in [specs](specs) and include
-* [SimpleSerialize (SSZ) spec](specs/simple_serialize.md)
+* [SimpleSerialize (SSZ) spec](specs/simple-serialize.md)
 * [BLS signature verification](specs/bls_signature.md)
 * [General test format](specs/test-format.md)
 

From 91eba49eebae8f5f95b950c907182a6687e503bb Mon Sep 17 00:00:00 2001
From: Danny Ryan <dannyjryan@gmail.com>
Date: Thu, 17 Jan 2019 13:00:29 -0600
Subject: [PATCH 18/29] fix slot for domain in verify_slashable_vote_data
 (#465)

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 0ba9b5bcc..6f970f8be 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1113,7 +1113,7 @@ def verify_slashable_vote_data(state: BeaconState, vote_data: SlashableVoteData)
         signature=vote_data.aggregate_signature,
         domain=get_domain(
             state.fork,
-            state.slot,
+            vote_data.data.slot,
             DOMAIN_ATTESTATION,
         ),
     )

From 063f6586971f54a2583d22683a7e659a5f7dbd52 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Fri, 18 Jan 2019 03:19:38 +0800
Subject: [PATCH 19/29] `bytes96` for Signature

---
 specs/bls_signature.md       |  6 ++--
 specs/core/0_beacon-chain.md | 58 +++++++++++-------------------------
 2 files changed, 20 insertions(+), 44 deletions(-)

diff --git a/specs/bls_signature.md b/specs/bls_signature.md
index bbb40e7c8..11264c251 100644
--- a/specs/bls_signature.md
+++ b/specs/bls_signature.md
@@ -110,7 +110,7 @@ Let `bls_aggregate_pubkeys(pubkeys: List[Bytes48]) -> Bytes48` return `pubkeys[0
 
 ### `bls_aggregate_signatures`
 
-Let `bls_aggregate_signatures(signatures: List[Tuple[Bytes48, Bytes48]]) -> Tuple[Bytes48, Bytes48]` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
+Let `bls_aggregate_signatures(signatures: List[Bytes96]) -> Bytes96` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve.
 
 ## Signature verification
 
@@ -124,7 +124,7 @@ g = Fq2(g_x, g_y)
 
 ### `bls_verify`
 
-Let `bls_verify(pubkey: Bytes48, message: Bytes32, signature: Tuple[Bytes48, Bytes48], domain: uint64) -> bool`:
+Let `bls_verify(pubkey: Bytes48, message: Bytes32, signature: Bytes96, domain: uint64) -> bool`:
 
 * Verify that `pubkey` is a valid G1 point.
 * Verify that `signature` is a valid G2 point.
@@ -132,7 +132,7 @@ Let `bls_verify(pubkey: Bytes48, message: Bytes32, signature: Tuple[Bytes48, Byt
 
 ### `bls_verify_multiple`
 
-Let `bls_verify_multiple(pubkeys: List[Bytes48], messages: List[Bytes32], signature: Tuple[Bytes48, Bytes48], domain: uint64) -> bool`:
+Let `bls_verify_multiple(pubkeys: List[Bytes48], messages: List[Bytes32], signature: Bytes96, domain: uint64) -> bool`:
 
 * Verify that each `pubkey` in `pubkeys` is a valid G1 point.
 * Verify that `signature` is a valid G2 point.
diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index f67c4b896..f97c19ac3 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -21,8 +21,6 @@
         - [Validator registry delta flags](#validator-registry-delta-flags)
         - [Signature domains](#signature-domains)
     - [Data structures](#data-structures)
-        - [Signatures](#signatures)
-            - [`Signature`](#signature)
         - [Beacon chain operations](#beacon-chain-operations)
             - [Proposer slashings](#proposer-slashings)
                 - [`ProposerSlashing`](#proposerslashing)
@@ -86,7 +84,6 @@
             - [`is_double_vote`](#is_double_vote)
             - [`is_surround_vote`](#is_surround_vote)
             - [`integer_squareroot`](#integer_squareroot)
-            - [`signature_to_tuple`](#signature_to_tuple)
             - [`bls_verify`](#bls_verify)
             - [`bls_verify_multiple`](#bls_verify_multiple)
             - [`bls_aggregate_pubkeys`](#bls_aggregate_pubkeys)
@@ -194,7 +191,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 | `GENESIS_START_SHARD` | `0` |
 | `FAR_FUTURE_SLOT` | `2**64 - 1` |
 | `ZERO_HASH` | `int_to_bytes32(0)` |
-| `EMPTY_SIGNATURE` | `Signature(int_to_bytes48(0), int_to_bytes48(0))` |
+| `EMPTY_SIGNATURE` | `int_to_bytes96(0)` |
 | `BLS_WITHDRAWAL_PREFIX_BYTE` | `int_to_bytes1(0)` |
 
 ### Time parameters
@@ -256,17 +253,6 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
 
 ## Data structures
 
-### Signatures
-
-#### `Signature`
-
-```python
-{
-    'z_1': 'bytes48',
-    'z_2': 'bytes48',
-}
-```
-
 ### Beacon chain operations
 
 #### Proposer slashings
@@ -280,11 +266,11 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # First proposal data
     'proposal_data_1': ProposalSignedData,
     # First proposal signature
-    'proposal_signature_1': Signature,
+    'proposal_signature_1': 'bytes96',
     # Second proposal data
     'proposal_data_2': ProposalSignedData,
     # Second proposal signature
-    'proposal_signature_2': Signature,
+    'proposal_signature_2': 'bytes96',
 }
 ```
 
@@ -312,7 +298,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Attestation data
     'data': AttestationData,
     # Aggregate signature
-    'aggregate_signature': Signature,
+    'aggregate_signature': 'bytes96',
 }
 ```
 
@@ -329,7 +315,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Custody bitfield
     'custody_bitfield': 'bytes',
     # BLS aggregate signature
-    'aggregate_signature': Signature,
+    'aggregate_signature': 'bytes96',
 }
 ```
 
@@ -408,7 +394,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Initial custody commitment
     'custody_commitment': 'bytes32',
     # A BLS signature of this `DepositInput`
-    'proof_of_possession': Signature,
+    'proof_of_possession': 'bytes96',
 }
 ```
 
@@ -423,7 +409,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     # Index of the exiting validator
     'validator_index': 'uint24',
     # Validator signature
-    'signature': Signature,
+    'signature': 'bytes96',
 }
 ```
 
@@ -439,7 +425,7 @@ Unless otherwise indicated, code appearing in `this style` is to be interpreted
     'state_root': 'bytes32',
     'randao_reveal': 'bytes32',
     'eth1_data': Eth1Data,
-    'signature': Signature,
+    'signature': 'bytes96',
 
     ## Body ##
     'body': BeaconBlockBody,
@@ -1124,7 +1110,7 @@ def verify_slashable_vote_data(state: BeaconState, vote_data: SlashableVoteData)
             hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, False)),
             hash_tree_root(AttestationDataAndCustodyBit(vote_data.data, True)),
         ],
-        signature=signature_to_tuple(vote_data.aggregate_signature),
+        signature=vote_data.aggregate_signature,
         domain=get_domain(
             state.fork,
             state.slot,
@@ -1187,16 +1173,6 @@ def integer_squareroot(n: int) -> int:
     return x
 ```
 
-#### `signature_to_tuple`
-
-```python
-def signature_to_tuple(signature: Signature) -> Tuple[bytes, bytes]:
-    """
-    Convert SSZ object ``signature`` to ``tuple`` type for BLS APIs.
-    """
-    return (signature.z_1, signature.z_2)
-```
-
 #### `bls_verify`
 
 `bls_verify` is a function for verifying a BLS signature, defined in the [BLS Signature spec](https://github.com/ethereum/eth2.0-specs/blob/master/specs/bls_signature.md#bls_verify).
@@ -1318,7 +1294,7 @@ First, a helper function:
 ```python
 def validate_proof_of_possession(state: BeaconState,
                                  pubkey: Bytes48,
-                                 proof_of_possession: Signature,
+                                 proof_of_possession: Bytes96,
                                  withdrawal_credentials: Bytes32,
                                  randao_commitment: Bytes32,
                                  custody_commitment: Bytes32) -> bool:
@@ -1333,7 +1309,7 @@ def validate_proof_of_possession(state: BeaconState,
     return bls_verify(
         pubkey=pubkey,
         message=hash_tree_root(proof_of_possession_data),
-        signature=signature_to_tuple(proof_of_possession),
+        signature=proof_of_possession,
         domain=get_domain(
             state.fork,
             state.slot,
@@ -1348,7 +1324,7 @@ Now, to add a [validator](#dfn-validator) or top up an existing [validator](#dfn
 def process_deposit(state: BeaconState,
                     pubkey: Bytes48,
                     amount: int,
-                    proof_of_possession: Signature,
+                    proof_of_possession: Bytes96,
                     withdrawal_credentials: Bytes32,
                     randao_commitment: Bytes32,
                     custody_commitment: Bytes32) -> None:
@@ -1494,7 +1470,7 @@ Below are the processing steps that happen at every `block`.
 
 * Let `block_without_signature_root` be the `hash_tree_root` of `block` where `block.signature` is set to `EMPTY_SIGNATURE`.
 * Let `proposal_root = hash_tree_root(ProposalSignedData(state.slot, BEACON_CHAIN_SHARD_NUMBER, block_without_signature_root))`.
-* Verify that `bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, message=proposal_root, signature=signature_to_tuple(block.signature), domain=get_domain(state.fork, state.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=state.validator_registry[get_beacon_proposer_index(state, state.slot)].pubkey, message=proposal_root, signature=block.signature, domain=get_domain(state.fork, state.slot, DOMAIN_PROPOSAL))`.
 
 ### RANDAO
 
@@ -1523,8 +1499,8 @@ For each `proposer_slashing` in `block.body.proposer_slashings`:
 * Verify that `proposer_slashing.proposal_data_1.shard == proposer_slashing.proposal_data_2.shard`.
 * Verify that `proposer_slashing.proposal_data_1.block_root != proposer_slashing.proposal_data_2.block_root`.
 * Verify that `proposer.penalized_slot > state.slot`.
-* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=signature_to_tuple(proposer_slashing.proposal_signature_1), domain=get_domain(state.fork, proposer_slashing.proposal_data_1.slot, DOMAIN_PROPOSAL))`.
-* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=signature_to_tuple(proposer_slashing.proposal_signature_2), domain=get_domain(state.fork, proposer_slashing.proposal_data_2.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_1), signature=proposer_slashing.proposal_signature_1, domain=get_domain(state.fork, proposer_slashing.proposal_data_1.slot, DOMAIN_PROPOSAL))`.
+* Verify that `bls_verify(pubkey=proposer.pubkey, message=hash_tree_root(proposer_slashing.proposal_data_2), signature=proposer_slashing.proposal_signature_2, domain=get_domain(state.fork, proposer_slashing.proposal_data_2.slot, DOMAIN_PROPOSAL))`.
 * Run `penalize_validator(state, proposer_slashing.proposer_index)`.
 
 #### Casper slashings
@@ -1558,7 +1534,7 @@ For each `attestation` in `block.body.attestations`:
 * `aggregate_signature` verification:
     * Let `participants = get_attestation_participants(state, attestation.data, attestation.aggregation_bitfield)`.
     * Let `group_public_key = bls_aggregate_pubkeys([state.validator_registry[v].pubkey for v in participants])`.
-    * Verify that `bls_verify(pubkey=group_public_key, message=hash_tree_root(AttestationDataAndCustodyBit(attestation.data, False)), signature=signature_to_tuple(attestation.aggregate_signature), domain=get_domain(state.fork, attestation.data.slot, DOMAIN_ATTESTATION))`.
+    * Verify that `bls_verify(pubkey=group_public_key, message=hash_tree_root(AttestationDataAndCustodyBit(attestation.data, False)), signature=attestation.aggregate_signature, domain=get_domain(state.fork, attestation.data.slot, DOMAIN_ATTESTATION))`.
 * [TO BE REMOVED IN PHASE 1] Verify that `attestation.data.shard_block_root == ZERO_HASH`.
 * Append `PendingAttestation(data=attestation.data, aggregation_bitfield=attestation.aggregation_bitfield, custody_bitfield=attestation.custody_bitfield, slot_included=state.slot)` to `state.latest_attestations`.
 
@@ -1608,7 +1584,7 @@ For each `exit` in `block.body.exits`:
 * Let `validator = state.validator_registry[exit.validator_index]`.
 * Verify that `validator.exit_slot > state.slot + ENTRY_EXIT_DELAY`.
 * Verify that `state.slot >= exit.slot`.
-* Verify that `bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=signature_to_tuple(exit.signature), domain=get_domain(state.fork, exit.slot, DOMAIN_EXIT))`.
+* Verify that `bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=exit.signature, domain=get_domain(state.fork, exit.slot, DOMAIN_EXIT))`.
 * Run `initiate_validator_exit(state, exit.validator_index)`.
 
 #### Custody

From c8597a0d49b1ea82592e53149746493fb36c9e66 Mon Sep 17 00:00:00 2001
From: Dean Eigenmann <dean@tokenate.io>
Date: Fri, 18 Jan 2019 02:11:54 +0100
Subject: [PATCH 20/29] Update 0_beacon-chain.md

---
 specs/core/0_beacon-chain.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 6f970f8be..38a973365 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1606,7 +1606,7 @@ All [validators](#dfn-validator):
 
 * Let `current_epoch_attestations = [a for a in state.latest_attestations if state.slot - EPOCH_LENGTH <= a.data.slot < state.slot]`. (Note: this is the set of attestations of slots in the epoch `state.slot-EPOCH_LENGTH...state.slot-1`, _not_ attestations that got included in the chain during the epoch `state.slot-EPOCH_LENGTH...state.slot-1`.)
 * Validators justifying the epoch boundary block at the start of the current epoch:
-  * Let `current_epoch_boundary_attestations = [a for a in current_epoch_attestations if a.data.epoch_boundary_root == get_block_root(state, state.slot-EPOCH_LENGTH) and a.justified_slot == state.justified_slot]`.
+  * Let `current_epoch_boundary_attestations = [a for a in current_epoch_attestations if a.data.epoch_boundary_root == get_block_root(state, state.slot-EPOCH_LENGTH) and a.data.justified_slot == state.justified_slot]`.
   * Let `current_epoch_boundary_attester_indices` be the union of the [validator](#dfn-validator) index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_boundary_attestations]`.
   * Let `current_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in current_epoch_boundary_attester_indices])`.
 
@@ -1616,7 +1616,7 @@ All [validators](#dfn-validator):
   * Let `previous_epoch_attestations = [a for a in state.latest_attestations if state.slot - 2 * EPOCH_LENGTH <= a.slot < state.slot - EPOCH_LENGTH]`.
   * Let `previous_epoch_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_attestations]`.
 * Validators targeting the previous justified slot:
-  * Let `previous_epoch_justified_attestations = [a for a in current_epoch_attestations + previous_epoch_attestations if a.justified_slot == state.previous_justified_slot]`.
+  * Let `previous_epoch_justified_attestations = [a for a in current_epoch_attestations + previous_epoch_attestations if a.data.justified_slot == state.previous_justified_slot]`.
   * Let `previous_epoch_justified_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_justified_attestations]`.
   * Let `previous_epoch_justified_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_justified_attester_indices])`.
 * Validators justifying the epoch boundary block at the start of the previous epoch:

From 96aac450b5b0360ac22c9d5d03527b5d9c8d762f Mon Sep 17 00:00:00 2001
From: Dean Eigenmann <dean@tokenate.io>
Date: Fri, 18 Jan 2019 02:22:12 +0100
Subject: [PATCH 21/29] Update 0_beacon-chain.md

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 6f970f8be..2f3c3461e 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1620,7 +1620,7 @@ All [validators](#dfn-validator):
   * Let `previous_epoch_justified_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_justified_attestations]`.
   * Let `previous_epoch_justified_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_justified_attester_indices])`.
 * Validators justifying the epoch boundary block at the start of the previous epoch:
-  * Let `previous_epoch_boundary_attestations = [a for a in previous_epoch_justified_attestations if a.epoch_boundary_root == get_block_root(state, state.slot - 2 * EPOCH_LENGTH)]`.
+  * Let `previous_epoch_boundary_attestations = [a for a in previous_epoch_justified_attestations if a.data.epoch_boundary_root == get_block_root(state, state.slot - 2 * EPOCH_LENGTH)]`.
   * Let `previous_epoch_boundary_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_boundary_attestations]`.
   * Let `previous_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_boundary_attester_indices])`.
 * Validators attesting to the expected beacon chain head during the previous epoch:

From 15fdea5b1bf70338436fc4913df0368c3de6babd Mon Sep 17 00:00:00 2001
From: Dean Eigenmann <dean@tokenate.io>
Date: Fri, 18 Jan 2019 02:36:37 +0100
Subject: [PATCH 22/29] `a.beacon_block_root` -> `a.data.beacon_block_root`

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 6f970f8be..b630cb0a0 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1624,7 +1624,7 @@ All [validators](#dfn-validator):
   * Let `previous_epoch_boundary_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_boundary_attestations]`.
   * Let `previous_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_boundary_attester_indices])`.
 * Validators attesting to the expected beacon chain head during the previous epoch:
-  * Let `previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.beacon_block_root == get_block_root(state, a.slot)]`.
+  * Let `previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.data.beacon_block_root == get_block_root(state, a.slot)]`.
   * Let `previous_epoch_head_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_head_attestations]`.
   * Let `previous_epoch_head_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_head_attester_indices])`.
 

From 690cf4b787585d9d0dea92a49b604a87dec0fe2c Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hwwang156@gmail.com>
Date: Fri, 18 Jan 2019 16:21:58 +0800
Subject: [PATCH 23/29] PR feedback

---
 specs/core/0_beacon-chain.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index f97c19ac3..006c4a54a 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1059,7 +1059,7 @@ def get_attestation_participants(state: BeaconState,
 
 #### `int_to_bytes1`, `int_to_bytes2`, ...
 
-`int_to_bytes1(x): return x.to_bytes(1, 'big')`, `int_to_bytes2(x): return x.to_bytes(2, 'big')`, and so on for all integers, particularly 1, 2, 3, 4, 8, 32.
+`int_to_bytes1(x): return x.to_bytes(1, 'big')`, `int_to_bytes2(x): return x.to_bytes(2, 'big')`, and so on for all integers, particularly 1, 2, 3, 4, 8, 32, 48, 96.
 
 #### `get_effective_balance`
 

From a7307b9236b95d607b24cb617d0502555f77ae1e Mon Sep 17 00:00:00 2001
From: Alex Stokes <r.alex.stokes@gmail.com>
Date: Fri, 18 Jan 2019 02:10:23 -0800
Subject: [PATCH 24/29] Sign full `Exit` message so that a validator can
 prevent tampering (#468)

---
 specs/core/0_beacon-chain.md | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 98df49eb9..002e53f97 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1585,7 +1585,8 @@ For each `exit` in `block.body.exits`:
 * Let `validator = state.validator_registry[exit.validator_index]`.
 * Verify that `validator.exit_slot > state.slot + ENTRY_EXIT_DELAY`.
 * Verify that `state.slot >= exit.slot`.
-* Verify that `bls_verify(pubkey=validator.pubkey, message=ZERO_HASH, signature=exit.signature, domain=get_domain(state.fork, exit.slot, DOMAIN_EXIT))`.
+* Let `exit_message = hash_tree_root(Exit(slot=exit.slot, validator_index=exit.validator_index, signature=EMPTY_SIGNATURE))`.
+* Verify that `bls_verify(pubkey=validator.pubkey, message=exit_message, signature=exit.signature, domain=get_domain(state.fork, exit.slot, DOMAIN_EXIT))`.
 * Run `initiate_validator_exit(state, exit.validator_index)`.
 
 #### Custody

From b493f850fce367b7818192caade9a85739ae1558 Mon Sep 17 00:00:00 2001
From: Danny Ryan <dannyjryan@gmail.com>
Date: Fri, 18 Jan 2019 09:40:15 -0600
Subject: [PATCH 25/29] cleanup crosslink committees iters

---
 specs/core/0_beacon-chain.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 002e53f97..272002d02 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1632,7 +1632,7 @@ All [validators](#dfn-validator):
 
 **Note**: `previous_epoch_boundary_attesting_balance` balance might be marginally different than `current_epoch_boundary_attesting_balance` during the previous epoch transition. Due to the tight bound on validator churn each epoch and small per-epoch rewards/penalties, the potential balance difference is very low and only marginally affects consensus safety.
 
-For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committee_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committee_at_slot`, compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * Let `shard_block_root` be `state.latest_crosslinks[shard].shard_block_root`
 * Let `attesting_validator_indices(crosslink_committee, shard_block_root)` be the union of the [validator](#dfn-validator) index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_attestations + previous_epoch_attestations if a.shard == shard and a.shard_block_root == shard_block_root]`.
@@ -1668,7 +1668,7 @@ Set `state.finalized_slot = state.previous_justified_slot` if any of the followi
 
 ### Crosslinks
 
-For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committee_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committee_at_slot`, compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * Set `state.latest_crosslinks[shard] = Crosslink(slot=state.slot, shard_block_root=winning_root(crosslink_committee))` if `3 * total_attesting_balance(crosslink_committee) >= 2 * total_balance(crosslink_committee)`.
 
@@ -1714,7 +1714,7 @@ For each `index` in `previous_epoch_attester_indices`, we determine the proposer
 
 #### Crosslinks
 
-For every `i in range(state.slot - 2 * EPOCH_LENGTH, state.slot - EPOCH_LENGTH)`, let `crosslink_committee_at_slot, start_shard = get_crosslink_committees_at_slot(i)`. For every `j in range(len(crosslink_committee_at_slot))`, let `crosslink_committee = crosslink_committee_at_slot[j]`, `shard = (start_shard + j) % SHARD_COUNT`, and compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot - EPOCH_LENGTH)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * If `index in attesting_validators(crosslink_committee)`, `state.validator_balances[index] += base_reward(state, index) * total_attesting_balance(crosslink_committee) // total_balance(crosslink_committee))`.
 * If `index not in attesting_validators(crosslink_committee)`, `state.validator_balances[index] -= base_reward(state, index)`.

From aa29b9a347ff9006a91aef493dd8d9fb135b7823 Mon Sep 17 00:00:00 2001
From: Danny Ryan <dannyjryan@gmail.com>
Date: Fri, 18 Jan 2019 09:55:32 -0600
Subject: [PATCH 26/29] add missing state param to calls to
 get_crosslink_committees_at_slots

---
 specs/core/0_beacon-chain.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 272002d02..5640340b6 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1632,7 +1632,7 @@ All [validators](#dfn-validator):
 
 **Note**: `previous_epoch_boundary_attesting_balance` balance might be marginally different than `current_epoch_boundary_attesting_balance` during the previous epoch transition. Due to the tight bound on validator churn each epoch and small per-epoch rewards/penalties, the potential balance difference is very low and only marginally affects consensus safety.
 
-For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * Let `shard_block_root` be `state.latest_crosslinks[shard].shard_block_root`
 * Let `attesting_validator_indices(crosslink_committee, shard_block_root)` be the union of the [validator](#dfn-validator) index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_attestations + previous_epoch_attestations if a.shard == shard and a.shard_block_root == shard_block_root]`.
@@ -1668,7 +1668,7 @@ Set `state.finalized_slot = state.previous_justified_slot` if any of the followi
 
 ### Crosslinks
 
-For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * Set `state.latest_crosslinks[shard] = Crosslink(slot=state.slot, shard_block_root=winning_root(crosslink_committee))` if `3 * total_attesting_balance(crosslink_committee) >= 2 * total_balance(crosslink_committee)`.
 
@@ -1714,7 +1714,7 @@ For each `index` in `previous_epoch_attester_indices`, we determine the proposer
 
 #### Crosslinks
 
-For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot - EPOCH_LENGTH)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
+For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot - EPOCH_LENGTH)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * If `index in attesting_validators(crosslink_committee)`, `state.validator_balances[index] += base_reward(state, index) * total_attesting_balance(crosslink_committee) // total_balance(crosslink_committee))`.
 * If `index not in attesting_validators(crosslink_committee)`, `state.validator_balances[index] -= base_reward(state, index)`.

From 8d5c75c6fded531f162a0265a40ba079ed36f93e Mon Sep 17 00:00:00 2001
From: Jonny Rhea <jonathan.rhea@gmail.com>
Date: Fri, 18 Jan 2019 15:39:28 -0600
Subject: [PATCH 27/29] Update bls_signature.md (#473)

fq2 param should be a list
---
 specs/bls_signature.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/bls_signature.md b/specs/bls_signature.md
index 8e0b76c6b..9dd5cac8b 100644
--- a/specs/bls_signature.md
+++ b/specs/bls_signature.md
@@ -119,7 +119,7 @@ In the following `e` is the pairing function and `g` is the G1 generator with th
 ```python
 g_x = 3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507
 g_y = 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569
-g = Fq2(g_x, g_y)
+g = Fq2([g_x, g_y])
 ```
 
 ### `bls_verify`

From 89f0543ab9ed5853017bdd0f83288737120223fe Mon Sep 17 00:00:00 2001
From: Danny Ryan <dannyjryan@gmail.com>
Date: Fri, 18 Jan 2019 17:13:23 -0700
Subject: [PATCH 28/29] add some missing `data` calls

---
 specs/core/0_beacon-chain.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 5640340b6..9ba2df843 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1615,7 +1615,7 @@ All [validators](#dfn-validator):
 [Validators](#dfn-Validator) attesting during the previous epoch:
 
 * Validators that made an attestation during the previous epoch:
-  * Let `previous_epoch_attestations = [a for a in state.latest_attestations if state.slot - 2 * EPOCH_LENGTH <= a.slot < state.slot - EPOCH_LENGTH]`.
+  * Let `previous_epoch_attestations = [a for a in state.latest_attestations if state.slot - 2 * EPOCH_LENGTH <= a.data.slot < state.slot - EPOCH_LENGTH]`.
   * Let `previous_epoch_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_attestations]`.
 * Validators targeting the previous justified slot:
   * Let `previous_epoch_justified_attestations = [a for a in current_epoch_attestations + previous_epoch_attestations if a.data.justified_slot == state.previous_justified_slot]`.
@@ -1626,7 +1626,7 @@ All [validators](#dfn-validator):
   * Let `previous_epoch_boundary_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_boundary_attestations]`.
   * Let `previous_epoch_boundary_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_boundary_attester_indices])`.
 * Validators attesting to the expected beacon chain head during the previous epoch:
-  * Let `previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.data.beacon_block_root == get_block_root(state, a.slot)]`.
+  * Let `previous_epoch_head_attestations = [a for a in previous_epoch_attestations if a.data.beacon_block_root == get_block_root(state, a.data.slot)]`.
   * Let `previous_epoch_head_attester_indices` be the union of the validator index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in previous_epoch_head_attestations]`.
   * Let `previous_epoch_head_attesting_balance = sum([get_effective_balance(state, i) for i in previous_epoch_head_attester_indices])`.
 
@@ -1635,7 +1635,7 @@ All [validators](#dfn-validator):
 For every `slot in range(state.slot - 2 * EPOCH_LENGTH, state.slot)`, let `crosslink_committees_at_slot = get_crosslink_committees_at_slot(state, slot)`. For every `(crosslink_committee, shard)` in `crosslink_committees_at_slot`, compute:
 
 * Let `shard_block_root` be `state.latest_crosslinks[shard].shard_block_root`
-* Let `attesting_validator_indices(crosslink_committee, shard_block_root)` be the union of the [validator](#dfn-validator) index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_attestations + previous_epoch_attestations if a.shard == shard and a.shard_block_root == shard_block_root]`.
+* Let `attesting_validator_indices(crosslink_committee, shard_block_root)` be the union of the [validator](#dfn-validator) index sets given by `[get_attestation_participants(state, a.data, a.aggregation_bitfield) for a in current_epoch_attestations + previous_epoch_attestations if a.data.shard == shard and a.data.shard_block_root == shard_block_root]`.
 * Let `winning_root(crosslink_committee)` be equal to the value of `shard_block_root` such that `sum([get_effective_balance(state, i) for i in attesting_validator_indices(crosslink_committee, shard_block_root)])` is maximized (ties broken by favoring lower `shard_block_root` values).
 * Let `attesting_validators(crosslink_committee)` be equal to `attesting_validator_indices(crosslink_committee, winning_root(crosslink_committee))` for convenience.
 * Let `total_attesting_balance(crosslink_committee) = sum([get_effective_balance(state, i) for i in attesting_validators(crosslink_committee)])`.

From cc18df11851092ed92160e8d010b1d56f6179854 Mon Sep 17 00:00:00 2001
From: Danny Ryan <dannyjryan@gmail.com>
Date: Fri, 18 Jan 2019 18:45:15 -0600
Subject: [PATCH 29/29] remove unused variable def

---
 specs/core/0_beacon-chain.md | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 5640340b6..ce8c55ea6 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -1363,8 +1363,7 @@ def process_deposit(state: BeaconState,
             penultimate_custody_reseed_slot=GENESIS_SLOT,
         )
 
-        # Note: In phase 2 registry indices that has been withdrawn for a long time will be recycled.
-        index = len(state.validator_registry)
+        # Note: In phase 2 registry indices that have been withdrawn for a long time will be recycled.
         state.validator_registry.append(validator)
         state.validator_balances.append(amount)
     else: