Updated phase 1: commitments (#579)

See #338 and #529 for discussion.

specs/core/1_shard-data-chains.md

@@ -20,6 +20,7 @@ Phase 1 depends upon all of the constants defined in [Phase 0](0_beacon-chain.md
 |------------------------|-----------------|-------|---------------|
 | `SHARD_CHUNK_SIZE`     | 2**5 (= 32)     | bytes |               |
 | `SHARD_BLOCK_SIZE`     | 2**14 (= 16384) | bytes |               |
+| `CROSSLINK_LOOKBACK`   | 2**5 (= 32)     | slots |               |
 
 ### Flags, domains, etc.
 
@@ -98,26 +99,39 @@ A node should sign a crosslink only if the following conditions hold. **If a nod
 
 First, the conditions must recursively apply to the crosslink referenced in `last_crosslink_root` for the same shard (unless `last_crosslink_root` equals zero, in which case we are at the genesis).
 
-Second, we verify the `shard_block_combined_data_root`. Let `h` be the slot _immediately after_ the slot of the shard block included by the last crosslink, and `h+n-1` be the slot number of the block directly referenced by the current `shard_block_root`. Let `B[i]` be the block at slot `h+i` in the shard chain. Let `bodies[0] .... bodies[n-1]` be the bodies of these blocks and `roots[0] ... roots[n-1]` the data roots. If there is a missing slot in the shard chain at position `h+i`, then `bodies[i] == b'\x00' * shard_block_maxbytes(state[i])` and `roots[i]` be the Merkle root of the empty data. Define `compute_merkle_root` be a simple Merkle root calculating function that takes as input a list of objects, where the list's length must be an exact power of two. We define the function for computing the combined data root as follows:
+Second, we verify the `shard_chain_commitment`.
+* Let `start_slot = state.latest_crosslinks[shard].epoch * EPOCH_LENGTH + EPOCH_LENGTH - CROSSLINK_LOOKBACK`.
+* Let `end_slot = attestation.data.slot - attestation.data.slot % EPOCH_LENGTH - CROSSLINK_LOOKBACK`.
+* Let `length = end_slot - start_slot`, `headers[0] .... headers[length-1]` be the serialized block headers in the canonical shard chain from the verifer's point of view (note that this implies that `headers` and `bodies` have been checked for validity).
+* Let `bodies[0] ... bodies[length-1]` be the bodies of the blocks.
+* Note: If there is a missing slot, then the header and body are the same as that of the block at the most recent slot that has a block.
+
+We define two helpers:
 
 ```python
-ZERO_ROOT = merkle_root(bytes([0] * SHARD_BLOCK_SIZE))
-
-def mk_combined_data_root(roots):
-    data = roots + [ZERO_ROOT for _ in range(len(roots), next_power_of_2(len(roots)))]
-    return compute_merkle_root(data)
+def pad_to_power_of_2(values: List[bytes]) -> List[bytes]:
+    while not is_power_of_two(len(values)):
+        values = values + [SHARD_BLOCK_SIZE]
+    return values
 ```
 
-This outputs the root of a tree of the data roots, with the data roots all adjusted to have the same height if needed. The tree can also be viewed as a tree of all of the underlying data concatenated together, appropriately padded. Here is an equivalent definition that uses bodies instead of roots [TODO: check equivalence]:
-
 ```python
-def mk_combined_data_root(depths, bodies):
-    data = b''.join(bodies)
-    data += bytes([0] * (next_power_of_2(len(data)) - len(data))
-    return compute_merkle_root([data[pos:pos+SHARD_CHUNK_SIZE] for pos in range(0, len(data), SHARD_CHUNK_SIZE)])
+def merkle_root_of_bytes(data: bytes) -> bytes:
+    return merkle_root([data[i:i+32] for i in range(0, len(data), 32)])
 ```
 
-Verify that the `shard_block_combined_data_root` is the output of these functions.
+We define the function for computing the commitment as follows:
+
+```python
+def compute_commitment(headers: List[ShardBlock], bodies: List[bytes]) -> Bytes32:
+    return hash(
+        merkle_root(pad_to_power_of_2([merkle_root_of_bytes(zpad(serialize(h), SHARD_BLOCK_SIZE)) for h in headers])),
+        merkle_root(pad_to_power_of_2([merkle_root_of_bytes(h) for h in bodies]))
+    )
+```
+
+The `shard_chain_commitment` is only valid if it equals `compute_commitment(headers, bodies)`.
+
 
 ### Shard block fork choice rule