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solo-chains/docs/bridge/etbereum-to-solo-chain.md

@@ -22,57 +22,26 @@
 - 2/3rds quorum needs to be reached
 - For now sync committee members are not slashed if the vote for a malicious header (although subject to change after EIP-7657)
 
+
+### Definitions and abreviations
+- SC: Current Sync Committee
+- NSC: Next Sync Committee
+- Attested header: the header for which we have the signatures by the sync committee
+- Finalized header: the latest finalized header, can be different to attested header (e.g., the latest finalized block is 5 but we have sync committee signatures for 7)
+- Slot: an interval of 12s
+- Period: A range of slots, in this case 8192 slots (around 27 hours)
 ### Verification PseudoCode
 
-```python
-def validate_light_client_update(snapshot: LightClientSnapshot,
-                                 update: LightClientUpdate,
-                                 genesis_validators_root: Root) -> None:
-    # Verify update slot is larger than snapshot slot
-    assert update.header.slot > snapshot.header.slot
-
-    # Verify update does not skip a sync committee period
-    snapshot_period = compute_epoch_at_slot(snapshot.header.slot) // EPOCHS_PER_SYNC_COMMITTEE_PERIOD
-    update_period = compute_epoch_at_slot(update.header.slot) // EPOCHS_PER_SYNC_COMMITTEE_PERIOD
-    assert update_period in (snapshot_period, snapshot_period + 1)
-
-    # Verify update header root is the finalized root of the finality header, if specified
-    if update.finality_header == BeaconBlockHeader():
-        signed_header = update.header
-        assert update.finality_branch == [Bytes32() for _ in range(floorlog2(FINALIZED_ROOT_INDEX))]
-    else:
-        signed_header = update.finality_header
-        assert is_valid_merkle_branch(
-            leaf=hash_tree_root(update.header),
-            branch=update.finality_branch,
-            depth=floorlog2(FINALIZED_ROOT_INDEX),
-            index=get_subtree_index(FINALIZED_ROOT_INDEX),
-            root=update.finality_header.state_root,
-        )
-
-    # Verify update next sync committee if the update period incremented
-    if update_period == snapshot_period:
-        sync_committee = snapshot.current_sync_committee
-        assert update.next_sync_committee_branch == [Bytes32() for _ in range(floorlog2(NEXT_SYNC_COMMITTEE_INDEX))]
-    else:
-        sync_committee = snapshot.next_sync_committee
-        assert is_valid_merkle_branch(
-            leaf=hash_tree_root(update.next_sync_committee),
-            branch=update.next_sync_committee_branch,
-            depth=floorlog2(NEXT_SYNC_COMMITTEE_INDEX),
-            index=get_subtree_index(NEXT_SYNC_COMMITTEE_INDEX),
-            root=update.header.state_root,
-        )
-
-    # Verify sync committee has sufficient participants
-    assert sum(update.sync_committee_bits) >= MIN_SYNC_COMMITTEE_PARTICIPANTS
-
-    # Verify sync committee aggregate signature
-    participant_pubkeys = [pubkey for (bit, pubkey) in zip(update.sync_committee_bits, sync_committee.pubkeys) if bit]
-    domain = compute_domain(DOMAIN_SYNC_COMMITTEE, update.fork_version, genesis_validators_root)
-    signing_root = compute_signing_root(signed_header, domain)
-    assert bls.FastAggregateVerify(participant_pubkeys, signing_root, update.sync_committee_signature)
-```
+The pseudo-code for the verification update can be found in [Altair](https://github.com/ethereum/annotated-spec/blob/master/altair/sync-protocol.md). We describe the pseudo-code in our own words in the following steps:
+
+1. First we verify that the update is meaningful, which basically means that the slot number provided is higher than the one stored in the light client
+2. Second we verify that the period is either the current one stored or the following one. OBviously an update for a past period cannot be provided,  neither we can skip period updates by injecting a period too far in the future.
+3. If we arre injecting an attested header (header for which the sync committee provided signatues) that is different than the finalized heaer (latest finalized header), a merkle proof verification is carried out to verify the validity of the finalized header in the attested header
+4. We take the sync committee against which we have to verify the signatues. In this case we have two options
+4.1 If the period of the attested header is the same one as we have stored, we take the current sync committe (SC), and verify the signatures against it.
+4.2 If the period of the attested header is higher (+1) of the latest one stored, then we take the next sync committee(NSC), and additionally, the prover needs to inject a merkle proof for the following next committee that needs to be validated.
+5. The aggreagated signature is verified against the selected committee.
+6. The header is accepted
 
 ## Transmiting Ethereum Headers to Starlight
 
@@ -90,12 +59,13 @@ def validate_light_client_update(snapshot: LightClientSnapshot,
 
 ### Overall Diagram ETH state validity Reception
 
+This is our own diagram, but a more low level diagram can be found in [Altair low-level diagram](https://github.com/ethereum/annotated-spec/raw/master/altair/lightsync.png)
+
 <p align="center">
   <img src="images/beacon_chain_altair.drawio.png" width="1000">
 </p>
 
 
-
 ## Proving symbiotic validator selection to starlight (or any other Ethereum state)
 
 ### The relayers job to transmit symbiotic validators