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notes/pbft.md

-#  Byzantine Consensus, and PBFT
-
-## Simple example: Two Generals
-
-![](byzTwoG.png)
-
-- one is a decider:
-- both need to attack same time
-- need to agree on:
-  - time: (easy: msg and an ack)
-  - agreement to attack (hard)
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-**Example:**
-A sends to B "attack at 10".
-But did B get it? Can't go unless sure.
-B sends an ack,
-but did A get the ack?
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-### Impossibility
-
-Look at sequence of msg-ack-ack......
-
-Assume there is some subset of i msgs that constitutes a proof, and
-that both would attack.  
-
-However, what if the last msg not delivered?
-  - receiver presumably would not attack
-  - sender, though, sees same msgs as the i-sequence, and so  attacks....
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-### Fix?
-- A sends a whole bunch, assume one gets through
-- A and B send and ack a while
-
-However, *provable agreement between even two parties in asynchronous
-environment not possible*.
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-## Two Lieutenants Problem
-
-*Safety*:
-- all loyal lieutenants make same decision
-- all loyal lieutenants follow loyal general
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-![](byzL.png)
-&nbsp;<p>
-Clearly impossible for both lieutenants to always make same decision, as neither knows if the fault
-  lies with the general or the other lieutenant.  **Therefore, no solution when *n = 3f*.**
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-![](byzL3.png)
-&nbsp;<p>
-Each lieutenant decides based on majority of input, done! **So *3f+1* works, at least in the case of *f=1*.**
-
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-&nbsp;<p>
-
-
-![](byzAlb.png)
-&nbsp;<p>
-
-
-
-Byzantine faults means byzantine protocols. *3f+1* nodes needed to tolerate *f*
-faults.
-- (Above) showed impossibility of *3* nodes to tolerate a single fault
-- Proof by contradiction:
-    - Assume we have a protocol, "BYZ3", that works even if one third of its members are faulty.
-    - Have *3* Byzantine generals each simulate *m* Albanian generals via the above protocol.
-      - Since only one Byzantine general is faulty, the other *2m*
-        generals are simulated correctly, so the situation is identical to
-        having *2f* non-faulty generals and *f* faulty generals.
-      - If BYZ3 is correct, this simulation will reach the correct conclusion.
-      - However, there are **really** only *3* generals, meaning we have
-        solved the problem for *3* generals with a single faulty node, which we
-        know to be impossible. 
-  - Contradiction!
-
-
-## Different Problems
-- Byz generals also called *byzantine broadcast*.
-  - sync: *f < n-1*
-  - async *f < n/3*
-- Byz agreement: trying to decide the truth by quorum consensus ("sky is blue" vs "sky is red")
-  - sync: *f < n/2*
-  - async *f < n/3*
-
-## Castro's proof of *3n + 1* for Byz Agreement
-- must be possible to make progress w/ only *n - f* replicas (because *f* might be faulty and choose not to respond)
-- however, the *f* not responding might be correct but slow (asynchrony), so there might be *f* bad responses out of the *n - f*
-  - implies *(n - f - f) > f*, or *n >= 3f + 1*
-
-
-
 # Practical Byzantine Fault Tolerence  (Castro and Liskov)
 
 ## Assumptions
 - operations are deterministic
 - replicas start in the same state
-Means that correct replicas will return identical results.
+  - i.e., correct replicas will return identical results.
 
 ## Differences from fail-stop consensus
-- 3*f*+1 replicas instead of 2*f*+1
+- 3*f*+1 replicas instead of 2*f*+1 (non-byzantine consensus):
+  - must be possible to make progress w/ only *n - f* replicas (because *f* might be faulty and choose not to respond)
+  - however, the *f* not responding might be correct but slow (asynchrony), so there might be *f* bad responses out of the *n - f*
+  - implies *(n - f - f) > f*, or *n >= 3f + 1*
 - 3 phases instead of two
 - cryptographic signatures
 
@@ -169,8 +49,9 @@ Means that correct replicas will return identical results.
 
 ## Phases
 - *pre-prepare* and *prepare* phases order request even if primary lying
+  - replica tells all others what the primary told it
 - a replica is *prepared* when:
-  - has received a proper pre-prepare from primary, sig, current view
+  - has received a proper pre-prepare from primary, signature, current *view*
   - 2*f* "prepares" from other replicas that match it
   - Such a replica then:
     - multicasts "commit" to other replicas
@@ -195,3 +76,5 @@ Means that correct replicas will return identical results.
   - rep *r* signs by encrypting msg digest w/ each other rep
   - vector of such sigs is the **authenticator**
 
+## Questions
+- how else can we do byzantine fault tolerance?