The document provides an overview of various consensus algorithms addressing Byzantine fault tolerance, including the 3m + 1 processors algorithm and practical Byzantine fault tolerance (PBFT). It discusses proof of work (PoW) challenges, such as the finality problem, and introduces Casper FFG, a proof-of-stake based finality system. Additionally, references to foundational works in the field and the complexities involved in achieving fault tolerance are included.

1. Basic
Consensus Algorithms
Mar. 2020
Sangmoon Oh (halfface@chollian.net)
v1.0

2. 2
목차
• Byzantine Fault
• Byzantine Fault Tolerance (BFT)
• 3m + 1 Processors Algorithm
• pBFT
• Proof of Work (PoW)
• Ethash
• PoW Finality Problem
• Casper FFG

3. 3
II. Consensus AlgorithmByzantine Fault (Byzantine General Problem)
Enemy
attack
retreat
retreat
retreat
retreat
retreat
attack
attack
attack
attack
Agree on Attack/Retreat
Protocol or Algorithm
 SIFT: design and analysis of a fault-tolerant computer for aircraft control (1978)
 




4. 4
II. Consensus AlgorithmByzantine Fault - Simple Example
2 Loyal Generals with 1 Unknown Traitor
1
3
2
Traitor
att
att
att
ret
att
att
(att,att,att)
Attack
(att,ret,att)
Attack
1
3
2
Traitor
att
ret
ret
ret
ret
ret
1
3
2
Traitor
att
att
att
ret
ret
ret
(ret,att,ret)
Retreat
(att,att,ret)
Attack
(ret,ret,ret)
Retreat
(att,ret,ret)
Retreat
Not Agreed !!
Case 1) Case 2) Case 3)

5. 5
II. Consensus AlgorithmByzantine Fault Tolerance (BFT)
Enemy
General
Traitor
General Traitor
General
General
Attack
G
T
G T
G
G
Attack
Attack
Attack Retreat
G
T
G T
G
G
Retreat Retreat
Retreat
Algorithm 1
Retreat
Attack
G
T
G T
G
G
Attack
Attack
Retreat
G
T
G T
G
G
Retreat Retreat
RetreatAlgorithm 2
Always
Agreement
No Agreement
Possible
A1: BTF
A2: Not BTF

6. 6
II. Consensus AlgorithmByzantine Fault Revisited
Enemy
 Synchronous or Asynchronous
Message Delivery
 Forgeable vs. Unforgeable Message
 Level of Connectivity
 Option Space
Is BTF Possible ?
What Algorithms ?

7. 7
II. Consensus Algorithm3m + 1 Processors Algorithm - Overview
With m faulty processors,
Min. (3m +1) processors are needed
to tolerate the fault.
Algorithm OM(m)
n  3m + 1
General
Traitor General
General General
GeneralTraitor
General General
General
General
Traitor
Traitor
General General
General
General
Traitor
Traitor
General

m=1, n=3 (< 3m+1) m=1, n=4 ( 3m+1) m=2, n=6 (< 3m+1) m=2, n=7 ( 3m+1)
(Synchronized message delivery, Forgeable message, Full connected)
OM(1) OM(2)

8. 8
II. Consensus Algorithm3m + 1 Processors Algorithm - Instructions
m nmin
Instruction
1 4 OM(0), OM(1)
2 7 OM(0), OM(1), OM(2)
3 10 OM(0), OM(1), OM(2), OM(3)









Recursive Relay
v:1:4
v:1:2:4
v:1:2
v:1:2:3 v:1:2:3:4
4
1
2
3
v:1:4
v:1:2:4
v:1:3:4
v:1:2:3:4
v:1:3:2:4
 1 + 2C1 + 2C11C1 = 5
v:1:::4

9. 9
II. Consensus Algorithm3m + 1 Processors Algorithm - Simplest Case
Case v:2:1 v:2:3:1 v:2:4:1 v:2:::1 v:2:3 v:2:1:3 v:2:4:3 v:2:::3 v:2:4 v2:1:4 v:2:3:4 v:2:::4
1) A A R 2A + R A A R 2A + R R A A 2A + R
2) A R A 2A + R R A A 2A + R A A R 2A + R
3) A R R A + 2R R A R A + 2R R A R A + 2R
4) R A A 2A + R A R A 2A + R A R A 2A + R
5) R A R A + 2R A R R A + 2R R R A A + 2R
6) R R A A + 2R R R A A + 2R A R R A + 2R

10. 10
II. Consensus Algorithm3m + 1 Processors Algorithm - Complexity
General General
GeneralTraitor
General General
General
General
Traitor
Traitor
General
Faulty Total Recursion Number of Message
1 4 OM(0), OM(1) 4*3 + 4*3*2
2 7 OM(0), OM(1), OM(2) 7*6 + 7*6*5 + 7*6*5*4
3 10 OM(0), OM(1), OM(2), OM(3) 10*9 + 10*9*8 + 10*9*8*7 + 10*9*8*7*6









m 3m + 1 OM(0), OM(1), , OM(m) (3m+1)*3m + (3m+1)*3m*(3m-1) +  + (3m+1)*3m2m
Complexity : O(nm)
General General
General
General
Traitor
Traitor

11. 11
II. Consensus AlgorithmState Machine Replication
a : 100
b : 100
c : 100
a : 120
b : 110
c : 70
a : 90
b : 100
c : 110
tx1
tx3
tx4
tx2
tx1 :
b a, 20
tx2 :
b c, 10
tx3 :
c b, 30
tx4 :
a c, 30
a : 100
b : 100
c : 100
a : 120
b : 110
c : 70
a : 90
b : 100
c : 110
tx1
tx3
tx4
tx2
a : 100
b : 100
c : 100
a : 120
b : 110
c : 70
a : 90
b : 100
c : 110
tx1
tx3
tx4
tx2
Ordering Txs
Decentralized way
Communication and
Corporation

12. 12
II. Consensus AlgorithmpBFT (Practical Byzantine Fault Tolerance)
1
2
3
4
client
Pre-prepare
(~ n)
Prepare
(~ n(n-1))
Commit
(~ n(n-1))
Primary Create and share next state
candidate
Backups Validate next state candidate
 Pre-prepare
Sender Share validated(accepted) next
state candidate
Receiver Collect the acceptance of the
other nodes
Commit if enough nodes
accepted ( >
2𝑛+1
3
)
 Prepare, Commit
tx

13. 13
II. Consensus AlgorithmProof of Work (PoW)
4 3
2
1
5
 Compute
 Solve
 Send
 Send
 Send
 Send
 Compute
 Compute
 Compute
 Verify
 Verify
 Verify
 Verify
 Compute
Try a puzzle hard to solve but easy to verify.
 Competition
 Reward

14. 14
II. Consensus Algorithm
4 3
2
1
5
 Compute
 Solve
 Send
 Send
 Send
 Send
 Compute
 Compute
 Compute
 Verify
 Verify
 Verify
 Verify
 Compute
Hash based PoW
Data + Nonce SHA-256(Data + Nonce)
Hello, world! 0x315f5bdb76d078c43b8ac0064e4a0164612b1fce77c869345bfc94c75894edd3
Hello, world!0 0x1312af178c253f84028d480a6adc1e25e81caa44c749ec81976192e2ec934c64
Hello, world!1 0xe9afc424b79e4f6ab42d99c81156d3a17228d6e1eef4139be78e948a9332a7d8
Hello, world!2 0xae37343a357a8297591625e7134cbea22f5928be8ca2a32aa475cf05fd4266b7
Hello, world!4248 0x6e110d98b388e77e9c6f042ac6b497cec46660deef75a55ebc7cfdf65cc0b965
Hello, world!4249 0xc004190b822f1669cac8dc37e761cb73652e7832fb814565702245cf26ebb9e6
Hello, world!4250 0x0000c3af42fc31103f1fdc0151fa747ff87349a4714df7cc52ea464e12dcd4e9
Puzzle: Find Nonce for SHA-256(Data, Nonce)
to Start with 0000

15. 15
II. Consensus AlgorithmEthash - PoW Algorithm for Ethereum 1.0
 https://github.com/ethereum/wiki/wiki/Ethash
Block # Date Difficulty Target
9,000,000 2019-11-26 2,573,664,196,528,490 0x0000000000001bff809762e524...
8,000,000 2019-06-21 2,037,888,242,889,388 0x000000000000235be453641714...
6,000,000 2018-07-21 3,483,739,548,912,554 0x00000000000014af19378e94eb
3,000,000 2017-01-16 103,975,266,902,792 0x000000000002b506bb4c196d40
4 3
2
1
5
 Compute
 Solve
 Send
 Send
 Send
 Send
 Compute
 Compute
 Compute
 Verify
 Verify
 Verify
 Verify
 Compute
Keccak(header, transactions, nonce) = 0x0000...
def mine(full_size, dataset, header, difficulty):
target = zpad(encode_int(2**256 // difficulty), 64)[::-1]
from random import randint
nonce = randint(0, 2**64)
while hashimoto_full(full_size, dataset, header, nonce) > target:
nonce = (nonce + 1) % 2**64
return nonce
difficulty x target = 2256

16. 16
II. Consensus AlgorithmPoW - Finality Problem
Solve Bm
Solve Bn
Bm
Two different miners may generate a block simultaneously.
Bm
Bn
Bn

17. 17
II. Consensus AlgorithmPoW - Finality Problem
Solve Bm2
Solve Bn2
Bm
Another two different miners may generate a block simultaneously again.
Bm
Bn
Bn
Bn2
Bn2Bm2 Bm2

18. 18
II. Consensus AlgorithmPoW - Finality Problem
Bm
Bn
Bm
Bn
Bm
Bn
Bm
Bn
Bn
  
  
txb(b a, 10)
txb(b a, 10)

19. 19
II. Consensus AlgorithmCasper FFG - Finality System
 "Casper FFG" paper by Vitalik and Virgil
PoS based Finality System
which Overlays an existing
PoW blockchain.
 Accountability : Penalty
 Dynamic validators
 Modular overlay
: a partial consensus mechanism

20. 20
II. Consensus AlgorithmCasper FFG - Process
1) Voting
 Vote for link(edge)
 2 Commandments
(Slashing conditions)
2) Justifying
  2/3 votes
3) Finalizing
 Canonical chain
1)
2)
3)

21. 21
AppendixReferences
 Consensus
 Byzantine fault
 Reaching Agreement in the Presence of Faults (1980, M. PEASE, R, SHOSTAK, AND L. LAMPORT)
 The Byzantine Generals Problem (1982, LESLIE LAMPORT, ROBERT SHOSTAK, and MARSHALL PEASE)
 Practical Byzantine Fault Tolerance (1999, Miguel Castro and Barbara Liskov)
 pBFT— Understanding the Consensus Algorithm
 Proof-of-Work, Explained
 Hashcash
 Ethash