How Bitcoin Solves the Byzantine Generals Problem

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๐Ÿฐ Picture a mighty city, surrounded by a bunch of Byzantine generals ๐Ÿ’‚โ€โ™‚๏ธ, trying to unite and plan an epic attack! But oh no! ๐Ÿ˜ฑ Some of these generals might be traitors ๐Ÿ•ต๏ธ, who’ll backstab the others and ruin the plan. They need a foolproof way to agree on a battle strategy while handling these sneaky tricksters.

๐ŸŒ This dilemma isn’t just for ancient generals, it’s also super relevant in computer networks called distributed systems. These networks have many participants, or “nodes” ๐Ÿ’ป, who need to agree on something (like data) in a process called consensus. But some nodes might be mischievous ๐Ÿ˜ˆ, spreading false info to confuse others.

๐Ÿค To solve this, we need a system that allows all honest nodes to reach the same decision, even if there are cheeky liars in the mix! The solution should be fault-tolerant, meaning that it keeps working even if some nodes are not playing nice. ๐Ÿ›ก๏ธ

๐Ÿค–๐Ÿ’ฐ Now let’s bring in our digital superhero, Bitcoin! It swoops in with its magical power called Proof of Work (PoW) ๐Ÿ”จ. Miners (the network’s participants) go head-to-head in a crazy math puzzle race ๐Ÿงฎ๐Ÿ. Whoever wins gets to add a new block of transactions to the super secure public diary called the blockchain ๐Ÿ“’.

๐ŸŒŸ PoW makes it super expensive for the bad guys ๐Ÿ˜ˆ to control the network, because they would need a whole lotta computing power ๐Ÿ’ป๐Ÿ’ช. So, the honest nodes prevail, and consensus is achieved! ๐Ÿฅณ

To sum it up, the Byzantine Generals Problem is all about getting everyone on the same page ๐Ÿ“„, even when some sneaky snakes ๐Ÿ are trying to cause chaos. Bitcoin’s Proof of Work saves the day, ensuring that everyone can trust the system and have a rockin’ good time! ๐ŸŽ‰๐Ÿš€

The Byzantine Generals Problem

๐Ÿฐ Imagine a bunch of Byzantine generals ๐Ÿ’‚โ€โ™‚๏ธ surrounding a city, trying to decide when to attack โš”๏ธ. They need to attack together to win, but oh no! ๐Ÿ˜ฑ Some of them might be traitors ๐Ÿ•ต๏ธ. So, how can they agree on when to strike without a trustworthy central party? ๐Ÿค”

๐Ÿ’ก The Byzantine Generals Problem is a challenge faced in distributed systems where nodes (or participants) need to agree on a specific action or piece of information (called consensus) while accounting for the possibility of some nodes being faulty or malicious. The problem is named after the analogy of Byzantine generals planning an attack on a city, who need to coordinate their actions without falling victim to traitors within their ranks.

๐Ÿ” Let’s look at some example scenarios:

1๏ธโƒฃ Scenario 1: The Disobedient General Imagine there are three Byzantine generals, A, B, and C. They need to decide whether to attack (1) or retreat (0). A and B are loyal, and they decide to attack (1). C, however, is a traitor and wants to cause confusion. C sends different messages to A and B, telling A to attack (1) and B to retreat (0). In this case, A and B receive contradicting information and cannot reach consensus. The attack fails because of the lack of coordination.

2๏ธโƒฃ Scenario 2: The Traitorous Messenger In this scenario, there are three loyal generals, A, B, and C, and a treacherous messenger, M. The generals agree to attack (1) and send their decisions through messenger M. However, M wants to sabotage the attack. M delivers the correct message (1) to general A, but changes the message to retreat (0) for general B. The generals receive conflicting information, leading to a failed consensus and a botched attack.

3๏ธโƒฃ Scenario 3: The Compromised Communication Channel Let’s say there are three loyal generals, A, B, and C, who need to reach consensus on attacking (1) or retreating (0). They communicate through a shared channel, but an enemy has compromised the channel and can intercept and alter their messages. When the generals send their decision to attack (1), the enemy changes the messages to retreat (0). As a result, the generals receive false information, fail to reach consensus, and the coordinated attack is unsuccessful.

4๏ธโƒฃ Scenario 4: The Double Agent General In this scenario, there are four generals: A, B, C, and D. Generals A, B, and C are loyal, but D is a double agent working for the enemy. The generals need to decide whether to attack (1) or retreat (0). The loyal generals agree to attack (1). However, general D sends a message to retreat (0) to A and B, while sending a message to attack (1) to C. The conflicting information creates confusion and prevents the generals from reaching a consensus, leading to a failed attack.

5๏ธโƒฃ Scenario 5: The Chain of Traitors Imagine there are five generals: A, B, C, D, and E. Generals A, B, and C are loyal, while D and E are traitors. They need to decide whether to attack (1) or retreat (0). The loyal generals agree to attack (1) and share their decision with the next general in line. However, general D, being a traitor, alters the message and sends a retreat (0) order to E. General E, also a traitor, forwards the retreat (0) order to A. Now A, B, and C receive contradictory information, causing them to fail in reaching consensus and undermining the coordinated attack.

6๏ธโƒฃ Scenario 6: The Traitors’ Majority In this scenario, there are five generals: A, B, C, D, and E. Generals A and B are loyal, but C, D, and E are traitors. The generals need to decide whether to attack (1) or retreat (0). The loyal generals agree to attack (1). However, the traitorous generals send messages to attack (1) to general A and retreat (0) to general B. Since there are more traitorous generals than loyal ones, it becomes impossible for the loyal generals to identify the correct course of action, leading to a failure in reaching consensus and a botched attack.

๐Ÿ’ฅ What makes this problem so tricky? It’s called Byzantine Fault Tolerance (BFT) ๐Ÿ›ก๏ธ.

These faults can happen for lots of reasons: software bugs ๐Ÿž, hardware malfunctions ๐Ÿ› ๏ธ, or even sneaky cyberattacks ๐Ÿฆนโ€โ™‚๏ธ. The challenge is making a system that can handle these faults without breaking a sweat. ๐Ÿ˜…

Formal Definition

The Byzantine Generals Problem is a fundamental challenge in distributed computing that addresses the issue of achieving consensus among distributed nodes or agents in the presence of faulty or malicious nodes. The formal definition of the problem can be stated as follows:

Given a distributed system with n nodes, some of which may be faulty or Byzantine, devise a protocol that enables the honest nodes to reach a common agreement or consensus on a specific value or action, under the following conditions:

  1. All honest nodes must agree on the same value or action.
  2. If the source node (the node that initiates the consensus process) is honest, then all honest nodes must agree on the value or action proposed by the source node.

The protocol should be robust against Byzantine faults, which are failures that can lead to arbitrary or unpredictable behaviors, including sending inconsistent information to different nodes or colluding with other faulty nodes to disrupt the consensus process.

The objective is to find a solution that guarantees consensus among the honest nodes, even when up to f nodes in the system are faulty or Byzantine, with f < n/3 for asynchronous systems and f < n/2 for synchronous systems.

Game Theory

๐ŸŽฒ Enter game theory, a way to think about social situations with competing players. It helps us understand the challenge of these generals, which also applies to computer systems ๐Ÿ–ฅ๏ธ. We need a way to make sure everyone can agree, even if there are sneaky liars around. ๐Ÿ

The Byzantine Generals Problem incorporates elements of game theory, a mathematical framework used to model situations where decision-makers interact strategically. In the context of the Byzantine Generals Problem, game theory helps analyze and understand the behavior of nodes (or participants) in a distributed system, considering the possible actions of faulty or malicious nodes.

The game theory concepts relevant to the Byzantine Generals Problem are as follows:

  1. Players: In the Byzantine Generals Problem, the players are the nodes (or generals) in the distributed system. Each player has its own strategy, which could be honest (loyal) or dishonest (traitorous).
  2. Strategies: Each node can choose to be honest and transmit accurate information or be malicious and send contradictory information to other nodes. The goal of the malicious nodes is to prevent the system from reaching consensus, while the honest nodes strive to achieve consensus despite the presence of faulty nodes.
  3. Payoffs: The payoff for the honest nodes is the successful achievement of consensus, leading to a coordinated action (e.g., a successful attack). The payoff for the malicious nodes is the disruption of consensus, causing confusion and preventing coordinated action.
  4. Equilibrium: In the context of the Byzantine Generals Problem, an equilibrium is reached when a consensus algorithm successfully enables the honest nodes to achieve consensus, despite the presence of faulty nodes. This equilibrium is considered a “solution” to the problem.

To devise a robust consensus algorithm that solves the Byzantine Generals Problem, game theory helps assess the incentives and potential actions of all players (nodes) in the system.

Famous Lamport Paper

๐Ÿ“œ Back in 1982, some super-smart researchers ๐Ÿง  (Lamport, Shostak, and Pease) published a paper ๐Ÿ“ that described the Byzantine Generals Problem and proposed solutions. They showed that it’s not just about military communication ๐Ÿ“ก, but it affects all kinds of computer systems too! ๐ŸŒ

The authors presented the problem using an analogy involving Byzantine generals who need to coordinate an attack or retreat while communicating only through messengers. They acknowledged the potential for some generals to be traitorous and send contradictory messages, creating confusion and preventing consensus.

The paper provided algorithms and proofs for various cases of the problem, depending on the number of total nodes and the number of faulty nodes. The authors demonstrated that, for a system to be fault-tolerant and reach consensus, the number of total nodes (n) must be at least 3 times the number of faulty nodes (f) (i.e., n > 3f) in an asynchronous system.

The Lamport et al. paper was a groundbreaking work that significantly advanced the understanding of distributed computing, fault tolerance, and consensus. The Byzantine Generals Problem continues to be an essential concept in the development of distributed systems, particularly in the context of blockchain technology and cryptocurrencies.

Satoshi’s Solution to the Byzantine Problem: Bitcoin

๐Ÿš€ In the world of decentralized systems, this problem is super important. Centralized systems have a big boss ๐Ÿข (like a bank or government) that can step in if something goes wrong. But decentralized systems don’t have that safety net, so they need to solve the Byzantine Generals Problem on their own. ๐ŸŒŸ

๐Ÿ’ฐ So, how can we create a trustworthy currency ๐Ÿ’ธ that everyone can agree on? Enter Bitcoin (BTC) ๐ŸŽ‰! It’s the first system to practically solve the Byzantine Generals Problem, making it a groundbreaking game-changer. ๐Ÿš€

Note: Bitcoin does not solve the Byzantine Generals Problem formally, i.e., mathematically, but it solves it practically, i.e., statistically. You are never 100% sure that the network has reached absolute consensus forever but the more blocks have been produced after your transaction was confirmed, the more sure you can be. Exponentially increasing certainty.

You can learn more about this in the following excellent video from a Bitcoin maximalist:

Did Satoshi Solve The Byzantine Generals Problem?

What If Bitcoin Wouldn’t Have Solved the Byzantine Generals Problem?

The following bad scenarios could’ve happened if Satoshi hadn’t fully solved the Byzantine Generals Problem. That’s why it was such a groundbreaking invention!

1๏ธโƒฃ Double Spending Spree ๐Ÿ›๏ธ๐Ÿ’ธ: If Bitcoin hadn’t solved the Byzantine Generals Problem, Alice could spend her bitcoin on a fancy new gadget ๐Ÿ“ฑ from Bob and then use the same bitcoin to buy a delicious pizza ๐Ÿ• from Carol. Without consensus, both transactions would be considered valid, leading to chaos in the Bitcoin network and undermining the trust in the currency. ๐Ÿ˜ฑ

2๏ธโƒฃ Miner Mayhem ๐Ÿšงโ›๏ธ: Miners competing to validate blocks might end up validating different versions of the blockchain ๐Ÿงฉ, creating multiple branches and causing confusion. Users wouldn’t know which branch to trust, making transactions risky and uncertain. ๐Ÿ˜•

3๏ธโƒฃ Transaction Tangle ๐Ÿ•ธ๏ธ๐Ÿ’ณ: If Bitcoin couldn’t reach consensus, Alice might send bitcoin to Bob, but Carol, who runs a node, might register a different transaction, sending the same bitcoin to Dave. This tangled web of conflicting transactions would create uncertainty and diminish trust in the system. ๐Ÿคฏ

4๏ธโƒฃ Cryptocurrency Carousel ๐ŸŽ ๐Ÿ’ฐ: Users might see their balances change unexpectedly, as faulty nodes could report different balances for the same wallet. One moment you’re a Bitcoin millionaire ๐Ÿค‘, and the next, you’ve got zilch. Talk about a wild ride! ๐ŸŽข

5๏ธโƒฃ Hacking Havoc ๐Ÿฆนโ€โ™‚๏ธ๐Ÿ–ฅ๏ธ: If Bitcoin hadn’t addressed the Byzantine Generals Problem, malicious actors could exploit the lack of consensus to manipulate the network. They might create fake transactions, falsify balances, or even steal bitcoins, causing panic and distrust among users. ๐Ÿ˜จ

6๏ธโƒฃ Slow-motion Meltdown ๐ŸŒ๐Ÿ”ฅ: Without solving the Byzantine Generals Problem, transactions would take ages to confirm, if they even confirm at all. Imagine waiting for your Bitcoin payment to go through while watching a sloth cross the road. ๐Ÿ›ฃ๏ธ Yep, it’d be that slow! And with such a sluggish system, nobody would want to use Bitcoin. ๐Ÿ™…โ€โ™€๏ธ

Luckily, Bitcoin’s blockchain technology and consensus algorithms like Proof of Work solved the Byzantine Generals Problem, ensuring trust, security, and reliability in the network. Phew! ๐ŸŽ‰๐Ÿฅณ

But How Does Bitcoin Solve The Problem?

Bitcoin solves the Byzantine Generals Problem through a combination of its blockchain technology and the Proof of Work (PoW) consensus algorithm.

Here’s a breakdown of how it works, with examples:

1๏ธโƒฃ Blockchain Structure ๐Ÿ“š๐Ÿ”—: Bitcoin’s transactions are grouped into blocks, and each block contains a reference to the previous block’s unique identifier (hash). This creates a chain of blocks that is difficult to tamper with, as altering any block would require changing all subsequent blocks.

Example: Alice sends 1 BTC to Bob and then tries to double-spend that same BTC by sending it to Carol. The blockchain structure ensures that the first transaction (to Bob) is permanently recorded and cannot be altered without changing all subsequent blocks.

2๏ธโƒฃ Proof of Work (PoW) โ›๏ธ๐Ÿ”: Miners compete to validate and add new blocks to the blockchain by solving complex mathematical puzzles. The first miner to solve the puzzle gets to add the new block and is rewarded with freshly minted bitcoins. This process requires considerable computational power, making it costly and difficult for any single actor to take control of the network.

Example: A malicious miner wants to manipulate the network by validating a fraudulent transaction. However, doing so would require outpacing the combined computational power of all other miners, which is highly improbable and expensive.

3๏ธโƒฃ Longest Chain Rule ๐Ÿ“๐Ÿ†: When multiple valid versions of the blockchain emerge (forks), Bitcoin follows the longest chain rule. Nodes consider the chain with the most accumulated PoW as the true version, eventually leading to a single, agreed-upon blockchain.

Example: Two miners, Miner A and Miner B, solve the PoW puzzle at nearly the same time, resulting in two competing branches of the blockchain. As other miners continue mining, one branch becomes longer than the other. Nodes eventually adopt the longer chain, reaching consensus on the valid blockchain.

4๏ธโƒฃ Network Incentives ๐Ÿ’ฐ๐Ÿ…: Bitcoin’s design rewards honest behavior and punishes dishonesty. Miners receive block rewards and transaction fees for validating transactions and adding blocks to the chain. Acting maliciously would require immense resources with little chance of success, making it economically unappealing.

Example: A group of malicious miners considers launching a 51% attack to control the network. However, they realize that the cost of acquiring the necessary computational power would outweigh any potential gains. Instead, they decide to mine honestly, contributing to the network’s security and stability.

By incorporating these features, Bitcoin successfully solves the Byzantine Generals Problem, ensuring consensus, trust, and security within the network.

Satoshi’s Bitcoin Whitepaper

Satoshi Nakamoto’s Bitcoin whitepaper, released in October 2008, laid the groundwork for solving the Byzantine Generals Problem, although the term itself wasn’t explicitly used. The solution was implemented with the launch of the Bitcoin network in January 2009.

Figure: Not at all what a “real” Bitcoin looks like. ๐Ÿ‘†

Nakamoto’s method involves cryptographic security and public-key encryption to address the Byzantine Generals Problem within a digital electronic network. Cryptographic security employs hashing to prevent data tampering, while public-key encryption verifies the identity of network users.

In the blockchain, transactions are secured within blocks connected by their hash values. All hashes can be traced back to the genesis block. The blockchain employs a Merkle Tree to verify hashes, with each block in the network considered valid if it originates from the genesis block.

Merkle trees, sometimes called Binary hash trees, are a popular kind of data structure in the world of computer science ๐Ÿ–ฅ๏ธ๐ŸŒณ. They play a crucial role in Bitcoin and other cryptocurrencies, making blockchain data encryption more efficient and secure ๐Ÿ”โœจ.

Miners validate blocks by competing to solve cryptographic puzzles as part of a Proof of Work (PoW) consensus mechanism.

Bitcoin resolves the Byzantine Generals Problem by using PoW to create an objective set of rules for the blockchain. A network participant must present proof of effort expended to add blocks to the blockchain. The cost of this work incentivizes accurate information sharing.

Objective rules ensure that there is no disagreement or tampering with the information on the Bitcoin network. The system for choosing who can mint new Bitcoin and the rules governing valid or invalid transactions are both objective. Moreover, once a block is added to the blockchain, it becomes immutable.

Miners in the blockchain are analogous to generals, with each node responsible for validating transactions, akin to the messages delivered to generals. Cryptographic security protects messages from potential attacks by malicious actors, such as hackers. Transactions are bundled into blocks and hashed to prevent tampering. By placing miners in a competition to validate blocks, Satoshi makes the process more decentralized and probabilistic, preventing any single miner from monopolizing validation.

Miners compete to solve a cryptographic puzzle using their computational power or hash rate. The higher the hash rate, the greater the chance of solving the puzzle. When a miner solves the puzzle, they broadcast the solution to the network, and other miners must verify or reject the value based on a difficulty target.

The Bitcoin network’s members can reach consensus on the blockchain’s state and all transactions at any given moment. Each node verifies block validity according to the PoW criterion and transaction validity based on additional criteria. If a network member tries to share misleading information, nodes will identify it as objectively invalid and disregard it. The trustless nature of Bitcoin eliminates the need for reliance on other network members since each node can independently verify all information.

The decentralized nature of the blockchain ensures that there is no single point of failure. Blocks are stored in a distributed database, replicated across the network. This redundancy also contributes to fault tolerance, guaranteeing that no single malfunctioning computer can bring down the entire system. This is akin to having multiple messengers to relay messages even if one gets ambushed, ensuring the message will not be lost as it is copied by other messengers.

Cryptography Mailing List Description of the Problem

Here’s Satoshi‘s rephrasing of the problem in his correspondence with the “Cryptography Mailing List” to which he initially launched the Bitcoin open-source project and whitepaper:

The proof-of-work chain is a solution to the Byzantine Generals' Problem. I'll try to rephrase it in that context.

A number of Byzantine Generals each have a computer and want to attack the King's wi-fi by brute forcing the password, which they've learned is a certain number of characters in length. Once they stimulate the network to generate a packet, they must crack the password within a limited time to break in and erase the logs, otherwise they will be discovered and get in trouble. They only have enough CPU power to crack it fast enough if a majority of them attack at the same time.

They don't particularly care when the attack will be, just that they all agree. It has been decided that anyone who feels like it will announce a time, and whatever time is heard first will be the official attack time. The problem is that the network is not instantaneous, and if two generals announce different attack times at close to the same time, some may hear one first and others hear the other first.

They use a proof-of-work chain to solve the problem. Once each general receives whatever attack time he hears first, he sets his computer to solve an extremely difficult proof-of-work problem that includes the attack time in its hash. The proof-of-work is so difficult, it's expected to take 10 minutes of them all working at once before one of them finds a solution. Once one of the generals finds a proof-of-work, he broadcasts it to the network, and everyone changes their current proof-of-work computation to include that proof-of-work in the hash they're working on. If anyone was working on a different attack time, they switch to this one, because its proof-of-work chain is now longer.

After two hours, one attack time should be hashed by a chain of 12 proofs-of-work. Every general, just by verifying the difficulty of the proof-of-work chain, can estimate how much parallel CPU power per hour was expended on it and see that it must have required the majority of the computers to produce that much proof-of-work in the allotted time. They had to all have seen it because the proof-of-work is proof that they worked on it. If the CPU power exhibited by the proof-of-work chain is sufficient to crack the password, they can safely attack at the agreed time.

The proof-of-work chain is how all the synchronisation, distributed database and global view problems you've asked about are solved.

In other words:

The Byzantine Generals have computers and want to attack the King’s Wi-Fi by cracking the password. ๐Ÿฐ๐Ÿ“ก๐Ÿ’ป They need to work together to break in and erase logs before getting caught. โฐ๐Ÿ”“๐Ÿ“‚ The generals don’t care when they attack but must agree on a time. โš”๏ธ๐Ÿ•’ They announce a time, but the network isn’t instant, causing confusion. ๐Ÿ˜•

They use a proof-of-work chain to agree. ๐Ÿ“Š๐Ÿ”— Each general starts solving a super hard problem, including the attack time in its hash. ๐Ÿค”๐Ÿ’ญ๐Ÿ’ก It’s so tough that it takes 10 minutes for one to find a solution. ๐Ÿ˜ฉโณ When a general solves it, he tells everyone, and they update their work. ๐Ÿ“ฃ๐Ÿ”„

After 2 hours, an attack time is confirmed by a chain of 12 proofs-of-work. ๐Ÿ•‘๐Ÿ”—โœ… Each general can see the CPU power spent and know that most computers worked on it. ๐Ÿ–ฅ๏ธ๐Ÿ’ช The proof-of-work chain helps them sync, share info, and see the big picture. ๐ŸŒ๐Ÿค With enough power, they can attack at the agreed time. ๐ŸŽฏ๐Ÿ•’


the Byzantine Generals Problem is a classic challenge in distributed systems, where trust and coordination are critical. Bitcoin, with its innovative proof-of-work mechanism, has successfully addressed this issue, paving the way for a decentralized and trustless digital currency. ๐Ÿš€๐Ÿ’ฐ

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