"GTO" (Game Theory Optimal) has been used a lot in poker recently. To clarify its meaning and application, the aim of this article is to define what it is, when to use it when playing, what strategies to apply for how to play GTO poker, for beginner and more advanced players!

Poker GTO

In poker, you can choose between two main winning strategies:

Exploitative play maximizes expected value (EV) by countering opponents' sub-optimal plays and weaker tendencies. Yes, playing this way opens you up to being exploited, but the weaker opponents you're targeting won't change their game to counteract this, allowing you to reap maximum profits over the long run.

GTO (Game-Theory Optimal): You try to play perfect poker, which forces your opponents to make mistakes (which is where almost all of your profit will be derived from). It mixes bluffs or semi-bluffs with value bets, clarifies bet sizes, and more.

GTO vs. Exploitative: Which is Better in Poker?

Before diving into GTO poker strategy, it's important to know which of these two play styles is more profitable for beginners and advanced players. The simple answer is probably both, but usually more exploitable.

Most players don't play GTO poker and can be exploited in their gameplay and strategy, allowing for more profits to be made using an exploitative approach. In fact, it’s only in some of the largest games at the highest stakes that GTO concepts are fully utilized and seen in practice, and even then, exploitative plays are still sometimes used.

Knowing, understanding, and applying GTO poker basics will create a solid foundation for your poker game. It's also important to understand GTO so you can deviate from it to maximize profits.

GTO Poker Strategy

According to Ed Miller's book "Poker's 1%," the most fundamental concept that only elite poker players understand is frequency, which could be in relation to cbets, bluffs, folds, calls, raises, etc.

GTO poker solvers (downloadable online software) give solutions for how to play optimally in any given spot and often recommend using mixed strategies based on select frequencies.

In a river situation, a solver may tell you to call 70% of the time and fold 30%. It may also suggest calling 50% of the time, folding 35% of the time, and raising 15% of the time (with a certain range of hands).

Frequencies are a fundamental and often unrecognized part of poker, but they run through these 5 GTO concepts.

1. Preflop ranges

To compensate for positional disadvantage, out-of-position players must open tighter hand ranges.

Premium starting hands aren't enough, though. Considering GTO poker ranges and principles, you want a good, balanced starting hand range from each position with at least some hands that can make a strong poker hand regardless of the flop texture (low, mid, high, disconnected, etc).

Below is a GTO preflop beginner poker chart for online 6-max play, showing which hand ranges one should open-raise with. Table positions are color-coded (see key below).

NOTE: For GTO play, it's advisable to use a mixed strategy for opening in the small blind, combining open-limps and open-raises for various hands. This cannot be illustrated with the color system used for the chart.

Choosing which hands to play is often a math problem, as discussed below.

Other preflop GTO poker charts include which hands to play after a raise, which to 3bet, etc. Solvers can help you decide which preflop hands to play (call, raise, re-raise, etc.).

2. Pot Odds

Always make +EV decisions that profit you as a poker player. Understanding pot odds (and equity) can help.

Postflop Pot Odds

Let’s say that we have JhTh on a board of 9h8h2s4c (open-ended straight-flush draw).  We have $40 left and $50 in the pot. He has you covered and goes all-in. As calling or folding are our only options, playing GTO involves calculating whether a call is +EV or –EV. (The hand was empty.)

Any remaining heart, Queen, or 7 wins the hand. This means we can improve 15 of 46 unknown cards, or 32.6% of the time.

What if our opponent has a set? The 4h or 2h could give us a flush, but it could also give the villain a boat. If we reduce outs from 15 to 14.5, our equity would be 31.5%.

We must now calculate pot odds.

(bet/(our bet+pot)) = pot odds

= $50 / ($40 + $90)

= $40 / $130

= 30.7%

To make a profitable call, we need at least 30.7% equity. This is a profitable call as we have 31.5% equity (even if villain has a set). Yes, we will lose most of the time, but we will make a small profit in the long run, making a call correct.

Pot odds aren't just for draws, either. If an opponent bets 50% pot, you get 3 to 1 odds on a call, so you must win 25% of the time to be profitable. If your current hand has more than 25% equity against your opponent's perceived range, call.

Preflop Pot Odds

Preflop, you raise to 3bb and the button 3bets to 9bb. You must decide how to act. In situations like these, we can actually use pot odds to assist our decision-making.

This pot is:

(our open+3bet size+small blind+big blind)

(3bb+9bb+0.5bb+1bb)

= 13.5

This means we must call 6bb to win a pot of 13.5bb, which requires 30.7% equity against the 3bettor's range.

Three additional factors must be considered:

Being out of position on our opponent makes it harder to realize our hand's equity, as he can use his position to put us in tough spots. To profitably continue against villain's hand range, we should add 7% to our equity.

Implied Odds / Reverse Implied Odds: The ability to win or lose significantly more post-flop (than pre-flop) based on our remaining stack.

While statistics on 3bet stats can be gained with a large enough sample size (i.e. 8% 3bet stat from button), the numbers don't tell us which 8% of hands villain could be 3betting with. Both polarized and depolarized charts below show 8% of possible hands.

7.4% of hands are depolarized.

Polarized Hand range (7.54%):

Each hand range has different contents. We don't know if he 3bets some hands and calls or folds others.

Using an exploitable strategy can help you play a hand range correctly. The next GTO concept will make things easier.

3. Minimum Defense Frequency:

This concept refers to the % of our range we must continue with (by calling or raising) to avoid being exploited by our opponents. This concept is most often used off-table and is difficult to apply in-game.

These beginner GTO concepts will help your decision-making during a hand, especially against aggressive opponents.

MDF formula:

MDF = POT SIZE/(POT SIZE+BET SIZE)

Here's a poker GTO chart of common bet sizes and minimum defense frequency.

Take the number of hand combos in your starting hand range and use the MDF to determine which hands to continue with. Choose hands with the most playability and equity against your opponent's betting range.

Say you open-raise HJ and BB calls. Qh9h6c flop. Your opponent leads you for a half-pot bet. MDF suggests keeping 67% of our range.

Using the above starting hand chart, we can determine that the HJ opens 254 combos:

We must defend 67% of these hands, or 170 combos, according to MDF. Hands we should keep include:

Flush draws

Open-Ended Straight Draws

Gut-Shot Straight Draws

Overcards

Any Pair or better

So, our flop continuing range could be:

Some highlights:

Fours and fives have little chance of improving on the turn or river.

We only continue with AX hearts (with a flush draw) without a pair or better.

We'll also include 4 AJo combos, all of which have the Ace of hearts, and AcJh, which can block a backdoor nut flush combo.

Let's assume all these hands are called and the turn is blank (2 of spades). Opponent bets full-pot. MDF says we must defend 50% of our flop continuing range, or 85 of 170 combos, to be unexploitable. This strategy includes our best flush draws, straight draws, and made hands.

Here, we keep combining:

Nut flush draws

Pair + flush draws

GS + flush draws

Second Pair, Top Kicker+

One combo of JJ that doesn’t block the flush draw or backdoor flush draw.

On the river, we can fold our missed draws and keep our best made hands. When calling with weaker hands, consider blocker effects and card removal to avoid overcalling and decide which combos to continue.

4. Poker GTO Bet Sizing

To avoid being exploited, balance your bluffs and value bets. Your betting range depends on how much you bet (in relation to the pot). This concept only applies on the river, as draws (bluffs) on the flop and turn still have equity (and are therefore total bluffs).

On the flop, you want a 2:1 bluff-to-value-bet ratio. On the flop, there won't be as many made hands as on the river, and your bluffs will usually contain equity. The turn should have a "bluffing" ratio of 1:1. Use the chart below to determine GTO river bluff frequencies (relative to your bet size):

This chart relates to your opponent's pot odds. If you bet 50% pot, your opponent gets 3:1 odds and must win 25% of the time to call. Poker GTO theory suggests including 25% bluff combinations in your betting range so you're indifferent to your opponent calling or folding.

Best river bluffs don't block hands you want your opponent to have (or not have). For example, betting with missed Ace-high flush draws is often a mistake because you block a missed flush draw you want your opponent to have when bluffing on the river (meaning that it would subsequently be less likely he would have it, if you held two of the flush draw cards). Ace-high usually has some river showdown value.

If you had a 3-flush on the river and wanted to raise, you could bluff raise with AX combos holding the bluff suit Ace. Blocking the nut flush prevents your opponent from using that combo.

5. Bet Sizes and Frequency

GTO beginner strategies aren't just bluffs and value bets. They show how often and how much to bet in certain spots. Top players have benefited greatly from poker solvers, which we'll discuss next.

GTO Poker Software

In recent years, various poker GTO solvers have been released to help beginner, intermediate, and advanced players play balanced/GTO poker in various situations.

PokerSnowie and PioSolver are popular GTO and poker study programs.

While you can't compute players' hand ranges and what hands to bet or check with in real time, studying GTO play strategies with these programs will pay off. It will improve your poker thinking and understanding.

Solvers can help you balance ranges, choose optimal bet sizes, and master cbet frequencies.

GTO Poker Tournament

Late-stage tournaments have shorter stacks than cash games. In order to follow GTO poker guidelines, Nash charts have been created, tweaked, and used for many years (and also when to call, depending on what number of big blinds you have when you find yourself shortstacked).

The charts are for heads-up push/fold. In a multi-player game, the "pusher" chart can only be used if play is folded to you in the small blind. The "caller" chart can only be used if you're in the big blind and assumes a small blind "pusher" (with a much wider range than if a player in another position was open-shoving).

Divide the pusher chart's numbers by 2 to see which hand to use from the Button. Divide the original chart numbers by 4 to find the CO's pushing range. Some of the figures will be impossible to calculate accurately for the CO or positions to the right of the blinds because the chart's highest figure is "20+" big blinds, which is also used for a wide range of hands in the push chart.

Both of the GTO charts below are ideal for heads-up play, but exploitable HU shortstack strategies can lead to more +EV decisions against certain opponents. Following the charts will make your play GTO and unexploitable.

Poker pro Max Silver created the GTO push/fold software SnapShove. (It's accessible online at www.snapshove.com or as iOS or Android apps.)

Players can access GTO shove range examples in the full version. (You can customize the number of big blinds you have, your position, the size of the ante, and many other options.)

In Conclusion

Due to the constantly changing poker landscape, players are always improving their skills. Exploitable strategies often yield higher profit margins than GTO-based approaches, but knowing GTO beginner and advanced concepts can give you an edge for a few reasons.

It creates a solid gameplay base.

Having a baseline makes it easier to exploit certain villains.

You can avoid leveling wars with your opponents by making sound poker decisions based on GTO strategy.

It doesn't require assuming opponents' play styles.

Not results-oriented.

This is just the beginning of GTO and poker theory. Consider investing in the GTO poker solver software listed above to improve your game.

I pitched 10 brands per workday for a total of 200.

How did I do?

It was difficult.

I've never pitched so much.

What did this challenge teach me?

• the superiority of quality over quantity

• When you need help, outsource

• Don't disregard burnout in order to complete a challenge because it exists.

First, pitching brands for brand deals requires quality. Find firms that align with your brand to expose to your audience.

If you associate with any company, you'll lose audience loyalty. I didn't lose sight of that, but I couldn't resist finishing the task.

Outsourcing.

Delegating work to teammates is effective.

I wish I'd done it.

Three people can pitch 200 companies a month significantly faster than one.

One person does research, one to two do outreach, and one to two do follow-up and negotiating.

Simple.

In 2022, I'll outsource everything.

Burnout.

I felt this, so I slowed down at the end of the month.

Thanksgiving week in November was slow.

I was buying and decorating for Christmas. First time putting up outdoor holiday lights was fun.

Much was happening.

I'm not perfect.

I'm being honest.

The Outcomes

Less than 50 brands pitched.

Result: A deal with 3 brands.

I hoped for 4 brands with reaching out to 200 companies, so three with under 50 is wonderful.

That’s a 6% conversion rate!

Whoo-hoo!

I needed 2%.

Here's a screenshot from one of the deals I booked.

These companies fit my company well. Each campaign is different, but I've booked $2,450 in brand work with a couple of pending transactions for December and January.

$2,450 in brand work booked!

How did I do? You tell me.

Is this something you’d try yourself?

It’s simply not conceivable, or at least not credible.

One of the most touted selling points of Crypto has always been this grandiose idea of decentralization. Bitcoin first arose in 2009 after the housing crisis and subsequent crash that came with it. It aimed to solve this supposed issue of centralization. Nobody “owns” Bitcoin in theory, so the idea then goes that it won’t be subject to the same downfalls that led to the 2008 crash or similarly speculative events that led to the 2008 disaster. The issue is the banks, not the human nature associated with the greedy individuals running them.

Subsequent blockchains have attempted to fix many of the issues of Bitcoin by increasing capacity, decreasing the costs and processing times associated with Bitcoin, and expanding what can be done with their blockchains. Since nobody owns Bitcoin, it hasn’t really been able to be expanded on. You have people like Vitalk Buterin, however, that actively work on Ethereum though.

The leap from Bitcoin to Ethereum was a massive leap toward centralization, and the trend has only gotten worse. In fact, crypto has since become almost exclusively centralized in recent years.

Decentralization is only good in theory

It’s a good idea. In fact, it’s a wonderful idea. However, like other utopian societies, individuals misjudge human nature and greed. In a perfect world, decentralization would certainly be a wonderful idea because sure, people may function as their own banks, move payments immediately, remain anonymous, and so on. However, underneath this are a couple issues:

• You can already send money instantaneously today.

• They are not decentralized.

• Decentralization is a bad idea.

• Being your own bank is a stupid move.

Let’s break these down. Some are quite simple, but lets have a look.

Sending money right away

One thing with crypto is the idea that you can send payments instantly. This has pretty much been entirely solved in current times. You can transmit significant sums of money instantly for a nominal cost and it’s instantaneously cleared. Venmo was launched in 2009 and has since increased to prominence, and currently is on most people's phones. I can directly send ANY amount of money quickly from my bank to another person's Venmo account.

Comparing that with ETH and Bitcoin, Venmo wins all around. I can send money to someone for free instantly in dollars and the only fee paid is optional depending on when you want it.

Both Bitcoin and Ethereum are subject to demand. If the blockchains have a lot of people trying to process transactions fee’s go up, and the time that it takes to receive your crypto takes longer. When Ethereum gets bad, people have reported spending several thousand of dollars on just 1 transaction.

These transactions take place via “miners” bundling and confirming transactions, then recording them on the blockchain to confirm that the transaction did indeed happen. They charge fees to do this and are also paid in Bitcoin/ETH. When a transaction is confirmed, it's then sent to the other users wallet. This within itself is subject to lots of controversy because each transaction needs to be confirmed 6 times, this takes massive amounts of power, and most of the power is wasted because this is an adversarial system in which the person that mines the transaction gets paid, and everyone else is out of luck. Also, these could theoretically be subject to a “51% attack” in which anyone with over 51% of the mining hash rate could effectively control all of the transactions, and reverse transactions while keeping the BTC resulting in “double spending”.

There are tons of other issues with this, but essentially it means: They rely on these third parties to confirm the transactions. Without people confirming these transactions, Bitcoin stalls completely, and if anyone becomes too dominant they can effectively control bitcoin.

Not to mention, these transactions are in Bitcoin and ETH, not dollars. So, you need to convert them to dollars still, and that's several more transactions, and likely to take several days anyway as the centralized exchange needs to send you the money by traditional methods.

They are not distributed

That takes me to the following point. This isn’t decentralized, at all. Bitcoin is the closest it gets because Satoshi basically closed it to new upgrades, although its still subject to:

• Whales

• Miners

It’s vital to realize that these are often the same folks. While whales aren’t centralized entities typically, they can considerably effect the price and outcome of Bitcoin. If the largest wallets holding as much as 1 million BTC were to sell, it’d effectively collapse the price perhaps beyond repair. However, Bitcoin can and is pretty much controlled by the miners. Further, Bitcoin is more like an oligarchy than decentralized. It’s been effectively used to make the rich richer, and both the mining and price is impacted by the rich. The overwhelming minority of those actually using it are retail investors. The retail investors are basically never the ones generating money from it either.

As far as ETH and other cryptos go, there is realistically 0 case for them being decentralized. Vitalik could not only kill it but even walking away from it would likely lead to a significant decline. It has tons of issues right now that Vitalik has promised to fix with the eventual Ethereum 2.0., and stepping away from it wouldn’t help.

Most tokens as well are generally tied to some promise of future developments and creators. The same is true for most NFT projects. The reason 99% of crypto and NFT projects fail is because they failed to deliver on various promises or bad dev teams, or poor innovation, or the founders just straight up stole from everyone. I could go more in-depth than this but go find any project and if there is a dev team, company, or person tied to it then it's likely, not decentralized. The success of that project is directly tied to the dev team, and if they wanted to, most hold large wallets and could sell it all off effectively killing the project. Not to mention, any crypto project that doesn’t have a locked contract can 100% be completely rugged and they can run off with all of the money.

Decentralization is undesirable

Even if they were decentralized then it would not be a good thing. The graphic above indicates this is effectively a rich person’s unregulated playground… so it’s exactly like… the very issue it tried to solve?

Not to mention, it’s supposedly meant to prevent things like 2008, but is regularly subjected to 50–90% drawdowns in value? Back when Bitcoin was only known in niche parts of the dark web and illegal markets, it would regularly drop as much as 90% and has a long history of massive drawdowns.

The majority of crypto is blatant scams, and ALL of crypto is a “zero” or “negative” sum game in that it relies on the next person buying for people to make money. This is not a good thing. This has yet to solve any issues around what caused the 2008 crisis. Rather, it seemingly amplified all of the bad parts of it actually. Crypto is the ultimate speculative asset and realistically has no valuation metric. People invest in Apple because it has revenue and cash on hand. People invest in crypto purely for speculation. The lack of regulation or accountability means this is amplified to the most extreme degree where anything goes: Fraud, deception, pump and dumps, scams, etc. This results in a pure speculative madhouse where, unsurprisingly, only the rich win. Not only that but the deck is massively stacked in against the everyday investor because you can’t do a pump and dump without money.

At the heart of all of this is still the same issues: greed and human nature. However, in setting out to solve the issues that allowed 2008 to happen, they made something that literally took all of the bad parts of 2008 and then amplified it. 2008, similarly, was due to greed and human nature but was allowed to happen due to lack of oversite, rich people's excessive leverage over the poor, and excessive speculation. Crypto trades SOLELY on human emotion, has 0 oversite, is pure speculation, and the power dynamic is just as bad or worse.

Why should each individual be their own bank?

This is the last one, and it's short and basic. Why do we want people functioning as their own bank? Everything we do relies on another person. Without the internet, and internet providers there is no crypto. We don’t have people functioning as their own home and car manufacturers or internet service providers. Sure, you might specialize in some of these things, but masquerading as your own bank is a horrible idea.

I am not in the banking industry so I don’t know all the issues with banking. Most people aren’t in banking or crypto, so they don’t know the ENDLESS scams associated with it, and they are bound to lose their money eventually.

If you appreciate this article and want to read more from me and authors like me, without any limits, consider buying me a coffee: buymeacoffee.com/calebnaysmith

The 2008 Bitcoin white paper solves the classic computer science consensus problem.

Issue Statement

The Byzantine Generals Problem (BGP) is called after an allegory in which several generals must collaborate and attack a city at the same time to win (figure 1-left). Any general who retreats at the last minute loses the fight (figure 1-right). Thus, precise messengers and no rogue generals are essential. This is difficult without a trusted central authority.

In their 1982 publication, Leslie Lamport, Robert Shostak, and Marshall Please termed this topic the Byzantine Generals Problem to simplify distributed computer systems.

Consensus in a distributed computer network is the issue. Reaching a consensus on which systems work (and stay in the network) and which don't makes maintaining a network tough (i.e., needs to be removed from network). Challenges include unreliable communication routes between systems and mis-reporting systems.

Solving BGP can let us construct machine learning solutions without single points of failure or trusted central entities. One server hosts model parameters while numerous workers train the model. This study describes fault-tolerant Distributed Byzantine Machine Learning.

Bitcoin invented a mechanism for a distributed network of nodes to agree on which transactions should go into the distributed ledger (blockchain) without a trusted central body. It solved BGP implementation. Satoshi Nakamoto, the pseudonymous bitcoin creator, solved the challenge by cleverly combining cryptography and consensus mechanisms.

Disclaimer

This is not financial advice. It discusses a unique computer science solution.

Bitcoin

Bitcoin's white paper begins:

“A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution.” Source: https://www.ussc.gov/sites/default/files/pdf/training/annual-national-training-seminar/2018/Emerging_Tech_Bitcoin_Crypto.pdf

Bitcoin's main parts:

1. The open-source and versioned bitcoin software that governs how nodes, miners, and the bitcoin token operate.

2. The native kind of token, known as a bitcoin token, may be created by mining (up to 21 million can be created), and it can be transferred between wallet addresses in the bitcoin network.

3. Distributed Ledger, which contains exact copies of the database (or "blockchain") containing each transaction since the first one in January 2009.

4. distributed network of nodes (computers) running the distributed ledger replica together with the bitcoin software. They broadcast the transactions to other peer nodes after validating and accepting them.

5. Proof of work (PoW) is a cryptographic requirement that must be met in order for a miner to be granted permission to add a new block of transactions to the blockchain of the cryptocurrency bitcoin. It takes the form of a valid hash digest. In order to produce new blocks on average every 10 minutes, Bitcoin features a built-in difficulty adjustment function that modifies the valid hash requirement (length of nonce). PoW requires a lot of energy since it must continually generate new hashes at random until it satisfies the criteria.

6. The competing parties known as miners carry out continuous computing processing to address recurrent cryptography issues. Transaction fees and some freshly minted (mined) bitcoin are the rewards they receive. The amount of hashes produced each second—or hash rate—is a measure of mining capacity.

Cryptography, decentralization, and the proof-of-work consensus method are Bitcoin's most unique features.

Bitcoin uses encryption

Bitcoin employs this established cryptography.

1. Hashing

2. digital signatures based on asymmetric encryption

Hashing (SHA-256) (SHA-256)

Hashing converts unique plaintext data into a digest. Creating the plaintext from the digest is impossible. Bitcoin miners generate new hashes using SHA-256 to win block rewards.

A new hash is created from the current block header and a variable value called nonce. To achieve the required hash, mining involves altering the nonce and re-hashing.

The block header contains the previous block hash and a Merkle root, which contains hashes of all transactions in the block. Thus, a chain of blocks with increasing hashes links back to the first block. Hashing protects new transactions and makes the bitcoin blockchain immutable. After a transaction block is mined, it becomes hard to fabricate even a little entry.

Asymmetric Cryptography Digital Signatures

Asymmetric cryptography (public-key encryption) requires each side to have a secret and public key. Public keys (wallet addresses) can be shared with the transaction party, but private keys should not. A message (e.g., bitcoin payment record) can only be signed by the owner (sender) with the private key, but any node or anybody with access to the public key (visible in the blockchain) can verify it. Alex will submit a digitally signed transaction with a desired amount of bitcoin addressed to Bob's wallet to a node to send bitcoin to Bob. Alex alone has the secret keys to authorize that amount. Alex's blockchain public key allows anyone to verify the transaction.

Solution

Now, apply bitcoin to BGP. BGP generals resemble bitcoin nodes. The generals' consensus is like bitcoin nodes' blockchain block selection. Bitcoin software on all nodes can:

Check transactions (i.e., validate digital signatures)

2. Accept and propagate just the first miner to receive the valid hash and verify it accomplished the task. The only way to guess the proper hash is to brute force it by repeatedly producing one with the fixed/current block header and a fresh nonce value.

Thus, PoW and a dispersed network of nodes that accept blocks from miners that solve the unfalsifiable cryptographic challenge solve consensus.

Suppose:

1. Unreliable nodes

2. Unreliable miners

Bitcoin accepts the longest chain if rogue nodes cause divergence in accepted blocks. Thus, rogue nodes must outnumber honest nodes in accepting/forming the longer chain for invalid transactions to reach the blockchain. As of November 2022, 7000 coordinated rogue nodes are needed to takeover the bitcoin network.

Dishonest miners could also try to insert blocks with falsified transactions (double spend, reverse, censor, etc.) into the chain. This requires over 50% (51% attack) of miners (total computational power) to outguess the hash and attack the network. Mining hash rate exceeds 200 million (source). Rewards and transaction fees encourage miners to cooperate rather than attack. Quantum computers may become a threat.

Visit my Quantum Computing post.

Quantum computers—what are they? Quantum computers will have a big influence. towardsdatascience.com

Nodes have more power than miners since they can validate transactions and reject fake blocks. Thus, the network is secure if honest nodes are the majority.

Summary

Table 1 compares three Byzantine Generals Problem implementations.

Bitcoin white paper and implementation solved the consensus challenge of distributed systems without central governance. It solved the illusive Byzantine Generals Problem.

Resources

Resources

1. https://en.wikipedia.org/wiki/Byzantine_fault

2. Source-code for Bitcoin Core Software — https://github.com/bitcoin/bitcoin

3. Bitcoin white paper — https://bitcoin.org/bitcoin.pdf

4. https://en.wikipedia.org/wiki/Bitcoin

5. https://www.microsoft.com/en-us/research/publication/byzantine-generals-problem/

6. https://www.microsoft.com/en-us/research/uploads/prod/2016/12/The-Byzantine-Generals-Problem.pdf

7. https://en.wikipedia.org/wiki/Hash_function

8. https://en.wikipedia.org/wiki/Merkle_tree

9. https://en.wikipedia.org/wiki/SHA-2

10. https://en.wikipedia.org/wiki/Public-key_cryptography

11. https://en.wikipedia.org/wiki/Digital_signature

12. https://en.wikipedia.org/wiki/Proof_of_work

13. https://en.wikipedia.org/wiki/Quantum_cryptography

14. https://dci.mit.edu/bitcoin-security-initiative

15. https://dci.mit.edu/51-attacks

16. Genuinely Distributed Byzantine Machine Learning, El-Mahdi El-Mhamdi et al., 2020. ACM, New York, NY, https://doi.org/10.1145/3382734.3405695