Zero-Knowledge Proofs for Beginners

Published here originally.

Introduction

I Spy—did you play as a kid? One person chose a room object, and the other had to guess it by answering yes or no questions. I Spy was entertaining, but did you know it could teach you cryptography?

Zero Knowledge Proofs let you show your pal you know what they picked without exposing how. Math replaces electronics in this secret spy mission. Zero-knowledge proofs (ZKPs) are sophisticated cryptographic tools that allow one party to prove they have particular knowledge without revealing it. This proves identification and ownership, secures financial transactions, and more. This article explains zero-knowledge proofs and provides examples to help you comprehend this powerful technology.

What is a Proof of Zero Knowledge?

Zero-knowledge proofs prove a proposition is true without revealing any other information. This lets the prover show the verifier that they know a fact without revealing it. So, a zero-knowledge proof is like a magician's trick: the prover proves they know something without revealing how or what. Complex mathematical procedures create a proof the verifier can verify.

Want to find an easy way to test it out? Try out with tis awesome example! ZK Crush

Describe it as if I'm 5

Alex and Jack found a cave with a center entrance that only opens when someone knows the secret. Alex knows how to open the cave door and wants to show Jack without telling him.

Alex and Jack name both pathways (let’s call them paths A and B).

1. In the first phase, Alex is already inside the cave and is free to select either path, in this case A or B.

2. As Alex made his decision, Jack entered the cave and asked him to exit from the B path.

3. Jack can confirm that Alex really does know the key to open the door because he came out for the B path and used it.

To conclude, Alex and Jack repeat:

1. Alex walks into the cave.

2. Alex follows a random route.

3. Jack walks into the cave.

4. Alex is asked to follow a random route by Jack.

5. Alex follows Jack's advice and heads back that way.

What is a Zero Knowledge Proof?

At a high level, the aim is to construct a secure and confidential conversation between the prover and the verifier, where the prover convinces the verifier that they have the requisite information without disclosing it. The prover and verifier exchange messages and calculate in each round of the dialogue.

The prover uses their knowledge to prove they have the information the verifier wants during these rounds. The verifier can verify the prover's truthfulness without learning more by checking the proof's mathematical statement or computation.

Zero knowledge proofs use advanced mathematical procedures and cryptography methods to secure communication. These methods ensure the evidence is authentic while preventing the prover from creating a phony proof or the verifier from extracting unnecessary information.

ZK proofs require examples to grasp. Before the examples, there are some preconditions.

Criteria for Proofs of Zero Knowledge

1. Completeness: If the proposition being proved is true, then an honest prover will persuade an honest verifier that it is true.

2. Soundness: If the proposition being proved is untrue, no dishonest prover can persuade a sincere verifier that it is true.

3. Zero-knowledge: The verifier only realizes that the proposition being proved is true. In other words, the proof only establishes the veracity of the proposition being supported and nothing more.

The zero-knowledge condition is crucial. Zero-knowledge proofs show only the secret's veracity. The verifier shouldn't know the secret's value or other details.

Example after example after example

To illustrate, take a zero-knowledge proof with several examples:

Initial Password Verification Example

You want to confirm you know a password or secret phrase without revealing it.

Use a zero-knowledge proof:

1. You and the verifier settle on a mathematical conundrum or issue, such as figuring out a big number's components.

2. The puzzle or problem is then solved using the hidden knowledge that you have learned. You may, for instance, utilize your understanding of the password to determine the components of a particular number.

3. You provide your answer to the verifier, who can assess its accuracy without knowing anything about your private data.

4. You go through this process several times with various riddles or issues to persuade the verifier that you actually are aware of the secret knowledge.

You solved the mathematical puzzles or problems, proving to the verifier that you know the hidden information. The proof is zero-knowledge since the verifier only sees puzzle solutions, not the secret information.

In this scenario, the mathematical challenge or problem represents the secret, and solving it proves you know it. The evidence does not expose the secret, and the verifier just learns that you know it.

My simple example meets the zero-knowledge proof conditions:

1. Completeness: If you actually know the hidden information, you will be able to solve the mathematical puzzles or problems, hence the proof is conclusive.

2. Soundness: The proof is sound because the verifier can use a publicly known algorithm to confirm that your answer to the mathematical conundrum or difficulty is accurate.

3. Zero-knowledge: The proof is zero-knowledge because all the verifier learns is that you are aware of the confidential information. Beyond the fact that you are aware of it, the verifier does not learn anything about the secret information itself, such as the password or the factors of the number. As a result, the proof does not provide any new insights into the secret.

Explanation #2: Toss a coin.

One coin is biased to come up heads more often than tails, while the other is fair (i.e., comes up heads and tails with equal probability). You know which coin is which, but you want to show a friend you can tell them apart without telling them.

Use a zero-knowledge proof:

1. One of the two coins is chosen at random, and you secretly flip it more than once.

2. You show your pal the following series of coin flips without revealing which coin you actually flipped.

3. Next, as one of the two coins is flipped in front of you, your friend asks you to tell which one it is.

4. Then, without revealing which coin is which, you can use your understanding of the secret order of coin flips to determine which coin your friend flipped.

5. To persuade your friend that you can actually differentiate between the coins, you repeat this process multiple times using various secret coin-flipping sequences.

In this example, the series of coin flips represents the knowledge of biased and fair coins. You can prove you know which coin is which without revealing which is biased or fair by employing a different secret sequence of coin flips for each round.

The evidence is zero-knowledge since your friend does not learn anything about which coin is biased and which is fair other than that you can tell them differently. The proof does not indicate which coin you flipped or how many times you flipped it.

The coin-flipping example meets zero-knowledge proof requirements:

1. Completeness: If you actually know which coin is biased and which is fair, you should be able to distinguish between them based on the order of coin flips, and your friend should be persuaded that you can.

2. Soundness: Your friend may confirm that you are correctly recognizing the coins by flipping one of them in front of you and validating your answer, thus the proof is sound in that regard. Because of this, your acquaintance can be sure that you are not just speculating or picking a coin at random.

3. Zero-knowledge: The argument is that your friend has no idea which coin is biased and which is fair beyond your ability to distinguish between them. Your friend is not made aware of the coin you used to make your decision or the order in which you flipped the coins. Consequently, except from letting you know which coin is biased and which is fair, the proof does not give any additional information about the coins themselves.

Figure out the prime number in Example #3.

You want to prove to a friend that you know their product n=pq without revealing p and q. Zero-knowledge proof?

Use a variant of the RSA algorithm. Method:

1. You determine a new number s = r2 mod n by computing a random number r.

2. You email your friend s and a declaration that you are aware of the values of p and q necessary for n to equal pq.

3. A random number (either 0 or 1) is selected by your friend and sent to you.

4. You send your friend r as evidence that you are aware of the values of p and q if e=0. You calculate and communicate your friend's s/r if e=1.

5. Without knowing the values of p and q, your friend can confirm that you know p and q (in the case where e=0) or that s/r is a legitimate square root of s mod n (in the situation where e=1).

This is a zero-knowledge proof since your friend learns nothing about p and q other than their product is n and your ability to verify it without exposing any other information. You can prove that you know p and q by sending r or by computing s/r and sending that instead (if e=1), and your friend can verify that you know p and q or that s/r is a valid square root of s mod n without learning anything else about their values. This meets the conditions of completeness, soundness, and zero-knowledge.

Zero-knowledge proofs satisfy the following:

1. Completeness: The prover can demonstrate this to the verifier by computing q = n/p and sending both p and q to the verifier. The prover also knows a prime number p and a factorization of n as p*q.

2. Soundness: Since it is impossible to identify any pair of numbers that correctly factorize n without being aware of its prime factors, the prover is unable to demonstrate knowledge of any p and q that do not do so.

3. Zero knowledge: The prover only admits that they are aware of a prime number p and its associated factor q, which is already known to the verifier. This is the extent of their knowledge of the prime factors of n. As a result, the prover does not provide any new details regarding n's prime factors.

Types of Proofs of Zero Knowledge

Each zero-knowledge proof has pros and cons. Most zero-knowledge proofs are:

1. Interactive Zero Knowledge Proofs: The prover and the verifier work together to establish the proof in this sort of zero-knowledge proof. The verifier disputes the prover's assertions after receiving a sequence of messages from the prover. When the evidence has been established, the prover will employ these new problems to generate additional responses.

2. Non-Interactive Zero Knowledge Proofs: For this kind of zero-knowledge proof, the prover and verifier just need to exchange a single message. Without further interaction between the two parties, the proof is established.

3. A statistical zero-knowledge proof is one in which the conclusion is reached with a high degree of probability but not with certainty. This indicates that there is a remote possibility that the proof is false, but that this possibility is so remote as to be unimportant.

4. Succinct Non-Interactive Argument of Knowledge (SNARKs): SNARKs are an extremely effective and scalable form of zero-knowledge proof. They are utilized in many different applications, such as machine learning, blockchain technology, and more. Similar to other zero-knowledge proof techniques, SNARKs enable one party—the prover—to demonstrate to another—the verifier—that they are aware of a specific piece of information without disclosing any more information about that information.

5. The main characteristic of SNARKs is their succinctness, which refers to the fact that the size of the proof is substantially smaller than the amount of the original data being proved. Because to its high efficiency and scalability, SNARKs can be used in a wide range of applications, such as machine learning, blockchain technology, and more.

Uses for Zero Knowledge Proofs

ZKP applications include:

1. Verifying Identity ZKPs can be used to verify your identity without disclosing any personal information. This has uses in access control, digital signatures, and online authentication.

2. Proof of Ownership ZKPs can be used to demonstrate ownership of a certain asset without divulging any details about the asset itself. This has uses for protecting intellectual property, managing supply chains, and owning digital assets.

3. Financial Exchanges Without disclosing any details about the transaction itself, ZKPs can be used to validate financial transactions. Cryptocurrency, internet payments, and other digital financial transactions can all use this.

4. By enabling parties to make calculations on the data without disclosing the data itself, Data Privacy ZKPs can be used to preserve the privacy of sensitive data. Applications for this can be found in the financial, healthcare, and other sectors that handle sensitive data.

5. By enabling voters to confirm that their vote was counted without disclosing how they voted, elections ZKPs can be used to ensure the integrity of elections. This is applicable to electronic voting, including internet voting.

6. Cryptography Modern cryptography's ZKPs are a potent instrument that enable secure communication and authentication. This can be used for encrypted messaging and other purposes in the business sector as well as for military and intelligence operations.

Proofs of Zero Knowledge and Compliance

Kubernetes and regulatory compliance use ZKPs in many ways. Examples:

1. Security for Kubernetes ZKPs offer a mechanism to authenticate nodes without disclosing any sensitive information, enhancing the security of Kubernetes clusters. ZKPs, for instance, can be used to verify, without disclosing the specifics of the program, that the nodes in a Kubernetes cluster are running permitted software.

2. Compliance Inspection Without disclosing any sensitive information, ZKPs can be used to demonstrate compliance with rules like the GDPR, HIPAA, and PCI DSS. ZKPs, for instance, can be used to demonstrate that data has been encrypted and stored securely without divulging the specifics of the mechanism employed for either encryption or storage.

3. Access Management Without disclosing any private data, ZKPs can be used to offer safe access control to Kubernetes resources. ZKPs can be used, for instance, to demonstrate that a user has the necessary permissions to access a particular Kubernetes resource without disclosing the details of those permissions.

4. Safe Data Exchange Without disclosing any sensitive information, ZKPs can be used to securely transmit data between Kubernetes clusters or between several businesses. ZKPs, for instance, can be used to demonstrate the sharing of a specific piece of data between two parties without disclosing the details of the data itself.

5. Kubernetes deployments audited Without disclosing the specifics of the deployment or the data being processed, ZKPs can be used to demonstrate that Kubernetes deployments are working as planned. This can be helpful for auditing purposes and for ensuring that Kubernetes deployments are operating as planned.

ZKPs preserve data and maintain regulatory compliance by letting parties prove things without revealing sensitive information. ZKPs will be used more in Kubernetes as it grows.

Cryptoshit. This thing is crazy to buy.

Charlie Munger is revered and powerful in finance.

Munger, vice chairman of Berkshire Hathaway, is noted for his wit, no-nonsense attitude to investment, and ability to spot promising firms and markets.

Munger's crypto views have upset some despite his reputation as a straight shooter.

“There’s only one correct answer for intelligent people, just totally avoid all the people that are promoting it.” — Charlie Munger

The Munger Interview on CNBC (4:48 secs)

This Monday, CNBC co-anchor Rebecca Quick interviewed Munger and brought up his 2007 statement, "I'm not allowed to have an opinion on this subject until I can present the arguments against my viewpoint better than the folks who are supporting it."

Great investing and life advice!

If you can't explain the opposing reasons, you're not informed enough to have an opinion.

In today's world, it's important to grasp both sides of a debate before supporting one.

Rebecca inquired:

Does your Wall Street Journal article on banning cryptocurrency apply? If so, would you like to present the counterarguments?

Mungers reply:

I don't see any viable counterarguments. I think my opponents are idiots, hence there is no sensible argument against my position.

Consider his words.

Do you believe Munger has studied both sides?

He said, "I assume my opponents are idiots, thus there is no sensible argument against my position."

This is worrisome, especially from a guy who once encouraged studying both sides before forming an opinion.

Munger said:

National currencies have benefitted humanity more than almost anything else.

Hang on, I think we located the perpetrator.

Munger thinks crypto will replace currencies.

False.

I doubt he studied cryptocurrencies because the name is deceptive.

He misread a headline as a Dollar destroyer.

Cryptocurrencies are speculations.

Like Tesla, Amazon, Apple, Google, Microsoft, etc.

Crypto won't replace dollars.

In the interview with CNBC, Munger continued:

“I’m not proud of my country for allowing this crap, what I call the cryptoshit. It’s worthless, it’s no good, it’s crazy, it’ll do nothing but harm, it’s anti-social to allow it.” — Charlie Munger

Not entirely inaccurate.

Daily cryptos are established solely to pump and dump regular investors.

Let's get into Munger's crypto aversion.

Rat poison is bitcoin.

Munger famously dubbed Bitcoin rat poison and a speculative bubble that would implode.

Partially.

But the bubble broke. Since 2021, the market has fallen.

Scam currencies and NFTs are being eliminated, which I like.

Whoa.

Why does Munger doubt crypto?

Mungers thinks cryptocurrencies has no intrinsic value.

He worries about crypto fraud and money laundering.

Both are valid issues.

Yet grouping crypto is intellectually dishonest.

Ethereum, Bitcoin, Solana, Chainlink, Flow, and Dogecoin have different purposes and values (not saying they’re all good investments).

Fraudsters who hurt innocents will be punished.

Therefore, complaining is useless.

Why not stop it? Repair rather than complain.

Regrettably, individuals today don't offer solutions.

Blind Areas for Mungers

As with everyone, Mungers' bitcoin views may be impacted by his biases and experiences.

OK.

But Munger has always advocated classic value investing and may be wary of investing in an asset outside his expertise.

Mungers' banking and insurance investments may influence his bitcoin views.

Could a coworker or acquaintance have told him crypto is bad and goes against traditional finance?

Right?

Takeaways

Do you respect Charlie Mungers?

Yes and no, like any investor or individual.

To understand Mungers' bitcoin beliefs, you must be critical.

Mungers is a successful investor, but his views about bitcoin should be considered alongside other viewpoints.

Munger’s success as an investor has made him an influencer in the space.

Influence gives power.

He controls people's thoughts.

Munger's ok. He will always be heard.

I'll do so cautiously.

Crypto and blockchain have rebranded as web3. They probably thought it sounded better and didn't want the baggage of scam ICOs, STOs, and skirted securities laws.

It was like Facebook becoming Meta. Crypto's biggest players wanted to change public (and regulator) perception away from pump-and-dump schemes.

After the 2018 ICO gold rush, it's understandable. Every project that raised millions (or billions) never shipped a meaningful product.

Like many crazes, charlatans took the money and ran.

Despite its grifter past, web3 is THE hot topic today as more founders, venture firms, and larger institutions look to build the future decentralized internet.

Supposedly.

How often have you heard: This will change the world, fix the internet, and give people power?

Why are most of web3's biggest proponents (and beneficiaries) the same rich, powerful players who built and invested in the modern internet? It's like they want to remake and own the internet.

Something seems off about that.

Why are insiders getting preferential presale terms before the public, allowing early investors and proponents to flip dirt cheap tokens and advisors shares almost immediately after the public sale?

It's a good gig with guaranteed markups, no risk or progress.

If it sounds like insider trading, it is, at least practically. This is clear when people talk about blockchain/web3 launches and tokens.

Fast money, quick flips, and guaranteed markups/returns are common.

Incentives-wise, it's hard to blame them. Who can blame someone for following the rules to win? Is it their fault or regulators' for not leveling the playing field?

It's similar to oil companies polluting for profit, Instagram depressing you into buying a new dress, or pharma pushing an unnecessary pill.

All of that is fair game, at least until we change the playbook, because people (and corporations) change for pain or love. Who doesn't love money?

belief based on money gain

Sinclair:

“It is difficult to get a man to understand something when his salary depends upon his not understanding it.”

Bitcoin, blockchain, and web3 analogies?

Most blockchain and web3 proponents are true believers, not cynical capitalists. They believe blockchain's inherent transparency and permissionless trust allow humanity to evolve beyond our reptilian ways and build a better decentralized and democratic world.

They highlight issues with the modern internet and monopoly players like Google, Facebook, and Apple. Decentralization fixes everything

If we could give power back to the people and get governments/corporations/individuals out of the way, we'd fix everything.

Blockchain solves supply chain and child labor issues in China.

To meet Paris climate goals, reduce emissions. Create a carbon token.

Fixing online hatred and polarization Web3 Twitter and Facebook replacement.

Web3 must just be the answer for everything… your “perfect” silver bullet.

Nothing fits everyone. Blockchain has pros and cons like everything else.

Blockchain's viral, ponzi-like nature has an MLM (mid level marketing) feel. If you bought Taylor Swift's NFT, your investment is tied to her popularity.

Probably makes you promote Swift more. Play music loudly.

Here's another example:

Imagine if Jehovah’s Witnesses (or evangelical preachers…) got paid for every single person they converted to their cause.

It becomes a self-fulfilling prophecy as their faith and wealth grow.

Which breeds extremism? Ultra-Orthodox Jews are an example. maximalists

Bitcoin and blockchain are causes, religions. It's a money-making movement and ideal.

We're good at convincing ourselves of things we want to believe, hence filter bubbles.

I ignore anything that doesn't fit my worldview and seek out like-minded people, which algorithms amplify.

Then what?

Is web3 merely a new scam?

No, never!

Blockchain has many crucial uses.

Sending money home/abroad without bank fees;

Like fleeing a war-torn country and converting savings to Bitcoin;

Like preventing Twitter from silencing dissidents.

Permissionless, trustless databases could benefit society and humanity. There are, however, many limitations.

Lost password?

What if you're cheated?

What if Trump/Putin/your favorite dictator incites a coup d'état?

What-ifs abound. Decentralization's openness brings good and bad.

No gatekeepers or firefighters to rescue you.

ISIS's fundraising is also frictionless.

Community-owned apps with bad interfaces and service.

Trade-offs rule.

So what compromises does web3 make?

What are your trade-offs? Decentralization has many strengths and flaws. Like Bitcoin's wasteful proof-of-work or Ethereum's political/wealth-based proof-of-stake.

To ensure the survival and veracity of the network/blockchain and to safeguard its nodes, extreme measures have been designed/put in place to prevent hostile takeovers aimed at altering the blockchain, i.e., adding money to your own wallet (account), etc.

These protective measures require significant resources and pose challenges. Reduced speed and throughput, high gas fees (cost to submit/write a transaction to the blockchain), and delayed development times, not to mention forked blockchain chains oops, web3 projects.

Protecting dissidents or rogue regimes makes sense. You need safety, privacy, and calm.

First-world life?

What if you assumed EVERYONE you saw was out to rob/attack you? You'd never travel, trust anyone, accomplish much, or live fully. The economy would collapse.

It's like an ant colony where half the ants do nothing but wait to be attacked.

Waste of time and money.

11% of the US budget goes to the military. Imagine what we could do with the $766B+ we spend on what-ifs annually.

Is so much hypothetical security needed?

Blockchain and web3 are similar.

Does your app need permissionless decentralization? Does your scooter-sharing company really need a proof-of-stake system and 1000s of nodes to avoid Russian hackers? Why?

Worst-case scenario? It's not life or death, unless you overstate the what-ifs. Web3 proponents find improbable scenarios to justify decentralization and tokenization.

Do I need a token to prove ownership of my painting? Unless I'm a master thief, I probably bought it.

despite losing the receipt.

I do, however, love Web 3.

Enough Web3 bashing for now. Understand? Decentralization isn't perfect, but it has huge potential when applied to the right problems.

I see many of the right problems as disrupting big tech's ruthless monopolies. I wrote several years ago about how tokenized blockchains could be used to break big tech's stranglehold on platforms, marketplaces, and social media.

Tokenomics schemes can be used for good and are powerful. Here’s how.

Before the ICO boom, I made a series of predictions about blockchain/crypto's future. It's still true.

Here's where I was then and where I see web3 going:

My 11 Big & Bold Predictions for Blockchain

In the near future, people may wear crypto cash rings or bracelets.

1. While some governments repress cryptocurrency, others will start to embrace it.

2. Blockchain will fundamentally alter voting and governance, resulting in a more open election process.

3. Money freedom will lead to a more geographically open world where people will be more able to leave when there is unrest.

4. Blockchain will make record keeping significantly easier, eliminating the need for a significant portion of government workers whose sole responsibility is paperwork.

5. Overrated are smart contracts.

6. Tokens will replace company stocks.

7. Blockchain increases real estate's liquidity, value, and volatility.

8. Healthcare may be most affected.

9. Crypto could end privacy and lead to Minority Report.

10. New companies with network effects will displace incumbents.

11. Soon, people will wear rings or bracelets with crypto cash.

Some have already happened, while others are still possible.

Time will tell if they happen.

And finally:

What will web3 be?

Who will be in charge?

Closing remarks

Hope you enjoyed this web3 dive. There's much more to say, but that's for another day.

We're writing history as we go.

Tech regulation, mergers, Bitcoin surge How will history remember us?

What about web3 and blockchain?

Is this a revolution or a tulip craze?

Remember, actions speak louder than words (share them in the comments).

Your turn.

Amazon's Rings of Power is the most costly TV series ever made. This is merely a down payment towards Amazon's grand goal.

Here's a video:

Amazon bought J.R.R. Tolkien's fantasy novels for $250 million in 2017. This agreement allows Amazon to create a Tolkien series for Prime Video.

The business spent years developing and constructing a Lord of the Rings prequel. Rings of Power premiered on September 2, 2022.

It drew 25 million global viewers in 24 hours. Prime Video's biggest debut.

An Exorbitant Budget

The most expensive. First season cost $750 million to $1 billion, making it the most costly TV show ever.

Jeff Bezos has spent years looking for the next Game of Thrones, a critically and commercially successful original series. Rings of Power could help.

Why would Amazon bet $1 billion on one series?

It's Not Just About the Streaming War

It's simple to assume Amazon just wants to win. Since 2018, the corporation has been fighting Hulu, Netflix, HBO, Apple, Disney, and NBC. Each wants your money, talent, and attention. Amazon's investment goes beyond rivalry.

Subscriptions Are the Bait

Audible, Amazon Music, and Prime Video are subscription services, although the company's fundamental business is retail. Amazon's online stores contribute over 50% of company revenue. Subscription services contribute 6.8%. The company's master plan depends on these subscriptions.

Streaming videos on Prime increases membership renewals. Free trial participants are more likely to join. Members buy twice as much as non-members.

Amazon Studios doesn't generate original programming to earn from Prime Video subscriptions. It aims to retain and attract clients.

Amazon can track what you watch and buy. Its algorithm recommends items and services. Mckinsey says you'll use more Amazon products, shop at Amazon stores, and watch Amazon entertainment.

In 2015, the firm launched the first season of The Man in the High Castle, a dystopian alternate history TV series depicting a world ruled by Nazi Germany and Japan after World War II.

This $72 million production earned two Emmys. It garnered 1.15 million new Prime users globally.

When asked about his Hollywood investment, Bezos said, "A Golden Globe helps us sell more shoes."

Selling more footwear

Amazon secured a deal with DirecTV to air Thursday Night Football in restaurants and bars. First streaming service to have exclusive NFL games.

This isn't just about Thursday night football, says media analyst Ritchie Greenfield. This sells t-shirts. This may be a ticket. Amazon does more than stream games.

The Rings of Power isn't merely a production showcase, either. This sells Tolkien's fantasy novels such Lord of the Rings, The Hobbit, and The Silmarillion.

This tiny commitment keeps you in Amazon's ecosystem.