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.

I've been pondering product strategy and how to articulate it. Frameworks helped guide our thinking.

If your teams aren't working together or there's no clear path to victory, your product strategy may not be well-articulated or communicated (if you have one).

Before diving into a product strategy's details, it's important to understand its role in the bigger picture — the pieces that move your organization forward.

the overall picture

A product strategy is crucial, in my opinion. It's part of a successful product or business. It's the showpiece.

To simplify, we'll discuss four main components:

1. Vision

2. Product Management

3. Goals

4. Roadmap

Vision

Your company's mission? Your company/product in 35 years? Which headlines?

The vision defines everything your organization will do in the long term. It shows how your company impacted the world. It's your organization's rallying cry.

An ambitious but realistic vision is needed.

Without a clear vision, your product strategy may be inconsistent.

Product Management

Our main subject. Product strategy connects everything. It fulfills the vision.

In Part 2, we'll discuss product strategy.

Goals

This component can be goals, objectives, key results, targets, milestones, or whatever goal-tracking framework works best for your organization.

These product strategy metrics will help your team prioritize strategies and roadmaps.

Your company's goals should be unified. This fuels success.

Roadmap

The roadmap is your product strategy's timeline. It provides a prioritized view of your team's upcoming deliverables.

A roadmap is time-bound and includes measurable goals for your company. Your team's steps and capabilities for executing product strategy.

If your team has trouble prioritizing or defining a roadmap, your product strategy or vision is likely unclear.

Formulation of a Product Strategy

Now that we've discussed where your product strategy fits in the big picture, let's look at a framework.

A product strategy should include challenges, an approach, and actions.

Challenges

First, analyze the problems/situations you're solving. It can be customer- or company-focused.

The analysis should explain the problems and why they're important. Try to simplify the situation and identify critical aspects.

Some questions:

• What issues are we attempting to resolve?

• What obstacles—internal or otherwise—are we attempting to overcome?

• What is the opportunity, and why should we pursue it, in your opinion?

Decided Method

Second, describe your approach. This can be a set of company policies for handling the challenge. It's the overall approach to the first part's analysis.

The approach can be your company's bets, the solutions you've found, or how you'll solve the problems you've identified.

Again, these questions can help:

• What is the value that we hope to offer to our clients?

• Which market are we focusing on first?

• What makes us stand out? Our benefit over rivals?

Actions

Third, identify actions that result from your approach. Second-part actions should be these.

Coordinate these actions. You may need to add products or features to your roadmap, acquire new capabilities through partnerships, or launch new marketing campaigns. Whatever fits your challenges and strategy.

Final questions:

• What skills do we need to develop or obtain?

• What is the chosen remedy? What are the main outputs?

• What else ought to be added to our road map?

Put everything together

… and iterate!

Strategy isn't one-and-done. Changes occur. Economies change. Competitors emerge. Customer expectations change.

One unexpected event can make strategies obsolete quickly. Muscle it. Review, evaluate, and course-correct your strategies with your teams. Quarterly works. In a new or unstable industry, more often.

Too much NFT, Discord, and shitposting.

How do you choose?

How do you recruit more members to join your NFT project?

In 2021, a successful NFT project required:

• Monkey/ape artwork

• Twitter and Discord bot-filled

• Roadmap overpromise

• Goal was quick cash.

2022 and the years after will change that.

These are 6 Rules for a Strong NFT Community in 2022:

THINK LONG TERM

This relates to roadmap planning. Hype and dumb luck may drive NFT projects (ahem, goblins) but rarely will your project soar.

Instead, consider sustainability.

Plan your roadmap based on your team's abilities.

Do what you're already doing, but with NFTs, make it bigger and better.

You shouldn't copy a project's roadmap just because it was profitable.

This will lead to over-promising, team burnout, and an RUG NFT project.

OFFER VALUE

Building a great community starts with giving.

Why are musicians popular?

Because they offer entertainment for everyone, a random person becomes a fan, and more fans become a cult.

That's how you should approach your community.

TEAM UP

A great team helps.

An NFT project could have 3 or 2 people.

Credibility trumps team size.

Make sure your team can answer community questions, resolve issues, and constantly attend to them.

Don't overwork and burn out.

Your community will be able to recognize that you are trying too hard and give up on the project.

BUILD A GREAT PRODUCT

Bored Ape Yacht Club altered the NFT space.

Cryptopunks transformed NFTs.

Many others did, including Okay Bears.

What made them that way?

Because they answered a key question.

What is my NFT supposed to be?

Before planning art, this question must be answered.

NFTs can't be just jpegs.

What does it represent?

Is it a Metaverse-ready project?

What blockchain are you going to be using and why?

Set some ground rules for yourself. This helps your project's direction.

These questions will help you and your team set a direction for blockchain, NFT, and Web3 technology.

EDUCATE ON WEB3

The more the team learns about Web3 technology, the more they can offer their community.

Think tokens, metaverse, cross-chain interoperability and more.

BUILD A GREAT COMMUNITY

Several projects mistreat their communities.

They treat their community like "customers" and try to sell them NFT.

Providing Whitelists and giveaways aren't your only community-building options.

Think bigger.

Consider them family and friends, not wallets.

Consider them fans.

These are some tips to start your NFT project.