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.

I analyze the top 200 DeFi crypto projects in this article.

This isn't a study. The findings benefit crypto investors.

Let’s go!

A set of data

I analyzed data from defillama.com. In my analysis, I used the top 200 DeFis by TVL in October 2022.

Total Locked Value

The chart below shows platform-specific locked value.

14 platforms had $1B+ TVL. 65 platforms have $100M-$1B TVL. The remaining 121 platforms had TVLs below $100 million, with the lowest being $23 million.

TVLs are distributed Pareto. Top 40% of DeFis account for 80% of TVLs.

Compliant Blockchains

Ethereum's blockchain leads DeFi. 96 of the examined projects offer services on Ethereum. Behind BSC, Polygon, and Avalanche.

Five platforms used 10+ blockchains. 36 between 2-10 159 used 1 blockchain.

Use Cases for DeFi

The chart below shows platform use cases. Each platform has decentralized exchanges, liquid staking, yield farming, and lending.

These use cases are DefiLlama's main platform features.

Which use case costs the most? Chart explains. Collateralized debt, liquid staking, dexes, and lending have high TVLs.

The DeFi Industry

I compared three high-TVL platforms (Maker DAO, Balancer, AAVE). The columns show monthly TVL and token price changes. The graph shows monthly Bitcoin price changes.

Each platform's market moves similarly.

Probably because most DeFi deposits are cryptocurrencies. Since individual currencies are highly correlated with Bitcoin, it's not surprising that they move in unison.

Takeaways

This analysis shows that the most common DeFi services (decentralized exchanges, liquid staking, yield farming, and lending) also have the highest average locked value.

Some projects run on one or two blockchains, while others use 15 or 20. Our analysis shows that a project's blockchain count has no correlation with its success.

It's hard to tell if certain use cases are rising. Bitcoin's price heavily affects the entire DeFi market.

TVL seems to be a good indicator of a DeFi platform's success and quality. Higher TVL platforms are cheaper. They're a better long-term investment because they gain or lose less value than DeFis with lower TVLs.

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’m a permaculture farmer. I want to create food-producing ecosystems. My hope is a world with easy access to a cuisine that nourishes consumers, supports producers, and leaves the Earth joyously habitable.

Permaculturists, natural farmers, plantsmen, and foodies share this ambition. I believe this group of green thumbs, stock-folk, and food champions is falling to tribalism, forgetting that rescuing the globe requires saving all of its inhabitants, even those who adore cheap burgers and Coke. We're digging foxholes and turning folks who disagree with us or don't understand into monsters.

Take Dr. Daphne Miller's comments at the end of her Slow Money Journal interview:

“Americans are going to fall into two camps when all is said and done: People who buy cheap goods, regardless of quality, versus people who are willing and able to pay for things that are made with integrity. We are seeing the limits of the “buying cheap crap” approach.”

This is one of the most judgmental things I've read outside the Bible. Consequences:

• People who purchase inexpensive things (food) are ignorant buffoons who prefer to choose fair trade coffee over fuel as long as the price is correct.

• It all depends on your WILL to buy quality or cheaply. Both those who are WILLING and those who ARE NOT exist. And able, too.

• People who are unwilling and unable are purchasing garbage. You're giving your kids bad food. Both the Earth and you are being destroyed by your actions. Your camp is the wrong one. You’re garbage! Disgrace to you.

Dr. Miller didn't say it, but words are worthless until interpreted. This interpretation depends on the interpreter's economic, racial, political, religious, family, and personal history. Complementary language insults another. Imagine how that Brown/Harvard M.D.'s comment sounds to a low-income household with no savings.

Dr. Miller's comment reflects the echo chamber into which nearly all clean food advocates speak. It asks easy questions and accepts non-solutions like raising food prices and eating less meat. People like me have cultivated an insular world unencumbered by challenges beyond the margins. We may disagree about technical details in rotationally-grazing livestock, but we short circuit when asked how our system could supply half the global beef demand. Most people have never seriously considered this question. We're so loved and affirmed that challenging ourselves doesn't seem necessary. Were generals insisting we don't need to study the terrain because God is on our side?

“Yes, the $8/lb ground beef is produced the way it should be. Yes, it’s good for my body. Yes it’s good for the Earth. But it’s eight freaking dollars, and my kid needs braces and protein. Bye Felicia, we’re going to McDonald’s.”

-Bobby Q. Homemaker

Funny clean foodies. People don't pay enough for food; they should value it more. Turn the concept of buying food with integrity into a wedge and drive it into the heart of America, dividing the willing and unwilling.

We go apeshit if you call our products high-end.

I've heard all sorts of gaslighting to defend a $10/lb pork chop as accessible (things I’ve definitely said in the past):

• At Whole Foods, it costs more.

• The steak at the supermarket is overly affordable.

• Pay me immediately or the doctor gets paid later.

I spoke with Timbercreek Market and Local Food Hub in front of 60 people. We were asked about local food availability.

They came to me last, after my co-panelists gave the same responses I would have given two years before.

I grumbled, "Our food is inaccessible." Nope. It's beyond the wallets of nearly everyone, and it's the biggest problem with sustainable food systems. We're criminally unserious about being leaders in sustainability until we propose solutions beyond economic relativism, wishful thinking, and insisting that vulnerable, distracted people do all the heavy lifting of finding a way to afford our food. And until we talk about solutions, all this preserve the world? False.

The room fell silent as if I'd revealed a terrible secret. Long, thunderous applause followed my other remarks. But I’m probably not getting invited back to any VNRLI events.

I make pricey cuisine. It’s high-end. I have customers who really have to stretch to get it, and they let me know it. They're forgoing other creature comforts to help me make a living and keep the Earth of my grandmothers alive, and they're doing it as an act of love. They believe in us and our work.

I remember it when I'm up to my shoulders in frigid water, when my vehicle stinks of four types of shit, when I come home covered in blood and mud, when I'm hauling water in 100-degree heat, when I'm herding pigs in a rainstorm and dodging lightning bolts to close the chickens. I'm reminded I'm not alone. Their enthusiasm is worth more than money; it helps me make a life and a living. I won't label that gift less than it is to make my meal seem more accessible.

Not everyone can sacrifice.

Let's not pretend we want to go back to peasant fare, despite our nostalgia. Industrial food has leveled what rich and poor eat. How food is cooked will be the largest difference between what you and a billionaire eat. Rich and poor have access to chicken, pork, and beef. You might be shocked how recently that wasn't the case. This abundance, particularly of animal protein, has helped vulnerable individuals.

Industrial food causes environmental damage, chronic disease, and distribution inequities. Clean food promotes non-industrial, artisan farming. This creates a higher-quality, more expensive product than the competition; we respond with aggressive marketing and the "people need to value food more" shtick geared at consumers who can spend the extra money.

The guy who is NOT able is rendered invisible by clean food's elitist marketing, which is bizarre given a.) clean food insists it's trying to save the world, yet b.) MOST PEOPLE IN THE WORLD ARE THAT GUY. No one can help him except feel-good charities. That's crazy.

Also wrong: a foodie telling a kid he can't eat a 99-cent fast food hamburger because it lacks integrity. Telling him how easy it is to save his ducketts and maybe have a grass-fed house burger at the end of the month as a reward, but in the meantime get your protein from canned beans you can't bake because you don't have a stove and, even if you did, your mom works two jobs and moonlights as an Uber driver so she doesn't have time to heat that shitup anyway.

A wealthy person's attitude toward the poor is indecent. It's 18th-century Versailles.

Human rights include access to nutritious food without social or environmental costs. As a food-forest-loving permaculture farmer, I no longer balk at the concept of cultured beef and hydroponics. My food is out of reach for many people, but access to decent food shouldn't be. Cultures and hydroponics could scale to meet the clean food affordability gap without externalities. If technology can deliver great, affordable beef without environmental negative effects, I can't reject it because it's new, unusual, or might endanger my business.

Why is your farm needed if cultured beef and hydroponics can feed the world? Permaculture food forests with trees, perennial plants, and animals are crucial to economically successful environmental protection. No matter how advanced technology gets, we still need clean air, water, soil, greenspace, and food.

Clean Food cultivated in/on live soil, minimally processed, and eaten close to harvest is part of the answer, not THE solution. Clean food advocates must recognize the conflicts at the intersection of environmental, social, and economic sustainability, the disproportionate effects of those conflicts on the poor and lower-middle classes, and the immorality and impracticality of insisting vulnerable people address those conflicts on their own and judging them if they don't.

Our clients, relatives, friends, and communities need an honest assessment of our role in a sustainable future. If we're serious about preserving the world, we owe honesty to non-customers. We owe our goal and sanity to honesty. Future health and happiness of the world left to the average person's pocketbook and long-term moral considerations is a dismal proposition with few parallels.

Let's make soil and grow food. Let the lab folks do their thing. We're all interdependent.