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.

Interpol has issued a red alert on Terraform Labs' CEO, South Korean prosecutors said.

After the May crash of Terra Luna revealed tax evasion issues, South Korean officials filed an arrest warrant for Do Kwon, but he is missing.

Do Kwon is now a fugitive in 195 countries after Seoul prosecutors placed him to Interpol's red list. Do Kwon hasn't commented since then. The red list allows any country's local authorities to apprehend Do Kwon.

Do Dwon and Terraform Labs were believed to have moved to Singapore days before the $40 billion wipeout, but Singapore authorities said he fled the country on September 17. Do Kwon tweeted that he wasn't on the run and cited privacy concerns.

Do Kwon was not on the red list at the time and said he wasn't "running," only to reply to his own tweet saying he hasn't jogged in a while and needed to trim calories.

Whether or not it makes sense to read too much into this, the reality is that Do Kwon is now on Interpol red list, despite the firmly asserts on twitter that he does absolutely nothing to hide.

UPDATE:

South Korean authorities are investigating alleged withdrawals of over $60 million U.S. and seeking to freeze these assets. Korean authorities believe a new wallet exchanged over 3000 BTC through OKX and Kucoin.

Do Kwon and the Luna Foundation Guard (of whom Do Kwon is a key member of) have declined all charges and dubbed this disinformation.

Singapore's Luna Foundation Guard (LFG) manages the Terra Ecosystem.

The Legal Situation

Multiple governments are searching for Do Kwon and five other Terraform Labs employees for financial markets legislation crimes.

South Korean authorities arrested a man suspected of tax fraud and Ponzi scheme.

The U.S. SEC is also examining Terraform Labs on how UST was advertised as a stablecoin. No legal precedent exists, so it's unclear what's illegal.

The future of Terraform Labs, Terra, and Terra 2 is unknown, and despite what Twitter shills say about LUNC, the company remains in limbo awaiting a decision that will determine its fate. This project isn't a wise investment.

Most don't understand the issue until it's too late.

My first startup was what entrepreneurs call the hardest. A two-sided marketplace.

Two-sided marketplaces are the hardest startups because founders must solve the chicken or the egg conundrum.

A two-sided marketplace needs suppliers and buyers. Without suppliers, buyers won't come. Without buyers, suppliers won't come. An empty marketplace and a founder striving to gain momentum result.

My first venture made me a struggling founder seeking to achieve traction for a two-sided marketplace. The company failed, and I vowed never to start another like it.

I didn’t. Unfortunately, my second venture was almost as hard. It failed like the second-hardest startup.

What kind of startup is the second-hardest?

The second-hardest startup, which is almost as hard to develop, is rarely discussed in the startup community. Because of this, I predict more founders fail each year trying to develop the second-toughest startup than the hardest.

Fairly, I have no proof. I see many startups, so I have enough of firsthand experience. From what I've seen, for every entrepreneur developing a two-sided marketplace, I'll meet at least 10 building this other challenging startup.

I'll describe a startup I just met with its two co-founders to explain the second hardest sort of startup and why it's so hard. They created a financial literacy software for parents of high schoolers.

The issue appears plausible. Children struggle with money. Parents must teach financial responsibility. Problems?

It's possible.

Buyers and users are different.

Buyer-user mismatch.

The financial literacy app I described above targets parents. The parent doesn't utilize the app. Child is end-user. That may not seem like much, but it makes customer and user acquisition and onboarding difficult for founders.

The difficulty of a buyer-user imbalance

The company developing a product faces a substantial operational burden when the buyer and end customer are different. Consider classic firms where the buyer is the end user to appreciate that responsibility.

Entrepreneurs selling directly to end users must educate them about the product's benefits and use. Each demands a lot of time, effort, and resources.

Imagine selling a financial literacy app where the buyer and user are different. To make the first sale, the entrepreneur must establish all the items I mentioned above. After selling, the entrepreneur must supply a fresh set of resources to teach, educate, or train end-users.

Thus, a startup with a buyer-user mismatch must market, sell, and train two organizations at once, requiring twice the work with the same resources.

The second hardest startup is hard for reasons other than the chicken-or-the-egg conundrum. It takes a lot of creativity and luck to solve the chicken-or-egg conundrum.

The buyer-user mismatch problem cannot be overcome by innovation or luck. Buyer-user mismatches must be solved by force. Simply said, when a product buyer is different from an end-user, founders have a lot more work. If they can't work extra, their companies fail.

I've spent over 50 hours researching one-person firms, which interest me. I've found countless one-person enterprises that made millions on the founder's determination and perseverance.

Throughout my investigation, I found three of the most outstanding one-person enterprises. These enterprises show that people who work hard and dedicate themselves to their ideas may succeed.

Eric Barone (@ConcernedApe) created Stardew Valley in 2011 to better his job prospects. Eric loved making the game, in which players inherit a farm, grow crops, raise livestock, make friends with the villagers, and form a family.

Eric handled complete game production, including 3D graphics, animations, and music, to maintain creative control. He stopped job hunting and worked 8-15 hours a day on the game.

Eric developed a Stardew Valley website and subreddit to engage with gamers and get feedback. Eric's devoted community helped him meet Steam's minimum vote requirement for single creators.

Stardew Valley sold 1 million copies in two months after Eric launched it for $15 in 2016. The game has sold 20 million copies and made $300 million.

The game's inexpensive price, outsourcing of PR, marketing, and publication, and loyal player base helped it succeed. Eric has turned down million-dollar proposals from Sony and Nintendo to sell the game and instead updates and improves it. Haunted Chocolatier is Eric's new game.

Is farming not profitable? Ask Stardew Valley creator Eric Barone.

Gary Brewer established BuiltWith to assist users find website technologies and services. BuiltWith boasts 3000 paying customers and $14 million in yearly revenue, making it a significant resource for businesses wishing to generate leads, do customer analytics, obtain business insight, compare websites, or search websites by keyword.

BuiltWith has one full-time employee, Gary, and one or two part-time contractors that help with the blog. Gary handles sales, customer service, and other company functions alone.

BuiltWith acquired popularity through blog promotions and a top Digg ranking. About Us, a domain directory, connected to BuiltWith on every domain page, boosting it. Gary introduced $295–$995 monthly subscriptions to search technology, keywords, and potential consumers in response to customer demand.

Gary uses numerous methods to manage a firm without staff. He spends one to two hours every day answering user queries, most of which are handled quickly by linking to BuiltWiths knowledge store. Gary creates step-by-step essays or videos for complex problems. Gary can focus on providing new features based on customer comments and requests since he makes it easy to unsubscribe.

BuiltWith is entirely automated and successful due to its unique approach and useful offerings. It works for Google, Meta, Amazon, and Twitter.

Digital Inspiration develops Google Documents, Sheets, and Slides plugins. Digital Inspiration, founded by Amit Agarwal, receives 5 million monthly visits and earns $10 million. 40 million individuals have downloaded Digital Inspirations plugins.

Amit started Digital Inspiration by advertising his blog at tech events and getting Indian filter blogs and other newspapers to promote his articles. Amit built plugins and promoted them on the blog once the blog acquired popularity, using ideas from comments, friends, and Reddit. Digital Inspiration has over 20 free and premium plugins.

Mail Merge, Notifications for Google Forms, YouTube Uploader, and Document Studio are some of Digital Inspiration's most popular plugins. Mail Merge allows users to send personalized emails in bulk and track email opens and clicks.

Since Amits manages Digital Inspiration alone, his success is astounding. Amit developed a successful company via hard work and creativity, despite platform dependence. His tale inspires entrepreneurs.

Calmness is a vital life skill.

It aids communication. It boosts creativity and performance.

I've studied calm people's habits for years. Commonalities:

Have learned to laugh at themselves.

Those who have something to protect can’t help but make it a very serious business, which drains the energy out of the room.

They are fixated on positive pursuits like making cool things, building a strong physique, and having fun with others rather than on depressing influences like the news and gossip.

Every day, spend at least 20 minutes moving, whether it's walking, yoga, or lifting weights.

Discover ways to take pleasure in life's challenges.

Since perspective is malleable, they change their view.

Set your own needs first.

Stressed people neglect themselves and wonder why they struggle.

Prioritize self-care.

Don't ruin your life to please others.

Make something.

Calm people create more than react.

They love creating beautiful things—paintings, children, relationships, and projects.

Hold your breath, please.

If you're stressed or angry, you may be surprised how much time you spend holding your breath and tightening your belly.

Release, breathe, and relax to find calm.

Stopped rushing.

Rushing is disadvantageous.

Calm people handle life better.

Are attuned to their personal dietary needs.

They avoid junk food and eat foods that keep them healthy, happy, and calm.

Don’t take anything personally.

Stressed people control everything.

Self-conscious.

Calm people put others and their work first.

Keep their surroundings neat.

Maintaining an uplifting and clutter-free environment daily calms the mind.

Minimise negative people.

Calm people are ruthless with their boundaries and avoid negative and drama-prone people.