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.

The easiest online explanation.

Web3 is growing. Crypto companies are growing.

Instagram, Adidas, and Stripe adopted cryptocurrency.

Bitcoin and other cryptocurrencies made web3 famous.

Most don't know where to start. Cryptocurrency, DeFi, etc. are investments.

Since we don't understand web3, I'll help you today.

Let’s go.

1. Web3

It is the third generation of the web, and it is built on the decentralization idea which means no one can control it.

There are static webpages that we can only read on the first generation of the web (i.e. Web 1.0).

Web 2.0 websites are interactive. Twitter, Medium, and YouTube.

Each generation controlled the website owner. Simply put, the owner can block us. However, data breaches and selling user data to other companies are issues.

They can influence the audience's mind since they have control.

Assume Twitter's CEO endorses Donald Trump. Result? Twitter would have promoted Donald Trump with tweets and graphics, enhancing his chances of winning.

We need a decentralized, uncontrollable system.

And then there’s Web3.0 to consider. As Bitcoin and Ethereum values climb, so has its popularity. Web3.0 is uncontrolled web evolution. It's good and bad.

Dapps, DeFi, and DAOs are here. It'll all be explained afterwards.

2. Cryptocurrencies:

No need to elaborate.

Bitcoin, Ethereum, Cardano, and Dogecoin are cryptocurrencies. It's digital money used for payments and other uses.

Programs must interact with cryptocurrencies.

3. Blockchain:

Blockchain facilitates bitcoin transactions, investments, and earnings.

This technology governs Web3. It underpins the web3 environment.

Let us delve much deeper.

Blockchain is simple. However, the name expresses the meaning.

Blockchain is a chain of blocks.

Let's use an image if you don't understand.

The graphic above explains blockchain. Think Blockchain. The block stores related data.

Here's more.

4. Smart contracts

Programmers and developers must write programs. Smart contracts are these blockchain apps.

That’s reasonable.

Decentralized web3.0 requires immutable smart contracts or programs.

5. NFTs

Blockchain art is NFT. Non-Fungible Tokens.

Explaining Non-Fungible Token may help.

Two sorts of tokens:

1. These tokens are fungible, meaning they can be changed. Think of Bitcoin or cash. The token won't change if you sell one Bitcoin and acquire another.

2. Non-Fungible Token: Since these tokens cannot be exchanged, they are exclusive. For instance, music, painting, and so forth.

Right now, Companies and even individuals are currently developing worthless NFTs.

The concept of NFTs is much improved when properly handled.

6. Dapp

Decentralized apps are Dapps. Instagram, Twitter, and Medium apps in the same way that there is a lot of decentralized blockchain app.

Curve, Yearn Finance, OpenSea, Axie Infinity, etc. are dapps.

7. DAOs

DAOs are member-owned and governed.

Consider it a company with a core group of contributors.

8. DeFi

We all utilize centrally regulated financial services. We fund these banks.

If you have $10,000 in your bank account, the bank can invest it and retain the majority of the profits.

We only get a penny back. Some banks offer poor returns. To secure a loan, we must trust the bank, divulge our information, and fill out lots of paperwork.

DeFi was built for such issues.

Decentralized banks are uncontrolled. Staking, liquidity, yield farming, and more can earn you money.

Web3 beginners should start with these resources.

After a decade in large tech, I know how software engineers were interviewed. I've seen outstanding engineers fail interviews because their responses were too vague.

So I wrote Nail A Coding Interview: Six-Step Mental Framework. Give candidates a mental framework for coding questions; help organizations better prepare candidates so they can calibrate traits.

Recently, I sold more than 100 books, something I never expected.

In this essay, I'll describe my publication journey, which included self-doubt and little triumphs. I hope this helps if you want to publish.

It was originally a Medium post.

How did I know to develop a coding interview book? Years ago, I posted on Medium.

Six steps to ace a coding interview Inhale. blog.devgenius.io

This story got a lot of attention and still gets a lot of daily traffic. It indicates this domain's value.

Converted the Medium article into an ebook

The Medium post contains strong bullet points, but it is missing the “flesh”. How to use these strategies in coding interviews, for example. I filled in the blanks and made a book.

I made the book cover for free. It's tidy.

Shared the article with my close friends on my social network WeChat.

I shared the book on Wechat's Friend Circle (朋友圈) after publishing it on Gumroad. Many friends enjoyed my post. It definitely triggered endorphins.

In Friend Circle, I presented a 100% off voucher. No one downloaded the book. Endorphins made my heart sink.

Several days later, my Apple Watch received a Gumroad notification. A friend downloaded it. I majored in finance, he subsequently said. My brother-in-law can get it? He downloaded it to cheer me up.

I liked him, but was disappointed that he didn't read it.

The Tipping Point: Reddit's Free Giving

I trusted the book. It's based on years of interviewing. I felt it might help job-hunting college students. If nobody wants it, it can still have value.

I posted the book's link on /r/leetcode. I told them to DM me for a free promo code.

Momentum shifted everything. Gumroad notifications kept coming when I was out with family. Following orders.

As promised, I sent DMs a promo code. Some consumers ordered without asking for a promo code. Some readers finished the book and posted reviews.

My book was finally on track.

A 5-Star Review, plus More

A reader afterwards DMed me and inquired if I had another book on system design interviewing. I said that was a good idea, but I didn't have one. If you write one, I'll be your first reader.

Later, I asked for a book review. Yes, but how? That's when I learned readers' reviews weren't easy. I built up an email pipeline to solicit customer reviews. Since then, I've gained credibility through ratings.

Learnings

I wouldn't have gotten 100 if I gave up when none of my pals downloaded. Here are some lessons.

• Your friends are your allies, but they are not your clients.

• Be present where your clients are

• Request ratings and testimonials

• gain credibility gradually

I did it, so can you. Follow me on Twitter @imgracehuang for my publishing and entrepreneurship adventure.

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d) The arbitration will permit the discovery or exchange of nonconfidential information pertinent to the Dispute. The arbitrator, INTΞGRITY, and you will maintain the confidentiality of all arbitration proceedings, judgments, and awards, as well as any information gathered, prepared, or presented for the purposes of the arbitration or relating to the Dispute(s) therein. Unless the law specifies otherwise, the arbitrator will have the right to make decisions that protect confidentiality. The duty of confidentiality does not apply where disclosure is required to prepare for or conduct the arbitration hearing on the merits, in connection with a court application for a preliminary remedy, in connection with a judicial challenge to an arbitration award or its enforcement, or where disclosure is otherwise required by law or judicial decision.

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(f) Any Dispute must be filed within one year after the relevant claim arose; otherwise, the Dispute is permanently barred, meaning that neither you nor INTΞGRITY will be able to assert the claim.

(g) You have the right to opt-out of binding arbitration within 30 days of the date you initially accepted the terms of this section by sending an email to hello@int3grity.com. For the opt-out notification to be effective, it must include your full name and address and clearly explain your intent to opt out of binding arbitration. By declining binding arbitration, you consent to the resolution of Disputes in accordance with "Governing Law and Venue" below.

(h) If any portion of this section is found to be unenforceable or unlawful for any reason: (1) the unenforceable or unlawful provision shall be severed from these Terms; (2) the severance of the unenforceable or unlawful provision shall have no effect whatsoever on the remainder of this section or the parties' ability to compel arbitration of any remaining claims on an individual basis pursuant to this section; and (3) to the extent that any claims must therefore proceed on an individual basis, the parties agree to arbitrate those claims on an individual basis. In addition, if it is determined that any portion of this section prohibits an individual claim seeking public injunctive relief, that provision will be null and void to the extent that such relief may be sought outside of arbitration, and the balance of this section will be enforceable.

Statute and Location

These Terms and any dispute that may arise between you and INTΞGRITY are governed by California law, excluding its conflict of law provisions. Any issue between the parties that is not arbitrable or cannot be heard in small claims court will be determined by the state or federal courts of California and the United States, sitting in San Francisco, California.

Some nations have regulations that require agreements to be controlled by the consumer's country's laws. These statutes are not overridden by this paragraph.

Amendments

Periodically, we may make modifications to these Terms. If we make modifications, we will notify you by sending an email to the address connected with your account, providing an in-product message, or amending the date at the top of these Terms. Unless we specify otherwise in our notification, the modified Terms will take effect immediately, and your continued use of our Services after we issue such notice indicates your acceptance of the changes. If you do not accept the updated Terms, you must cease using our services.

Severability

If any section or portion of a provision of these Terms is determined to be unlawful, void, or unenforceable, that provision or part of the provision shall be deemed severable from these Terms and shall not affect the validity and enforceability of the other terms.

Miscellaneous INTΞGRITY’s omission to assert or enforce any right or term of these Terms is not a waiver of such right or provision. These Terms and the terms and policies specified in the Other Terms and Policies that May Apply to You Section constitute the complete agreement between the parties pertaining to the subject matter hereof and supersede all prior agreements, statements, and understandings between the parties. The section headings in these Terms are for convenience only and have no legal or contractual significance. The use of the word "including" shall be taken to mean "including without limitation." Unless otherwise specified, these Terms are intended solely for the benefit of the parties and are not intended to confer third-party beneficiary rights on any other person or entity. You consent to the use of electronic means for our communications and transactions.

A must-read for everyone

Jobs is well-known. You probably know the tall, thin guy who wore the same clothing every day. His influence is unavoidable. In fewer than 40 years, Jobs' innovations have impacted computers, movies, cellphones, music, and communication.

Steve Jobs may be more imaginative than the typical person, but if we can use some of his ingenuity, ambition, and good traits, we'll be successful. This essay explains how to follow his guidance and success secrets.

1. Repetition is necessary for success.

Be patient and diligent to master something. Practice makes perfect. This is why older workers are often more skilled.

When should you repeat a task? When you're confident and excited to share your product. It's when to stop tweaking and repeating.

Jobs stated he'd make the crowd sh** their pants with an iChat demo.

Use this in your daily life.

• Start with the end in mind. You can put it in writing and be as detailed as you like with your plan's schedule and metrics. For instance, you have a goal of selling three coffee makers in a week.

• Break it down, break the goal down into particular tasks you must complete, and then repeat those tasks. To sell your coffee maker, you might need to make 50 phone calls.

• Be mindful of the amount of work necessary to produce the desired results. Continue doing this until you are happy with your product.

2. Acquire the ability to add and subtract.

How did Picasso invent cubism? Pablo Picasso was influenced by stylised, non-naturalistic African masks that depict a human figure.

Artists create. Constantly seeking inspiration. They think creatively about random objects. Jobs said creativity is linking things. Creative people feel terrible when asked how they achieved something unique because they didn't do it all. They saw innovation. They had mastered connecting and synthesizing experiences.

Use this in your daily life.

• On your phone, there is a note-taking app. Ideas for what you desire to learn should be written down. It may be learning a new language, calligraphy, or anything else that inspires or intrigues you.

• Note any ideas you have, quotations, or any information that strikes you as important.

• Spend time with smart individuals, that is the most important thing. Jim Rohn, a well-known motivational speaker, has observed that we are the average of the five people with whom we spend the most time.

• Learning alone won't get you very far. You need to put what you've learnt into practice. If you don't use your knowledge and skills, they are useless.

3. Develop the ability to refuse.

Steve Jobs deleted thousands of items when he created Apple's design ethic. Saying no to distractions meant upsetting customers and partners.

John Sculley, the former CEO of Apple, said something like this. According to Sculley, Steve’s methodology differs from others as he always believed that the most critical decisions are things you choose not to do.

Use this in your daily life.

• Never be afraid to say "no," "I won't," or "I don't want to." Keep it simple. This method works well in some situations.

• Give a different option. For instance, X might be interested even if I won't be able to achieve it.

• Control your top priority. Before saying yes to anything, make sure your work schedule and priority list are up to date.

4. Follow your passion

“Follow your passion” is the worst advice people can give you. Steve Jobs didn't start Apple because he suddenly loved computers. He wanted to help others attain their maximum potential.

Great things take a lot of work, so quitting makes sense if you're not passionate. Jobs learned from history that successful people were passionate about their work and persisted through challenges.

Use this in your daily life.

• Stay away from your passion. Allow it to develop daily. Keep working at your 9-5-hour job while carefully gauging your level of desire and endurance. Less risk exists.

• The truth is that if you decide to work on a project by yourself rather than in a group, it will take you years to complete it instead of a week. Instead, network with others who have interests in common.

• Prepare a fallback strategy in case things go wrong.

Success, this small two-syllable word eventually gives your life meaning, a perspective. What is success?  For most, it's achieving their ambitions. However, there's a catch. Successful people aren't always happy.

Furthermore, where do people’s goals and achievements end? It’s a never-ending process. Success is a journey, not a destination. We wish you not to lose your way on this journey.