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.

and it’s easy to build.

A landing page with good copy is a money-maker.

Let's be honest, page-builder templates are garbage.

They can help you create a nice-looking landing page, but not persuasive writing.

Over the previous 90 days, I've examined 200+ landing pages.

What's crazy?

Top digital entrepreneurs use a 7-part strategy to bring in email subscribers, generate prospects, and (passively) sell their digital courses.

Steal this 7-part landing page architecture to maximize digital product sales.

The offer

Landing pages require offers.

Newsletter, cohort, or course offer.

Your reader should see this offer first. Includind:

• Headline

• Imagery

• Call-to-action

Clear, persuasive, and simplicity are key. Example: the Linkedin OS course home page of digital entrepreneur Justin Welsh offers:

A distinctly defined problem

Everyone needs an enemy.

You need an opponent on your landing page. Problematic.

Next, employ psychology to create a struggle in your visitor's thoughts.

Don't be clever here; label your customer's problem. The more particular you are, the bigger the situation will seem.

When you build a clear monster, you invite defeat. I appreciate Theo Ohene's Growth Roadmaps landing page.

Exacerbation of the effects

Problem identification doesn't motivate action.

What would an unresolved problem mean?

This is landing page copy. When you describe the unsolved problem's repercussions, you accomplish several things:

• You write a narrative (and stories are remembered better than stats)

• You cause the reader to feel something.

• You help the reader relate to the issue

Important!

My favorite script is:

"Sure, you can let [problem] go untreated. But what will happen if you do? Soon, you'll begin to notice [new problem 1] will start to arise. That might bring up [problem 2], etc."

Take the copywriting course, digital writer and entrepreneur Dickie Bush illustrates below when he labels the problem (see: "poor habit") and then illustrates the repercussions.

The tale of transformation

Every landing page needs that "ah-ha!" moment.

Transformation stories do this.

Did you find a solution? Someone else made the discovery? Have you tested your theory?

Next, describe your (or your subject's) metamorphosis.

Kieran Drew nails his narrative (and revelation) here. Right before the disclosure, he introduces his "ah-ha!" moment:

Testimonials

Social proof completes any landing page.

Social proof tells the reader, "If others do it, it must be worthwhile."

This is your argument.

Positive social proof helps (obviously).

Offer "free" training in exchange for a testimonial if you need social evidence. This builds social proof.

Most social proof is testimonies (recommended). Kurtis Hanni's creative take on social proof (using a screenshot of his colleague) is entertaining.

Bravo.

Reveal your offer

Now's the moment to act.

Describe the "bundle" that provides the transformation.

Here's:

• Course

• Cohort

• Ebook

Whatever you're selling.

Include a product or service image, what the consumer is getting ("how it works"), the price, any "free" bonuses (preferred), and a CTA ("buy now").

Clarity is key. Don't make a cunning offer. Make sure your presentation emphasizes customer change (benefits). Dan Koe's Modern Mastery landing page makes an offer. Consider:

An ultimatum

Offering isn't enough.

You must give your prospect an ultimatum.

1. They can buy your merchandise from you.

2. They may exit the webpage.

That’s it.

It's crucial to show what happens if the reader does either. Stress the consequences of not buying (again, a little consequence amplification). Remind them of the benefits of buying.

I appreciate Charles Miller's product offer ending:

The top online creators use a 7-part landing page structure:

1. Offer the service

2. Describe the problem

3. Amplify the consequences

4. Tell the transformational story

5. Include testimonials and social proof.

6. Reveal the offer (with any bonuses if applicable)

7. Finally, give the reader a deadline to encourage them to take action.

Sequence these sections to develop a landing page that (essentially) prints money.

Kim Kardashian created Skky Partners. Consumer products, internet & e-commerce, consumer media, hospitality, and luxury are company targets.

Some call this another Kardashian publicity gimmick.

This maneuver is brilliance upon closer inspection. Why?

1) Kim has amassed a sizable social media fan base:

Over 320 million Instagram and 70 million Twitter users follow Kim Kardashian.

Kim Kardashian's Instagram account ranks 8th. Three Kardashians in top 10 is ridiculous.

This gives her access to consumer data. She knows what people are discussing. Investment firms need this data.

Quality, not quantity, of her followers matters. Studies suggest that her following are more engaged than Selena Gomez and Beyonce's.

Kim's followers are worth roughly $500 million to her brand, according to a research. They trust her and buy what she recommends.

2) She has a special aptitude for identifying trends.

Kim Kardashian can sense trends.

She's always ahead of fashion and beauty trends. She's always trying new things, too. She doesn't mind making mistakes when trying anything new. Her desire to experiment makes her a good business prospector.

Kim has also created a lifestyle brand that followers love. Kim is an entrepreneur, mom, and role model, not just a reality TV star or model. She's established a brand around her appearance, so people want to buy her things.

Her fragrance collection has sold over $100 million since its 2009 introduction, and her Sears apparel line did over $200 million in its first year.

SKIMS is her latest $3.2bn brand. She can establish multibillion-dollar firms with her enormous distribution platform.

Early founders would kill for Kim Kardashian's network.

Making great products is hard, but distribution is more difficult. — David Sacks, All-in-Podcast

3) She can delegate the financial choices to Jay Sammons, one of the greatest in the industry.

Jay Sammons is well-suited to develop Kim Kardashian's new private equity fund.

Sammons has 16 years of consumer investing experience at Carlyle. This will help Kardashian invest in consumer-facing enterprises.

Sammons has invested in Supreme and Beats Electronics, both of which have grown significantly. Sammons' track record and competence make him the obvious choice.

Kim Kardashian and Jay Sammons have joined forces to create a new business endeavor. The agreement will increase Kardashian's commercial empire. Sammons can leverage one of the world's most famous celebrities.

“Together we hope to leverage our complementary expertise to build the next generation consumer and media private equity firm” — Kim Kardashian

Kim Kardashian is a successful businesswoman. She developed an empire by leveraging social media to connect with fans. By developing a global lifestyle brand, she has sold things and experiences that have made her one of the world's richest celebrities.

She's a shrewd entrepreneur who knows how to maximize on herself and her image.

Imagine how much interest Kim K will bring to private equity and venture capital.

I'm curious about the company's growth.

> Say Hello To The Lean-Hedge Model

People think starting a business requires significant debt and investment. Like Shark Tank, you need a world-changing idea. I'm not saying to avoid investors or brilliant ideas.

Investing is essential to build a genuinely profitable company. Think Apple or Starbucks.

Entrepreneurship is risky because many people go bankrupt from debt. As starters, we shouldn't do it. Instead, use lean-hedge.

Simply defined, you construct a cash-flow business to hedge against long-term investment-heavy business expenses.

What the “fx!$rench-toast” is the lean-hedge model?

When you start a business, your money should move down, down, down, then up when it becomes profitable.

Many people don't survive the business's initial losses and debt. What if, we created a cash-flow business BEFORE we started our Starbucks to hedge against its initial expenses?

Lean-hedge has two sections. Start a cash-flow business. A cash-flow business takes minimal investment and usually involves sweat and time.

Let’s take a look at some examples:

A Translation company

Personal portfolio website (you make a site then you do cold e-mail marketing)

FREELANCE (UpWork, Fiverr).

Educational business.

Infomarketing. (You design a knowledge-based product. You sell the info).

Online fitness/diet/health coaching ($50-$300/month, calls, training plan)

Amazon e-book publishing. (Medium writers do this)

YouTube, cash-flow channel

A web development agency (I'm a dev, but if you're not, a graphic design agency, etc.) (Sell your time.)

Digital Marketing

Online paralegal (A million lawyers work in the U.S).

Some dropshipping (Organic Tik Tok dropshipping, where you create content to drive traffic to your shopify store instead of spend money on ads).

(Disclaimer: My first two cash-flow enterprises, which were language teaching, failed terribly. My translation firm is now booming because B2B e-mail marketing is easy.)

Crossover occurs. Your long-term business starts earning more money than your cash flow business.

My cash-flow business (freelancing, translation) makes $7k+/month.

I’ve decided to start a slightly more investment-heavy digital marketing agency

Here are the anticipated business's time- and money-intensive investments:

1. ($$$) Top Front-End designer's Figma/UI-UX design (in negotiation)

2. (Time): A little copywriting (I will do this myself)

3. ($$) Creating an animated webpage with HTML (in negotiation)

4. Backend Development (Duration) (I'll carry out this myself using Laravel.)

5. Logo Design ($$)

6. Logo Intro Video for $

7. Video Intro (I’ll edit this myself with Premiere Pro)

etc.

Then evaluate product, place, price, and promotion. Consider promotion and pricing.

The lean-hedge model's point is:

Don't gamble. Avoid debt. First create a cash-flow project, then grow it steadily.

Check read my previous posts on “Nightmare Mode” (which teaches you how to make work as interesting as video games) and Why most people can't escape a 9-5 to learn how to develop a cash-flow business.