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.

Discover How We Used Facebook Ads to Grow a New Mobile App from $0 to $15K MRR in Just 6 Months and Our Strategy to Hit $100K a Month.

Our client introduced a mobile app for Poshmark resellers in December and wanted as many to experience it and subscribe to the monthly plan.

An Error We Committed

We initiated a Facebook ad campaign with a "awareness" goal, not "installs." This sent them to a landing page that linked to the iPhone App Store and Android Play Store. Smart, right?

We got some installs, but we couldn't tell how many came from the ad versus organic/other channels because the objective we chose only reported landing page clicks, not app installs.

We didn't know which interest groups/audiences had the best cost per install (CPI) to optimize and scale our budget.

After spending $700 without adequate data (installs and trials report), we stopped the campaign and worked with our client's app developer to set up app events tracking.

This allowed us to create an installs campaign and track installs, trials, and purchases (in some cases).

Finding a Successful Audience

Once we knew what ad sets brought in what installs at what cost, we began optimizing and testing other interest groups and audiences, growing the profitable low CPI ones and eliminating the high CPI ones.

We did all our audience testing using an ABO campaign (Ad Set Budget Optimization), spending $10 to $30 on each ad set for three days and optimizing afterward. All ad sets under $30 were moved to a CBO campaign (Campaign Budget Optimization).

We let Facebook's AI decide how much to spend on each ad set, usually the one most likely to convert at the lowest cost.

If the CBO campaign maintains a nice CPI, we keep increasing the budget by $50 every few days or duplicating it sometimes in order to double the budget. This is how we've scaled to $400/day profitably.

Finding Successful Creatives

Per campaign, we tested 2-6 images/videos. Same ad copy and CTA. There was no clear winner because some images did better with some interest groups.

The image above with mail packages, for example, got us a cheap CPI of $9.71 from our Goodwill Stores interest group but, a high $48 CPI from our lookalike audience. Once we had statistically significant data, we turned off the high-cost ad.

New marketers who are just discovering A/B testing may assume it's black and white — winner and loser. However, Facebook ads' machine learning and reporting has gotten so sophisticated that it's hard to call a creative a flat-out loser, but rather a 'bad fit' for some audiences, and perfect for others.

You can see how each creative performs across age groups and optimize.

How Many Installs Did It Take Us to Earn $15K Per Month?

Six months after paying $25K, we got 1,940 app installs, 681 free trials, and 522 $30 monthly subscriptions. 522 * $30 gives us $15,660 in monthly recurring revenue (MRR).

Next, what? $100K per month

The conversation above is with the app's owner. We got on a 30-minute call where I shared how I plan to get the app to be making $100K a month like I’ve done for other businesses.

Reverse Engineering $100K

Formula:

For $100K/month, we need 3,334 people to pay $30/month. 522 people pay that. We need 2,812 more paid users.

522 paid users from 1,940 installs is a 27% conversion rate. To hit $100K/month, we need 10,415 more installs. Assuming...

With a $400 daily ad spend, we average 40 installs per day. This means that if everything stays the same, it would take us 260 days (around 9 months) to get to $100K a month (MRR).

Conclusion

You must market your goods to reach your income objective (without waiting forever). Paid ads is the way to go if you hate knocking on doors or irritating friends and family (who aren’t scalable anyways).

You must also test and optimize different angles, audiences, interest groups, and creatives.

One great idea might make you rich.

Imagine having a million-dollar concept in college that made a million.

2005 precisely.

Alex Tew, 21, from Wiltshire, England, created The Million Dollar Homepage in August 2005. The idea is basic but beyond the ordinary, which is why it worked.

Alex built a 1,000,000-pixel webpage.

Each website pixel would cost $1. Since pixels are hard to discern, he sold 10x10 squares for $100.

He'd make a million if all the spots sold.

He may have thought about NFTs and the Metaverse decades ago.

MillionDollarHomepage.com launched in 2005.

Businesses and individuals could buy a website spot and add their logo, website link, and tagline. You bought an ad, but nobody visited the website.

If a few thousand people visited the website, it could drive traffic to your business's site.

Alex promised buyers the website would be up for 5 years, so it was a safe bet.

Alex's friend with a music website was the first to buy real estate on the site. Within two weeks, 4,700 pixels sold, and a tracker showed how many were sold and available.

Word-of-mouth marketing got the press's attention quickly. Everyone loves reading about new ways to make money, so it was a good news story.

By September, over 250,000 pixels had been sold, according to a BBC press release.

Alex and the website gained more media and public attention, so traffic skyrocketed. Two months after the site launched, 1,400 customers bought more than 500,000 pixels.

Businesses bought online real estate. They heard thousands visited the site, so they could get attention cheaply.

Unless you bought a few squares, I'm not sure how many people would notice your ad or click your link.

A sponge website owner emailed Alex:

“We tried Million Dollar Homepage because we were impressed at the level of ingenuity and the sheer simplicity of it. If we’re honest, we didn’t expect too much from it. Now, as a direct result, we are pitching for £18,000 GBP worth of new clients and have seen our site traffic increase over a hundred-fold. We’re even going to have to upgrade our hosting facility! It’s been exceptional.”

Web.archive.org screenshots show how the website changed.

“The idea is to create something of an internet time capsule: a homepage that is unique and permanent. Everything on the internet keeps changing so fast, it will be nice to have something that stays solid and permanent for many years. You can be a part of that!” Alex Tew, 2005

The last 1,000 pixels were sold on January 1, 2006.

By then, the homepage had hundreds of thousands of monthly visitors. Alex put the last space on eBay due to high demand.

MillionDollarWeightLoss.com won the last pixels for $38,100, bringing revenue to $1,037,100 in 4 months.

Many have tried to replicate this website's success. They've all failed.

This idea only worked because no one had seen this website before.

This winner won't be repeated, but it should inspire you to try something new and creative.

Still popular, you could buy one of the linked domains. You can't buy pixels, but you can buy an expired domain.

One link I clicked costs $59,888.

You'd own a piece of internet history if you spent that much on a domain.

Someone bought stablesgallery.co.uk after the domain expired and restored it.

Many of the linked websites have expired or been redirected, but some still link to the original. I couldn't find sponge's website. Can you?

This is a great example of how a simple creative idea can go viral.

Comment on this amazing success story.