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.

Offices without books are like bodies without souls.

After 10 years, my CEO sold his company for $30 million. I've shared many of his lessons on medium. You could ask him anything at his always-open office. He also said we could use his office for meetings while he was away. When I used his office for work, I was always struck by how many books he had.

Books are useful in almost every aspect of learning. Building a business, improving family relationships, learning a new language, a new skill... Books teach, guide, and structure. Whether fiction or nonfiction, books inspire, give ideas, and develop critical thinking skills.

My CEO prefers non-fiction and attends a Friday book club. This article discusses 5 books I found in his office that impacted my life/business. My CEO sold his company for $30 million, but I've built a steady business through blogging and video making.

I recall events and lessons I learned from my CEO and how they relate to each book, and I explain how I applied the book's lessons to my business and life.

Note: This post has no affiliate links.

1. The One Thing — Gary Keller

Gary Keller, a real estate agent, wanted more customers. So he and his team brainstormed ways to get more customers. They decided to write a bestseller about work and productivity. The more people who saw the book, the more customers they'd get.

Gary Keller focused on writing the best book on productivity, work, and efficiency for months. His business experience. Keller's business grew after the book's release.

The author summarizes the book in one question.

"What's the one thing that will make everything else easier or unnecessary?"

When I started my blog and business alongside my 9–5, I quickly identified my one thing: writing. My business relied on it, so it had to be great. Without writing, there was no content, traffic, or business.

My CEO focused on funding when he started his business. Even in his final years, he spent a lot of time on the phone with investors, either to get more money or to explain what he was doing with it. My CEO's top concern was money, and the other super important factors were handled by separate teams.

• Product tech and design

• Incredible customer support team

• Excellent promotion team

• Profitable sales team

My CEO didn't always focus on one thing and ignore the rest. He was on all of those teams when I started my job. He'd start his day in tech, have lunch with marketing, and then work in sales. He was in his office on the phone at night.

He eventually realized his errors. Investors told him he couldn't do everything for the company. If needed, he had to change internally. He learned to let go, mind his own business, and focus for the next four years. Then he sold for $30 million.

The bigger your project/company/idea, the more you'll need to delegate to stay laser-focused. I started something new every few months for 10 years before realizing this. So much to do makes it easy to avoid progress. Once you identify the most important aspect of your project and enlist others' help, you'll be successful.

2. Eat That Frog — Brian Tracy

The author quote sums up book's essence:

Mark Twain said that if you eat a live frog in the morning, it's probably the worst thing that will happen to you all day. Your "frog" is the biggest, most important task you're most likely to procrastinate on.

"Frog" and "One Thing" are both about focusing on what's most important. Eat That Frog recommends doing the most important task first thing in the morning.

I shared my CEO's calendar in an article 10 months ago. Like this:

CEO's average week (some information crossed out for confidentiality)

Notice anything about 8am-8:45am? Almost every day is the same (except Friday). My CEO started his day with a management check-in for 2 reasons:

• Checking in with all managers is cognitively demanding, and my CEO is a morning person.

• In a young startup where everyone is busy, the morning management check-in was crucial. After 10 am, you couldn't gather all managers.

When I started my blog, writing was my passion. I'm a morning person, so I woke up at 6 am and started writing by 6:30 am every day for a year. This allowed me to publish 3 articles a week for 52 weeks to build my blog and audience. After 2 years, I'm not stopping.

3. Deep Work — Cal Newport

Deep work is focusing on a cognitively demanding task without distractions (like a morning management meeting). It helps you master complex information quickly and produce better results faster. In a competitive world 10 or 20 years ago, focus wasn't a huge advantage. Smartphones, emails, and social media made focus a rare, valuable skill.

Most people can't focus anymore. Screens light up, notifications buzz, emails arrive, Instagram feeds... Many people don't realize they're interrupted because it's become part of their normal workflow.

Cal Newport mentions Bill Gates' "Think Weeks" in Deep Work.

Microsoft CEO Bill Gates would isolate himself (often in a lakeside cottage) twice a year to read and think big thoughts.

Inside Bill's Brain on Netflix shows Newport's lakeside cottage. I've always wanted a lakeside cabin to work in. My CEO bought a lakehouse after selling his company, but now he's retired.

As a company grows, you can focus less on it. In a previous section, I said investors told my CEO to get back to basics and stop micromanaging. My CEO's commitment and ability to get work done helped save the company. His deep work and new frameworks helped us survive the corona crisis (more on this later).

The ability to deep work will be a huge competitive advantage in the next century. Those who learn to work deeply will likely be successful while everyone else is glued to their screens, Bluetooth-synced to their watches, and playing Candy Crush on their tablets.

4. The 7 Habits of Highly Effective People — Stephen R. Covey

It took me a while to start reading this book because it seemed like another shallow self-help bible. I kept finding this book when researching self-improvement. I tried it because it was everywhere.

Stephen Covey taught me 2 years ago to have a personal mission statement.

A 7 Habits mission statement describes the life you want to lead, the character traits you want to embody, and the impact you want to have on others. shortform.com

I've had many lunches with my CEO and talked about Vipassana meditation and Sunday forest runs, but I've never seen his mission statement. I'm sure his family is important, though. In the above calendar screenshot, you can see he always included family events (in green) so we could all see those time slots. We couldn't book him then. Although he never spent as much time with his family as he wanted, he always made sure to be on time for his kid's birthday rather than a conference call.

My CEO emphasized his company's mission. Your mission statement should answer 3 questions.

• What does your company do?

• How does it do it?

• Why does your company do it?

As a graphic designer, I had to create mission-statement posters. My CEO hung posters in each office.

5. Measure What Matters — John Doerr

This book is about Andrew Grove's OKR strategy, developed in 1968. When he joined Google's early investors board, he introduced it to Larry Page and Sergey Brin. Google still uses OKR.

Objective Key Results

• Objective: It explains your goals and desired outcome. When one goal is reached, another replaces it. OKR objectives aren't technical, measured, or numerical. They must be clear.

• Key Result should be precise, technical, and measurable, unlike the Objective. It shows if the Goal is being worked on. Time-bound results are quarterly or yearly.

Our company almost sank several times. Sales goals were missed, management failed, and bad decisions were made. On a Monday, our CEO announced we'd implement OKR to revamp our processes.

This was a year before the pandemic, and I'm certain we wouldn't have sold millions or survived without this change. This book impacted the company the most, not just management but all levels. Organization and transparency improved. We reached realistic goals. Happy investors. We used the online tool Gtmhub to implement OKR across the organization.

My CEO's company went from near bankruptcy to being acquired for $30 million in 2 years after implementing OKR.

I hope you enjoyed this booklist. Here's a recap of the 5 books and the lessons I learned from each.

1. The 7 Habits of Highly Effective People — Stephen R. Covey

Have a mission statement that outlines your goals, character traits, and impact on others.

1. Deep Work — Cal Newport

Focus is a rare skill; master it. Deep workers will succeed in our hyper-connected, distracted world.

1. The One Thing — Gary Keller

What can you do that will make everything else easier or unnecessary? Once you've identified it, focus on it.

1. Eat That Frog — Brian Tracy

Identify your most important task the night before and do it first thing in the morning. You'll have a lighter day.

1. Measure What Matters — John Doerr

On a timeline, divide each long-term goal into chunks. Divide those slices into daily tasks (your goals). Time-bound results are quarterly or yearly. Objectives aren't measured or numbered.

Thanks for reading. Enjoy the ride!

AI images are like funhouse mirrors.

Google's AI Blog introduced the puppy-slug in the summer of 2015.

Puppy-slug isn't a single image or character. "Puppy-slug" refers to Google's DeepDream's unsettling psychedelia. This tool uses convolutional neural networks to train models to recognize dataset entities. If researchers feed the model millions of dog pictures, the network will learn to recognize a dog.

DeepDream used neural networks to analyze and classify image data as well as generate its own images. DeepDream's early examples were created by training a convolutional network on dog images and asking it to add "dog-ness" to other images. The models analyzed images to find dog-like pixels and modified surrounding pixels to highlight them.

Puppy-slugs and other DeepDream images are ugly. Even when they don't trigger my trypophobia, they give me vertigo when my mind tries to reconcile familiar features and forms in unnatural, physically impossible arrangements. I feel like I've been poisoned by a forbidden mushroom or a noxious toad. I'm a Lovecraft character going mad from extradimensional exposure. They're gross!

Is this really how AIs see the world? This is possibly an even more unsettling topic that DeepDream raises than the blatant abjection of the images.

When these photographs originally circulated online, many friends were startled and scandalized. People imagined a computer's imagination would be literal, accurate, and boring. We didn't expect vivid hallucinations and organic-looking formations.

DeepDream's images didn't really show the machines' imaginations, at least not in the way that scared some people. DeepDream displays data visualizations. DeepDream reveals the "black box" of convolutional network training.

Some of these images look scary because the models don't "know" anything, at least not in the way we do.

These images are the result of advanced algorithms and calculators that compare pixel values. They can spot and reproduce trends from training data, but can't interpret it. If so, they'd know dogs have two eyes and one face per head. If machines can think creatively, they're keeping it quiet.

You could be forgiven for thinking otherwise, given OpenAI's Dall-impressive E's results. From a technological perspective, it's incredible.

Arthur C. Clarke once said, "Any sufficiently advanced technology is indistinguishable from magic." Dall-magic E's requires a lot of math, computer science, processing power, and research. OpenAI did a great job, and we should applaud them.

Dall-E and similar tools match words and phrases to image data to train generative models. Matching text to images requires sorting and defining the images. Untold millions of low-wage data entry workers, content creators optimizing images for SEO, and anyone who has used a Captcha to access a website make these decisions. These people could live and die without receiving credit for their work, even though the project wouldn't exist without them.

This technique produces images that are less like paintings and more like mirrors that reflect our own beliefs and ideals back at us, albeit via a very complex prism. Due to the limitations and biases that these models portray, we must exercise caution when viewing these images.

The issue was succinctly articulated by artist Mimi Onuoha in her piece "On Algorithmic Violence":

As we continue to see the rise of algorithms being used for civic, social, and cultural decision-making, it becomes that much more important that we name the reality that we are seeing. Not because it is exceptional, but because it is ubiquitous. Not because it creates new inequities, but because it has the power to cloak and amplify existing ones. Not because it is on the horizon, but because it is already here.