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.

Massive Layoffs, record low VC investment, debate over crash... why is it happening and what’s the endgame?

This article generalizes a diverse industry. For objectivity, specific tech company challenges like growing competition within named segments won't be considered. Please comment on the posts.

According to Layoffs.fyi, nearly 120.000 people have been fired from startups since March 2020. More than 700 startups have fired 1% to 100% of their workforce. "The tech market is crashing"

Venture capital investment dropped 19% QoQ in the first four months of 2022, a 2018 low. Since January 2022, Nasdaq has dropped 27%. Some believe the tech market is collapsing.

It's bad, but nothing has crashed yet. We're about to get super technical, so buckle up!

I've written a follow-up article about what's next. For a more optimistic view of the crisis' aftermath, see: Tech Diaspora and Silicon Valley crisis

What happened?

Insanity reigned. Last decade, everyone became a unicorn. Seed investments can be made without a product or team. While the "real world" economy suffered from the pandemic for three years, tech companies enjoyed the "new normal."

COVID sped up technology adoption on several fronts, but this "new normal" wasn't so new after many restrictions were lifted. Worse, it lived with disrupted logistics chains, high oil prices, and WW3. The consumer market has felt the industry's boom for almost 3 years. Inflation, unemployment, mental distress...what looked like a fast economic recovery now looks like unfulfilled promises.

People rethink everything they eat. Paying a Netflix subscription instead of buying beef is moronic if you can watch it for free on your cousin’s account. No matter how great your real estate app's UI is, buying a house can wait until mortgage rates drop. PLGProduct Led Growth (PLG) isn't the go-to strategy when consumers have more basic expense priorities.

Exponential growth and investment

Until recently, tech companies believed that non-exponential revenue growth was fatal. Exponential growth entails doing more with less. From Salim Ismail words:

An Exponential Organization (ExO) has 10x the impact of its peers.

Many tech companies' theories are far from reality.

Investors have funded (sometimes non-exponential) growth. Scale-driven companies throw people at problems until they're solved. Need an entire closing team because you’ve just bought a TV prime time add? Sure. Want gold-weight engineers to colorize buttons? Why not?

Tech companies don't need cash flow to do it; they can just show revenue growth and get funding. Even though it's hard to get funding, this was the market's momentum until recently.

The graph at the beginning of this section shows how industry heavyweights burned money until 2020, despite being far from their market-share seed stage. Being big and being sturdy are different things, and a lot of the tech startups out there are paper tigers. Without investor money, they have no foundation.

A little bit about interest rates

Inflation-driven high interest rates are said to be causing tough times. Investors would rather leave money in the bank than spend it (I myself said it some days ago). It’s not wrong, but it’s also not that simple.

The USA central bank (FED) is a good proxy of global economics. Dollar treasury bonds are the safest investment in the world. Buying U.S. debt, the only country that can print dollars, guarantees payment.

The graph above shows that FED interest rates are low and 10+ year bond yields are near 2018 levels. Nobody was firing at 2018. What’s with that then?

Full explanation is too technical for this article, so I'll just summarize: Bond yields rise due to lack of demand or market expectations of longer-lasting inflation. Safe assets aren't a "easy money" tactic for investors. If that were true, we'd have seen the current scenario before.

Long-term investors are protecting their capital from inflation.

Not a crash, a landing

I bombarded you with info... Let's review:

• Consumption is down, hurting revenue.

• Tech companies of all ages have been hiring to grow revenue at the expense of profit.

• Investors expect inflation to last longer, reducing future investment gains.

Inflation puts pressure on a wheel that was rolling full speed not long ago. Investment spurs hiring, growth, and more investment. Worried investors and consumers reduce the cycle, and hiring follows.

Long-term investors back startups. When the invested company goes public or is sold, it's ok to burn money. What happens when the payoff gets further away? What if all that money sinks? Investors want immediate returns.

Why isn't the market crashing? Technology is not losing capital. It’s expecting change. The market realizes it threw moderation out the window and is reversing course. Profitability is back on the menu.

People solve problems and make money, but they also cost money. Huge cost for the tech industry. Engineers, Product Managers, and Designers earn up to 100% more than similar roles. Businesses must be careful about who they keep and in what positions to avoid wasting money.

What the future holds

From here on, it's all speculation. I found many great articles while researching this piece. Some are cited, others aren't (like this and this). We're in an adjustment period that may or may not last long.

Big companies aren't laying off many workers. Netflix firing 100 people makes headlines, but it's only 1% of their workforce. The biggest seem to prefer not hiring over firing.

Smaller startups beyond the seeding stage may be hardest hit. Without structure or product maturity, many will die.

I expect layoffs to continue for some time, even at Meta or Amazon. I don't see any industry names falling like they did during the .com crisis, but the market will shrink.

If you are currently employed, think twice before moving out and where to.
If you've been fired, hurry, there are still many opportunities.
If you're considering a tech career, wait.
If you're starting a business, I respect you. Good luck.

I've spent the last 10 years of my career mastering time management. I've tried different approaches and followed multiple people and sources. My knowledge is summarized.

Great people, including entrepreneurs, master time management. I learned time management in college. I was studying Computer Science and Finance and leading Tanzanian students in Bangalore, India. I had 24 hours per day to do this and enjoy campus. I graduated and received several awards. I've learned to maximize my time. These tips and tools help me finish quickly.

Eisenhower-Box

I don't remember when I read the article. James Clear, one of my favorite bloggers, introduced me to the Eisenhower Box, which I've used for years. Eliminate waste to master time management. By grouping your activities by importance and urgency, the tool helps you prioritize what matters and drop what doesn't. If it's urgent, do it. Delegate if it's urgent but not necessary. If it's important but not urgent, reschedule it; otherwise, drop it. I integrated the tool with Trello to manage my daily tasks. Since 2007, I've done this.

James Clear's article mentions Eisenhower Box.

Essentialism rules

Greg McKeown's book Essentialism introduced me to disciplined pursuit of less. I once wrote about this. I wasn't sure what my career's real opportunities and distractions were. A non-essentialist thinks everything is essential; you want to be everything to everyone, and your life lacks satisfaction. Poor time management starts it all. Reading and applying this book will change your life.

Essential vs non-essential

Life Calendar

Most of us make corporate calendars. Peter Njonjo, founder of Twiga Foods, said he manages time by putting life activities in his core calendars. It includes family retreats, weddings, and other events. He joked that his wife always complained to him to avoid becoming a calendar item. It's key. "Time Masters" manages life's four burners, not just work and corporate life. There's no "work-life balance"; it's life.

Health, Family, Work, and Friends.

The Brutal No

In a culture where people want to look good, saying "NO" to a favor request seems rude. In reality, the crime is breaking a promise. "Time Masters" have mastered "NO".  More "YES" means less time, and more "NO" means more time for tasks and priorities. Brutal No doesn't mean being mean to your coworkers; it means explaining kindly and professionally that you have other priorities.

To-Do vs. MITs

Most people are productive with a routine to-do list. You can't be effective by just checking boxes on a To-do list. When was the last time you completed all of your daily tasks? Never. You must replace the to-do list with Most Important Tasks (MITs). MITs allow you to focus on the most important tasks on your list. You feel progress and accomplishment when you finish these tasks. MITs don't include ad-hoc emails, meetings, etc.

Journal Mapped

Most people don't journal or plan their day in the developing South. I've learned to plan my day in my journal over time. I have multiple sections on one page: MITs (things I want to accomplish that day), Other Activities (stuff I can postpone), Life (health, faith, and family issues), and Pop-Ups (things that just pop up). I leave the next page blank for notes. I reflected on the blocks to identify areas to improve the next day. You will have bad days, but at least you'll realize it was due to poor time management.

Buy time/delegate

Time or money? When you make enough money, you lose time to make more. The smart buy "Time." I resisted buying other people's time for years. I regret not hiring an assistant sooner. Learn to buy time from others and pay for time-consuming tasks. Sometimes you think you're saving money by doing things yourself, but you're actually losing money.

This post is a summary. See the full post here.

How to overcome public speaking fear and give a killer presentation

"Public speaking is people's biggest fear, according to studies. Death's second. The average person is better off in the casket than delivering the eulogy."  — American comedian, actor, writer, and producer Jerry Seinfeld

People fear public speaking, according to research. Public speaking can be intimidating.

Most professions require public speaking, whether to 5, 50, 500, or 5,000 people. Your career will require many presentations. In a small meeting, company update, or industry conference.

You can improve your public speaking skills. You can reduce your anxiety, improve your performance, and feel more comfortable speaking in public.

“If I returned to college, I'd focus on writing and public speaking. Effective communication is everything.” — 38th president Gerald R. Ford

You can deliver a great presentation despite your fear of public speaking. There are ways to stay calm while speaking and become a more effective public speaker.

Seven tips to improve your public speaking today. Let's help you overcome your fear (no pun intended).

Know your audience.

"You're not being judged; the audience is." — Entrepreneur, author, and speaker Seth Godin

Understand your audience before speaking publicly. Before preparing a presentation, know your audience. Learn what they care about and find useful.

Your presentation may depend on where you're speaking. A classroom is different from a company meeting.

Determine your audience before developing your main messages. Learn everything about them. Knowing your audience helps you choose the right words, information (thought leadership vs. technical), and motivational message.

2. Be Observant

Observe others' speeches to improve your own. Watching free TED Talks on education, business, science, technology, and creativity can teach you a lot about public speaking.

What worked and what didn't?

• What would you change?

• Their strengths

• How interesting or dull was the topic?

Note their techniques to learn more. Studying the best public speakers will amaze you.

Learn how their stage presence helped them communicate and captivated their audience. Please note their pauses, humor, and pacing.

3. Practice

"A speaker should prepare based on what he wants to learn, not say." — Author, speaker, and pastor Tod Stocker

Practice makes perfect when it comes to public speaking. By repeating your presentation, you can find your comfort zone.

When you've practiced your presentation many times, you'll feel natural and confident giving it. Preparation helps overcome fear and anxiety. Review notes and important messages.

When you know the material well, you can explain it better. Your presentation preparation starts before you go on stage.

Keep a notebook or journal of ideas, quotes, and examples. More content means better audience-targeting.

4. Self-record

Videotape your speeches. Check yourself. Body language, hands, pacing, and vocabulary should be reviewed.

Best public speakers evaluate their performance to improve.

Write down what you did best, what you could improve and what you should stop doing after watching a recording of yourself. Seeing yourself can be unsettling. This is how you improve.

5. Remove text from slides

"Humans can't read and comprehend screen text while listening to a speaker. Therefore, lots of text and long, complete sentences are bad, bad, bad.” —Communications expert Garr Reynolds

Presentation slides shouldn't have too much text. 100-slide presentations bore the audience. Your slides should preview what you'll say to the audience.

Use slides to emphasize your main point visually.

If you add text, use at least 40-point font. Your slides shouldn't require squinting to read. You want people to watch you, not your slides.

6. Body language

"Body language is powerful." We had body language before speech, and 80% of a conversation is read through the body, not the words." — Dancer, writer, and broadcaster Deborah Bull

Nonverbal communication dominates. Our bodies speak louder than words. Don't fidget, rock, lean, or pace.

Relax your body to communicate clearly and without distraction through nonverbal cues. Public speaking anxiety can cause tense body language.

Maintain posture and eye contact. Don’t put your hand in your pockets, cross your arms, or stare at your notes. Make purposeful hand gestures that match what you're saying.

7. Beginning/ending Strong

Beginning and end are memorable. Your presentation must start strong and end strongly. To engage your audience, don't sound robotic.

Begin with a story, stat, or quote. Conclude with a summary of key points. Focus on how you will start and end your speech.

You should memorize your presentation's opening and closing. Memorize something naturally. Excellent presentations start and end strong because people won't remember the middle.

Bringing It All Together

Seven simple yet powerful ways to improve public speaking. Know your audience, study others, prepare and rehearse, record yourself, remove as much text as possible from slides, and start and end strong.

Follow these tips to improve your speaking and audience communication. Prepare, practice, and learn from great speakers to reduce your fear of public speaking.

"Speaking to one person or a thousand is public speaking." — Vocal coach Roger Love