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.

Key Takeaways:

Another hack. This time was different. This is a doozy.

Why? Nomad got exploited for $190m. It was crypto's 5th-biggest hack. Ouch.

It wasn't hackers, but random folks. What happened:

A Nomad smart contract flaw was discovered. They couldn't drain the funds at once, so they tried numerous transactions. Rookie!

People noticed and copied the attack.

They just needed to discover a working transaction, substitute the other person's address with theirs, and run it.

In a two-and-a-half-hour attack, $190M was siphoned from Nomad Bridge.

Nomad is a novel approach to blockchain interoperability that leverages an optimistic mechanism to increase the security of cross-chain communication.  — nomad.xyz

This hack was permissionless, therefore anyone could participate.

After the fatal blow, people fought over the scraps.

Cross-chain bridges remain a DeFi weakness and exploit target. When they collapse, it's typically total.

$190M...gobbled.

Unbacked assets are hurting Nomad-dependent chains. Moonbeam, EVMOS, and Milkomeda's TVLs dropped.

This incident is every-man-for-himself, although numerous whitehats exploited the issue... 

But what triggered the feeding frenzy?

How did so many pick the bones?

After a normal upgrade in June, the bridge's Replica contract was initialized with a severe security issue. The  0x00 address was a trusted root, therefore all messages were valid by default.

After a botched first attempt (costing $350k in gas), the original attacker's exploit tx called process() without first 'proving' its validity.

The process() function executes all cross-chain messages and checks the merkle root of all messages (line 185).

The upgrade caused transactions with a'messages' value of 0 (invalid, according to old logic) to be read by default as 0x00, a trusted root, passing validation as 'proven'

Any process() calls were valid. In reality, a more sophisticated exploiter may have designed a contract to drain the whole bridge.

Copycat attackers simply copied/pasted the same process() function call using Etherscan, substituting their address.

The incident was a wild combination of crowdhacking, whitehat activities, and MEV-bot (Maximal Extractable Value) mayhem.

For example, 🍉🍉🍉. eth stole $4M from the bridge, but claims to be whitehat.

Others stood out for the wrong reasons. Repeat criminal Rari Capital (Artibrum) exploited over $3M in stablecoins, which moved to Tornado Cash.

The top three exploiters (with 95M between them) are:

$47M: 0x56D8B635A7C88Fd1104D23d632AF40c1C3Aac4e3

$40M: 0xBF293D5138a2a1BA407B43672643434C43827179

$8M: 0xB5C55f76f90Cc528B2609109Ca14d8d84593590E

Here's a list of all the exploiters:

The project conducted a Quantstamp audit in June; QSP-19 foreshadowed a similar problem.

The auditor's comments that "We feel the Nomad team misinterpreted the issue" speak to a troubling attitude towards security that the project's "Long-Term Security" plan appears to confirm:

Concerns were raised about the team's response time to a live, public exploit; the team's official acknowledgement came three hours later.

"Removing the Replica contract as owner" stopped the exploit, but it was too late to preserve the cash.

Closed blockchain systems are only as strong as their weakest link.

The Harmony network is in turmoil after its bridge was attacked and lost $100M in late June.

What's next for Nomad's ecosystems?

Moonbeam's TVL is now $135M, EVMOS's is $3M, and Milkomeda's is $20M.

Loss of confidence may do more damage than $190M.

Cross-chain infrastructure is difficult to secure in a new, experimental sector. Bridge attacks can pollute an entire ecosystem or more.

Nomadic liquidity has no permanent home, so consumers will always migrate in pursuit of the "next big thing" and get stung when attentiveness wanes.

DeFi still has easy prey...

Sources: rekt.news & The Milk Road.

4 Pricey Tools That Are Valuable

I pay based on ROI (return on investment).

If a $20/month tool or $500 online course doubles my return, I'm in.

Investing helps me build wealth.

Canva Pro

I initially refused to pay.

My course content needed updating a few months ago. My Google Docs text looked cleaner and more professional in Canva.

I've used it to:

• product cover pages

• eBook covers

• Product page infographics

See my Google Sheets vs. Canva product page graph.

Google Sheets vs Canva

Yesterday, I used it to make a LinkedIn video thumbnail. It took less than 5 minutes and improved my video.

In 30 hours, the video had 39,000 views.

Here's more.

HypeFury

Hypefury rocks!

It builds my brand as I sleep. What else?

Because I'm traveling this weekend, I planned tweets for 10 days. It took me 80 minutes.

So while I travel or am absent, my content mill keeps producing.

Also I like:

• I can reach hundreds of people thanks to auto-DMs. I utilize it to advertise freebies; for instance, leave an emoji remark to receive my checklist. And they automatically receive a message in their DM.

• Scheduled Retweets: By appearing in a different time zone, they give my tweet a second chance.

It helps me save time and expand my following, so that's my favorite part.

It’s also super neat:

Zoom Pro

My course involves weekly and monthly calls for alumni.

Google Meet isn't great for group calls. The interface isn't great.

Zoom Pro is expensive, and the monthly payments suck, but it's necessary.

It gives my students a smooth experience.

Previously, we'd do 40-minute meetings and then reconvene.

Zoom's free edition limits group calls to 40 minutes.

This wouldn't be a good online course if I paid hundreds of dollars.

So I felt obligated to help.

YouTube Premium

My laptop has an ad blocker.

I bought an iPad recently.

When you're self-employed and work from home, the line between the two blurs. My bed is only 5 steps away!

When I read or watched videos on my laptop, I'd slide into work mode. Only option was to view on phone, which is awkward.

YouTube premium handles it. No more advertisements and I can listen on the move.

3 Expensive Tools That Aren't Valuable

Marketing strategies are sometimes aimed to make you feel you need 38474 cool features when you don’t.

Certain tools are useless.

I found it useless.

Depending on your needs. As a writer and creator, I get no return.

They could for other jobs.

Shield Analytics

It tracks LinkedIn stats, like:

• follower growth

• trend chart for impressions

• Engagement, views, and comment stats for posts

• and much more.

Middle-tier creator costs $12/month.

I got a 25% off coupon but canceled my free trial before writing this. It's not worth the discount.

Why?

LinkedIn provides free analytics. See:

Not thorough and won't show top posts.

I don't need to see my top posts because I love experimenting with writing.

Slack Premium

Slack was my classroom. Slack provided me a premium trial during the prior cohort.

I skipped it.

Sure, voice notes are better than a big paragraph. I didn't require pro features.

Marketing methods sometimes make you think you need 38474 amazing features. Don’t fall for it.

Calendly Pro

This may be worth it if you get many calls.

I avoid calls. During my 9-5, I had too many pointless calls.

I don't need:

• ability to schedule calls for 15, 30, or 60 minutes: I just distribute each link separately.

• I have a Gumroad consultation page with a payment option.

• follow-up emails: I hardly ever make calls, so

• I just use one calendar, therefore I link to various calendars.

I'll admit, the integrations are cool. Not for me.

If you're a coach or consultant, the features may be helpful. Or book meetings.

Conclusion

Investing is spending to make money.

Use my technique — put money in tools that help you make money. This separates it from being an investment instead of an expense.

Try free versions of these tools before buying them since everyone else is.

Learn these subtle cues to gain influence.

Hate being ignored?

As a 24-year-old, I struggled at work. Attention-getting tips How to avoid being judged by my size, gender, and lack of wrinkles or gray hair?

I've learned seniority hacks. Influence. Within two years as a product manager, I led a team. I'm a Stanford MBA student.

These communication hacks can make you look senior and influential.

1. Slowly speak

We speak quickly because we're afraid of being interrupted.

When I doubt my ideas, I speak quickly. How can we slow down? Jamie Chapman says speaking slowly saps our energy.

Chapman suggests emphasizing certain words and pausing.

2. Interrupted? Stop the stopper

Someone interrupt your speech?

Don't wait. "May I finish?" No pause needed. Stop interrupting. I first tried this in Leadership Laboratory at Stanford. How quickly I gained influence amazed me.

Next time, try “May I finish?” If that’s not enough, try these other tips from Wendy R.S. O’Connor.

3. Context

Others don't always see what's obvious to you.

Through explanation, you help others see the big picture. If a senior knows it, you help them see where your work fits.

4. Don't ask questions in statements

“Your statement lost its effect when you ended it on a high pitch,” a group member told me. Upspeak, it’s called. I do it when I feel uncertain.

Upspeak loses influence and credibility. Unneeded. When unsure, we can say "I think." We can even ask a proper question.

Someone else's boasting is no reason to be dismissive. As leaders and colleagues, we should listen to our colleagues even if they use this speech pattern.

Give your words impact.

5. Signpost structure

Signposts improve clarity by providing structure and transitions.

Communication coach Alexander Lyon explains how to use "first," "second," and "third" He explains classic and summary transitions to help the listener switch topics.

Signs clarify. Clarity matters.

6. Eliminate email fluff

“Fine. When will the report be ready? — Jeff.”

Notice how senior leaders write short, direct emails? I often use formalities like "dear," "hope you're well," and "kind regards"

Formality is (usually) unnecessary.

7. Replace exclamation marks with periods

See how junior an exclamation-filled email looks:

Hi, all!
Hope you’re as excited as I am for tomorrow! We’re celebrating our accomplishments with cake! Join us tomorrow at 2 pm!
See you soon!

Why the exclamation points? Why not just one?

Hi, all.
Hope you’re as excited as I am for tomorrow. We’re celebrating our accomplishments with cake. Join us tomorrow at 2 pm!
See you soon.

8. Take space

"Playing high" means having an open, relaxed body, says Stanford professor and author Deborah Gruenfield.

Crossed legs or looking small? Relax. Get bigger.

Internet's fantastic, right?

We've never had a better way to share our creativity.

I can now draw on my iPad and tweet or Instagram it to thousands. I may get some likes.

With such a great, free tool, you're not alone.

Millions more bright-eyed artists are sharing their work online.

The issue is getting innovative work noticed, not sharing it.

In a world where creators want attention, attention is valuable.

We build for attention.

Attention helps us establish a following, make money, get notoriety, and make a difference.

Most of us require attention to stay sane while creating wonderful things.

I know how hard it is to work hard and receive little views.

How do we receive more attention, more often, in a sea of talent?

Advertising and celebrity endorsements are options. These may work temporarily.

To attract true, organic, and long-term attention, you must create in high quality, high volume, and consistency.

Adapting Steve Martin's Be so amazing, they can't ignore you (with a mention to Dan Norris in his great book Create or Hate for the reminder)

Create a lot.

Eventually, your effort will gain traction.

Traction shows your work's influence.

Traction is when your product sells more. Traction is exponential user growth. Your work is shared more.

No matter how good your work is, it will always have minimal impact on the world.

Your work can eventually dent or puncture. Daily, people work to dent.

To achieve this tipping point, you must consistently produce exceptional work.

Expect traction after hundreds of outputs.

Dilbert creator Scott Adams says repetition persuades. If you don't stop, you can persuade practically anyone with anything.

Volume lends believability. So make more.

I worked as an illustrator for at least a year and a half without any recognition. After 150 illustrations on iStockphoto, my work started selling.

With 350 illustrations on iStock, I started getting decent client commissions.

Producing often will improve your craft and draw attention.

It's the only way to succeed. More creation means better results and greater attention.

Austin Kleon says you can improve your skill in relative anonymity before you become famous. Before obtaining traction, generate a lot and become excellent.

Most artists, even excellent ones, don't create consistently enough to get traction.

It may hurt. For makers who don't love and flow with their work, it's extremely difficult.

Your work must bring you to life.

To generate so much that others can't ignore you, decide what you'll accomplish every day (or most days).

Commit and be patient.

Prepare for zero-traction.

Anticipating this will help you persevere and create.

My online guru Grant Cardone says: Anything worth doing is worth doing every day.

Do.