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.

I've written a lot about Y Combinator's success, but as any startup founder or investor knows, many startups fail.

Rebel Fund invests in the top 5-10% of new Y Combinator startups each year, so we focus on identifying and supporting the most promising technology startups in our ecosystem. Given the power law dynamic and asymmetric risk/return profile of venture capital, we worry more about our successes than our failures. Since the latter still counts, this essay will focus on the proportion of YC startups that fail.

Since YC's launch in 2005, the figure below shows the percentage of active, inactive, and public/acquired YC startups by batch.

As more startups finish, the blue bars (active) decrease significantly. By 12 years, 88% of startups have closed or exited. Only 7% of startups reach resolution each year.

YC startups by status after 12 years:

Half the startups have failed, over one-third have exited, and the rest are still operating.

In venture investing, it's said that failed investments show up before successful ones. This is true for YC startups, but only in their early years.

Below, we only present resolved companies from the first chart. Some companies fail soon after establishment, but after a few years, the inactive vs. public/acquired ratio stabilizes around 55:45. After a few years, a YC firm is roughly as likely to quit as fail, which is better than I imagined.

I prepared this post because Rebel investors regularly question me about YC startup failure rates and how long it takes for them to exit or shut down.

Early-stage venture investors can overlook it because 100x investments matter more than 0x investments.

YC founders can ignore it because it shouldn't matter if many of their peers succeed or fail ;)

He built a $40M hip-hop empire from street drug dealing.

50 Cent was nearly killed by 9mm bullets.

Before 50 Cent, Curtis Jackson sold drugs.

He sold coke to worried addicts after being orphaned at 8.

Pursuing police. Murderous hustlers and gangs. Unwitting informers.

Despite his hard life, his hip-hop career was a success.

An assassination attempt ended his career at the start.

What sane producer would want to deal with a man entrenched in crime?

Most would have drowned in self-pity and drank themselves to death.

But 50 Cent isn't most people. Life on the streets had given him fearlessness.

“Having a brush with death, or being reminded in a dramatic way of the shortness of our lives, can have a positive, therapeutic effect. So it is best to make every moment count, to have a sense of urgency about life.” ― 50 Cent, The 50th Law

50 released a series of mixtapes that caught Eminem's attention and earned him a $50 million deal!

50 Cents turned death into life.

Things happen; that is life.

We want problems solved.

Every human has problems, whether it's Jeff Bezos swimming in his billions, Obama in his comfortable retirement home, or Dan Bilzerian with his hired bikini models.

All problems.

Problems churn through life. solve one, another appears.

It's harsh. Life's unfair. We can face reality or run from it.

The latter will worsen your issues.

“The firmer your grasp on reality, the more power you will have to alter it for your purposes.” — 50 Cent, The 50th Law

In a fantasy-obsessed world, 50 Cent loves reality.

Wish for better problem-solving skills rather than problem-free living.

Don't wish, work.

We All Have the True Power of Alchemy

Humans are arrogant enough to think the universe cares about them.

That things happen as if the universe notices our nanosecond existences.

Things simply happen. Period.

By changing our perspective, we can turn good things bad.

The alchemists' search for the philosopher's stone may have symbolized the ability to turn our lead-like perceptions into gold.

Negativity bias tints our perceptions.

Normal sparring broke your elbow? Rest and rethink your training. Fired? You can improve your skills and get a better job.

Consider Curtis if he had fallen into despair.

The legend we call 50 Cent wouldn’t have existed.

The Best Lesson in Life Ever?

Neither avoid nor fear your reality.

That simple sentence contains every self-help tip and life lesson on Earth.

When reality is all there is, why fear it? avoidance?

Or worse, fleeing?

To accept reality, we must eliminate the words should be, could be, wish it were, and hope it will be.

It is. Period.

Only by accepting reality's chaos can you shape your life.

“Behind me is infinite power. Before me is endless possibility, around me is boundless opportunity. My strength is mental, physical and spiritual.” — 50 Cent

Wasn't that a get-rich-quick scheme?

OpenSea processed $2.7 billion in NFT transactions in May 2021.

Fueled by a crypto bull run, rumors of unfathomable riches, and FOMO, Bored Apes, Crypto Punks, and other JPEG-format trash projects flew off the virtual shelves, snatched up by retail investors and celebrities alike.

Over a year later, those shelves are overflowing and warehouses are backlogged. Since March, I've been writing less. In May and June, the bubble was close to bursting.

Apparently, the boom has finally peaked.

This bubble has punctured, and deflation has begun. On Aug. 28, OpenSea processed $9.34 million.

From that euphoric high of $2.7 billion, $9.34 million represents a spectacular decline of 99%.

OpenSea contradicts the data. A trading platform spokeswoman stated the comparison is unfair because it compares the site's highest and lowest trading days. They're the perfect two data points to assess the drop. OpenSea chooses to use ETH volume measures, which ignore crypto's shifting price. Since January 2022, monthly ETH volume has dropped 140%, according to Dune.

Unconvincing counterargument.

Further OpenSea indicators point to declining NFT demand:

• Since January 2022, daily user visits have decreased by 50%.

• Daily transactions have decreased by 50% since the beginning of the year in the same manner.

Off-platform, the floor price of Bored Apes has dropped from 145 ETH to 77 ETH. (At $4,800, a reduction from $700,000 to $370,000). Google search data shows waning popular interest.

It is a trend that will soon vanish, just like laser eyes.

NFTs haven't moved since the new year. Eminem and Snoop Dogg can utilize their apes in music videos or as 3D visuals to perform at the VMAs, but the reality is that NFTs have lost their public appeal and the market is trying to regain its footing.

They've lost popularity because?

Breaking records. The technology still lacks genuine use cases a year and a half after being popular.

They're pricey prestige symbols that have made a few people rich through cunning timing or less-than-savory scams or rug pulling. Over $10.5 billion has been taken through frauds, most of which are NFT enterprises promising to be the next Bored Apes, according to Web3 is going wonderfully. As the market falls, many ordinary investors realize they purchased into a self-fulfilling ecosystem that's halted. Many NFTs are sold between owner-held accounts to boost their price, data suggests. Most projects rely on social media excitement to debut with a high price before the first owners sell and chuckle to the bank. When they don't, the initiative fails, leaving investors high and dry.

NFTs are fading like laser eyes. Most people pushing the technology don't believe in it or the future it may bring. No, they just need a Kool-Aid-drunk buyer.

Everybody wins. When your JPEGs are worth 99% less than when you bought them, you've lost.

When demand reaches zero, many will lose.