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.

Alex Mashinsky, CEO of Celsius, speaks at Web Summit 2021 in Lisbon. 

Celsius Network filed for Chapter 11 bankruptcy a month after freezing customer withdrawals, joining other crypto casualties.

Celsius took the step to stabilize its business and restructure for all stakeholders. The filing was done in the Southern District of New York.

The company, which amassed more than $20 billion by offering 18% interest on cryptocurrency deposits, paused withdrawals and other functions in mid-June, citing "extreme market conditions."

As the Fed raises interest rates aggressively, it hurts risk sentiment and squeezes funding costs. Voyager Digital Ltd. filed for Chapter 11 bankruptcy this month, and Three Arrows Capital has called in liquidators.

Celsius called the pause "difficult but necessary." Without the halt, "the acceleration of withdrawals would have allowed certain customers to be paid in full while leaving others to wait for Celsius to harvest value from illiquid or longer-term asset deployment activities," it said.

Celsius declined to comment. CEO Alex Mashinsky said the move will strengthen the company's future.

The company wants to keep operating. It's not requesting permission to allow customer withdrawals right now; Chapter 11 will handle customer claims. The filing estimates assets and liabilities between $1 billion and $10 billion.

Celsius is advised by Kirkland & Ellis, Centerview Partners, and Alvarez & Marsal.

Yield-promises

Celsius promised 18% returns on crypto loans. It lent those coins to institutional investors and participated in decentralized-finance apps.

When TerraUSD (UST) and Luna collapsed in May, Celsius pulled its funds from Terra's Anchor Protocol, which offered 20% returns on UST deposits. Recently, another large holding, staked ETH, or stETH, which is tied to Ether, became illiquid and discounted to Ether.

The lender is one of many crypto companies hurt by risky bets in the bear market. Also, Babel halted withdrawals. Voyager Digital filed for bankruptcy, and crypto hedge fund Three Arrows Capital filed for Chapter 15 bankruptcy.

According to blockchain data and tracker Zapper, Celsius repaid all of its debt in Aave, Compound, and MakerDAO last month.

Celsius charged Symbolic Capital Partners Ltd. 2,000 Ether as collateral for a cash loan on June 13. According to company filings, Symbolic was charged 2,545.25 Ether on June 11.

In July 6 filings, it said it reshuffled its board, appointing two new members and firing others.

Most don't understand the issue until it's too late.

My first startup was what entrepreneurs call the hardest. A two-sided marketplace.

Two-sided marketplaces are the hardest startups because founders must solve the chicken or the egg conundrum.

A two-sided marketplace needs suppliers and buyers. Without suppliers, buyers won't come. Without buyers, suppliers won't come. An empty marketplace and a founder striving to gain momentum result.

My first venture made me a struggling founder seeking to achieve traction for a two-sided marketplace. The company failed, and I vowed never to start another like it.

I didn’t. Unfortunately, my second venture was almost as hard. It failed like the second-hardest startup.

What kind of startup is the second-hardest?

The second-hardest startup, which is almost as hard to develop, is rarely discussed in the startup community. Because of this, I predict more founders fail each year trying to develop the second-toughest startup than the hardest.

Fairly, I have no proof. I see many startups, so I have enough of firsthand experience. From what I've seen, for every entrepreneur developing a two-sided marketplace, I'll meet at least 10 building this other challenging startup.

I'll describe a startup I just met with its two co-founders to explain the second hardest sort of startup and why it's so hard. They created a financial literacy software for parents of high schoolers.

The issue appears plausible. Children struggle with money. Parents must teach financial responsibility. Problems?

It's possible.

Buyers and users are different.

Buyer-user mismatch.

The financial literacy app I described above targets parents. The parent doesn't utilize the app. Child is end-user. That may not seem like much, but it makes customer and user acquisition and onboarding difficult for founders.

The difficulty of a buyer-user imbalance

The company developing a product faces a substantial operational burden when the buyer and end customer are different. Consider classic firms where the buyer is the end user to appreciate that responsibility.

Entrepreneurs selling directly to end users must educate them about the product's benefits and use. Each demands a lot of time, effort, and resources.

Imagine selling a financial literacy app where the buyer and user are different. To make the first sale, the entrepreneur must establish all the items I mentioned above. After selling, the entrepreneur must supply a fresh set of resources to teach, educate, or train end-users.

Thus, a startup with a buyer-user mismatch must market, sell, and train two organizations at once, requiring twice the work with the same resources.

The second hardest startup is hard for reasons other than the chicken-or-the-egg conundrum. It takes a lot of creativity and luck to solve the chicken-or-egg conundrum.

The buyer-user mismatch problem cannot be overcome by innovation or luck. Buyer-user mismatches must be solved by force. Simply said, when a product buyer is different from an end-user, founders have a lot more work. If they can't work extra, their companies fail.

The VP has revealed a secret that should frighten the rest of the EV world.

Tesla led the EV revolution. Elon Musk's invention offers a viable alternative to gas-guzzlers. Tesla has lost ground in recent years. VW, BMW, Mercedes, and Ford offer EVs with similar ranges, charging speeds, performance, and cost. Tesla's next-generation 4680 battery pack, Roadster, Cybertruck, and Semi were all delayed. CATL offers superior batteries than the 4680. Martin Viecha, Tesla's Vice President, recently told Business Insider something that startled the EV world and will establish Tesla as the EV king.

Viecha mentioned that Tesla's production costs have dropped 57% since 2017. This isn't due to cheaper batteries or devices like Model 3. No, this is due to amazing factory efficiency gains.

Musk wasn't crazy to want a nearly 100% automated production line, and Tesla's strategy of sticking with one model and improving it has paid off. Others change models every several years. This implies they must spend on new R&D, set up factories, and modernize service and parts systems. All of this costs a ton of money and prevents them from refining production to cut expenses.

Meanwhile, Tesla updates its vehicles progressively. Everything from the backseats to the screen has been enhanced in a 2022 Model 3. Tesla can refine, standardize, and cheaply produce every part without changing the production line.

In 2017, Tesla's automobile production averaged $84,000. In 2022, it'll be $36,000.

Mr. Viecha also claimed that new factories in Shanghai and Berlin will be significantly cheaper to operate once fully operating.

Tesla's hand is visible. Tesla selling $36,000 cars for $60,000 This barely beats the competition. Model Y long-range costs just over $60,000. Tesla makes $24,000+ every sale, giving it a 40% profit margin, one of the best in the auto business.

VW I.D4 costs about the same but makes no profit. Tesla's rivals face similar challenges. Their EVs make little or no profit.

Tesla costs the same as other EVs, but they're in a different league.

But don't forget that the battery pack accounts for 40% of an EV's cost. Tesla may soon fully utilize its 4680 battery pack.

The 4680 battery pack has larger cells and a unique internal design. This means fewer cells are needed for a car, making it cheaper to assemble and produce (per kWh). Energy density and charge speeds increase slightly.

Tesla underestimated the difficulty of making this revolutionary new cell. Each time they try to scale up production, quality drops and rejected cells rise.

Tesla recently installed this battery pack in Model Ys and is scaling production. If they succeed, Tesla battery prices will plummet.

Tesla's Model Ys 2170 battery costs $11,000. The same size pack with 4680 cells costs $3,400 less. Once scaled, it could be $5,500 (50%) less. The 4680 battery pack could reduce Tesla production costs by 20%.

With these cost savings, Tesla could sell Model Ys for $40,000 while still making a profit. They could offer a $25,000 car.

Even with new battery technology, it seems like other manufacturers will struggle to make EVs profitable.

Teslas cost about the same as competitors, so don't be fooled. Behind the scenes, they're still years ahead, and the 4680 battery pack and new factories will only increase that lead. Musk faces a first. He could sell Teslas at current prices and make billions while other manufacturers struggle. Or, he could massively undercut everyone and crush the competition once and for all. Tesla and Elon win.

This is arguably the best three-minute demonstration I've ever seen of anything. Derek Sivers turns a shaky video of a lone dancing guy at a music festival into a leadership lesson.

A leader must have the courage to stand alone and appear silly. But what he's doing is so straightforward that it's almost instructive. This is critical. You must be simple to follow!

Now comes the first follower, who plays an important role: he publicly demonstrates how to follow. The leader embraces him as an equal, so it's no longer about the leader — it's about them, plural. He's inviting his friends to join him. It takes courage to be the first follower! You stand out and dare to be mocked. Being a first follower is a style of leadership that is underappreciated. The first follower elevates a lone nut to the position of leader. If the first follower is the spark that starts the fire, the leader is the flint.

This link was sent to me by @ottmark, who noted its resemblance to Kurt Vonnegut's three categories of specialists required for revolution.

The rarest of these specialists, he claims, is an actual genius – a person capable generating seemingly wonderful ideas that are not widely known. "A genius working alone is generally dismissed as a crazy," he claims.

The second type of specialist is much easier to find: a highly intellectual person in good standing in his or her community who understands and admires the genius's new ideas and can attest that the genius is not insane. "A person like him working alone can only crave loudly for changes, but fail to say what their shapes should be," Slazinger argues.

Jeff Veen reduced the three personalities to "the inventor, the investor, and the evangelist" on Twitter.