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.

In this essay, I studied REAL NFT use examples and their potential uses.

Knowledge of the Hype Cycle

Gartner's Hype Cycle.

It proposes 5 phases for disruptive technology.

1. Technology Trigger: the emergence of potentially disruptive technology.

2. Peak of Inflated Expectations: Early publicity creates hype. (Ex: 2021 Bubble)

3. Trough of Disillusionment: Early projects fail to deliver on promises and the public loses interest. I suspect NFTs are somewhere around this trough of disillusionment now.

4. Enlightenment slope: The tech shows successful use cases.

5. Plateau of Productivity: Mainstream adoption has arrived and broader market applications have proven themselves. Here’s a more detailed visual of the Gartner Hype Cycle from Wikipedia.

In the speculative NFT bubble of 2021, @beeple sold Everydays: the First 5000 Days for $69 MILLION in 2021's NFT bubble.

@nbatopshot sold millions in video collectibles.

This is when expectations peaked.

Let's examine NFTs' real-world applications.

Watch this video if you're unfamiliar with NFTs.

Online Art

Most people think NFTs are rich people buying worthless JPEGs and MP4s.

Digital artwork and collectibles are revolutionary for creators and enthusiasts.

NFT Profile Pictures

You might also have seen NFT profile pictures on Twitter.

My profile picture is an NFT I coined with @skogards factoria app, which helps me avoid bogus accounts.

Profile pictures are a good beginning point because they're unique and clearly yours.

NFTs are a way to represent proof-of-ownership. It’s easier to prove ownership of digital assets than physical assets, which is why artwork and pfps are the first use cases.

They can do much more.

NFTs can represent anything with a unique owner and digital ownership certificate. Domains and usernames.

Usernames & Domains

@unstoppableweb, @ensdomains, @rarible sell NFT domains.

NFT domains are transferable, which is a benefit.

Godaddy and other web2 providers have difficult-to-transfer domains. Domains are often leased instead of purchased.

Tickets

NFTs can also represent concert tickets and event passes.

There's a limited number, and entry requires proof.

NFTs can eliminate the problem of forgery and make it easy to verify authenticity and ownership.

NFT tickets can be traded on the secondary market, which allows for:

1. marketplaces that are uniform and offer the seller and buyer security (currently, tickets are traded on inefficient markets like FB & craigslist)

2. unbiased pricing

3. Payment of royalties to the creator

4. Historical ticket ownership data implies performers can airdrop future passes, discounts, etc.

5. NFT passes can be a fandom badge.

The $30B+ online tickets business is increasing fast.

NFT-based ticketing projects:

• @GetProtocol

• @SeatlabNFT

• @YellowHeartNFT

• @GUTStickets

• @WicketEvents

Gaming Assets

NFTs also help in-game assets.

Imagine someone spending five years collecting a rare in-game blade, then outgrowing or quitting the game. Gamers value that collectible.

The gaming industry is expected to make $200 BILLION in revenue this year, a significant portion of which comes from in-game purchases.

Royalties on secondary market trading of gaming assets encourage gaming businesses to develop NFT-based ecosystems.

Digital assets are the start. On-chain NFTs can represent real-world assets effectively.

Real estate has a unique owner and requires ownership confirmation.

Real Estate

Tokenizing property has many benefits.

1. Can be fractionalized to increase access, liquidity

2. Can be collateralized to increase capital efficiency and access to loans backed by an on-chain asset

3. Allows investors to diversify or make bets on specific neighborhoods, towns or cities +++

I've written about this thought exercise before.

I made an animated video explaining this.

We've just explored NFTs for transferable assets. But what about non-transferrable NFTs?

SBTs are Soul-Bound Tokens. Vitalik Buterin (Ethereum co-founder) blogged about this.

NFTs are basically verifiable digital certificates.

Diplomas & Degrees

That fits Degrees & Diplomas. These shouldn't be marketable, thus they can be non-transferable SBTs.

Anyone can verify the legitimacy of on-chain credentials, degrees, abilities, and achievements.

The same goes for other awards.

For example, LinkedIn could give you a verified checkmark for your degree or skills.

Authenticity Protection

NFTs can also safeguard against counterfeiting.

Counterfeiting is the largest criminal enterprise in the world, estimated to be $2 TRILLION a year and growing.

Anti-counterfeit tech is valuable.

This is one of @ORIGYNTech's projects.

Identity

Identity theft/verification is another real-world problem NFTs can handle.

In the US, 15 million+ citizens face identity theft every year, suffering damages of over $50 billion a year.

This isn't surprising considering all you need for US identity theft is a 9-digit number handed around in emails, documents, on the phone, etc.

Identity NFTs can fix this.

• NFTs are one-of-a-kind and unforgeable.

• NFTs offer a universal standard.

• NFTs are simple to verify.

• SBTs, or non-transferrable NFTs, are tied to a particular wallet.

• In the event of wallet loss or theft, NFTs may be revoked.

This could be one of the biggest use cases for NFTs.

Imagine a global identity standard that is standardized across countries, cannot be forged or stolen, is digital, easy to verify, and protects your private details.

Since your identity is more than your government ID, you may have many NFTs.

@0xPolygon and @civickey are developing on-chain identity.

Memberships

NFTs can authenticate digital and physical memberships.

Voting

NFT IDs can verify votes.

If you remember 2020, you'll know why this is an issue.

Online voting's ease can boost turnout.

Informational property

NFTs can protect IP.

This can earn creators royalties.

NFTs have 2 important properties:

• Verifiability IP ownership is unambiguously stated and publicly verified.

• Platforms that enable authors to receive royalties on their IP can enter the market thanks to standardization.

Content Rights

Monetization without copyrighting = more opportunities for everyone.

This works well with the music.

Spotify and Apple Music pay creators very little.

Crowdfunding

Creators can crowdfund with NFTs.

NFTs can represent future royalties for investors.

This is particularly useful for fields where people who are not in the top 1% can’t make money. (Example: Professional sports players)

Mirror.xyz allows blog-based crowdfunding.

Financial NFTs

This introduces Financial NFTs (fNFTs). Unique financial contracts abound.

Examples:

• a person's collection of assets (unique portfolio)

• A loan contract that has been partially repaid with a lender

• temporal tokens (ex: veCRV)

Legal Agreements

Not just financial contracts.

NFT can represent any legal contract or document.

Messages & Emails

What about other agreements? Verbal agreements through emails and messages are likewise unique, but they're easily lost and fabricated.

Health Records

Medical records or prescriptions are another types of documentation that has to be verified but isn't.

Medical NFT examples:

• Immunization records

• Covid test outcomes

• Prescriptions

• health issues that may affect one's identity

• Observations made via health sensors

Existing systems of proof by paper / PDF have photoshop-risk.

I tried to include most use scenarios, but this is just the beginning.

NFTs have many innovative uses.

For example: @ShaanVP minted an NFT called “5 Minutes of Fame” 👇

Here are 2 Twitter threads about NFTs:

1. This piece of gold by @chriscantino

2. This conversation between @punk6529 and @RaoulGMI on @RealVision“The World According to @punk6529”

If you're wondering why NFTs are better than web2 databases for these use scenarios, see this Twitter thread I wrote:

If you liked this, please share it.

I apologize if this sounds basic, but throughout my career, I've always been clear that a company's activities are shaped by its founder(s)' morality.

I consider Palantir, owned by PayPal founder Peter Thiel, evil. He got $5 billion tax-free by hacking a statute to help middle-class savings. That may appear clever, but I think it demonstrates a shocking lack of solidarity with society. As a result of this and other things he has said and done, I early on dismissed Peter Thiel as someone who could contribute anything positive to society, and events soon proved me right: we are talking about someone who clearly considers himself above everyone else and who does not hesitate to set up a company, Palantir, to exploit the data of the little people and sell it to the highest bidder, whoever that is and whatever the consequences.

The German courts have confirmed my warnings concerning Palantir. The problem is that politicians love its surveillance tools because they think knowing more about their constituents gives them power. These are ideal for dictatorships who want to snoop on their populace. Hence, Silicon Valley's triumphalist dialectic has seduced many governments at many levels and collected massive volumes of data to hold forever.

Dangerous company. There are many more. My analysis of the moral principles that disclose company management changed my opinion of Facebook, now Meta, and anyone with a modicum of interest might deduce when that happened, a discovery that leaves you dumbfounded. TikTok was easy because its lack of morality was revealed early when I saw the videos it encouraged minors to post and the repercussions of sharing them through its content recommendation algorithm. When you see something like this, nothing can convince you that the firm can change its morals and become good. Nothing. You know the company is awful and will fail. Speak it, announce it, and change it. It's like a fingerprint—unchangeable.

Some of you who read me frequently make its Facebook today jokes when I write about these firms, and that's fine: they're my moral standards, those of an elderly professor with thirty-five years of experience studying corporations and discussing their cases in class, but you don't have to share them. Since I'm writing this and don't have to submit to any editorial review, that's what it is: when you continuously read a person, you have to assume that they have moral standards and that sometimes you'll agree with them and sometimes you won't. Morality accepts hierarchies, nuances, and even obsessions. I know not everyone shares my opinions, but at least I can voice them. One day, one of those firms may sue me (as record companies did some years ago).

Palantir is incredibly harmful. Limit its operations. Like Meta and TikTok, its business strategy is shaped by its founders' immorality. Such a procedure can never be beneficial.