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.

My growth-killing mistake

When you initially start developing your audience, you need guidance.

What does my audience like? What do they not like? How can I grow more?

When I started writing two years ago, I inquired daily. Taking cues from your audience to develop more valuable content is a good concept, but it's simple to let them destroy your growth.

A small audience doesn't represent the full picture.

When I had fewer than 100 YouTube subscribers, I tried several video styles and topics. I looked to my audience for what to preserve and what to change.

If my views, click-through rate, or average view % dropped, that topic or style was awful. Avoiding that style helped me grow.

Vlogs, talking head videos on writing, and long-form tutorials didn't fare well.

Since I was small, I've limited the types of films I make. I have decided to make my own videos.

Surprisingly, the videos I avoided making meet or exceed my views, CTR, and audience retention.

A limited audience can't tell you what your tribe wants. Therefore, limiting your innovation will prohibit you from reaching the right audience. Finding them may take longer.

Large Creators Experience The Same Issue

In the last two years, I've heard Vanessa Lau and Cathrin Manning say they felt pigeonholed into generating videos they didn't want to do.

Why does this happen over and over again?

Once you have a popular piece of content, your audience will grow. So when you publish inconsistent material, fewer of your new audience will view it. You interpret the drop in views as a sign that your audience doesn't want the content, so you stop making it.

Repeat this procedure a few times, and you'll create stuff you're not passionate about because you're frightened to publish it.

How to Manage Your Creativity and Audience Development

I'm not recommending you generate random content.

Instead of feeling trapped by your audience, you can cultivate a diverse audience.

Create quality material on a range of topics and styles as you improve. Be creative until you get 100 followers. Look for comments on how to improve your article.

If you observe trends in the types of content that expand your audience, focus 50-75% of your material on those trends. Allow yourself to develop 25% non-performing material.

This method can help you expand your audience faster with your primary trends and like all your stuff. Slowly, people will find 25% of your material, which will boost its performance.

How to Expand Your Audience Without Having More Limited Content

Follow these techniques to build your audience without feeling confined.

• Don't think that you need restrict yourself to what your limited audience prefers.

• Don't let the poor performance of your desired material demotivate you.

• You shouldn't restrict the type of content you publish or the themes you cover when you have less than 100 followers.

• When your audience expands, save 25% of your content for your personal interests, regardless of how well it does.

Here's how to be clear.

“I have only made this letter longer because I have not had the time to make it shorter.” — French mathematician, physicist, inventor, philosopher, and writer Blaise Pascal

Concise.

People want this. We tend to repeat ourselves and use unnecessary words.

Being vague frustrates readers. It focuses their limited attention span on figuring out what you're saying rather than your message.

Edit carefully.

“Examine every word you put on paper. You’ll find a surprising number that don’t serve any purpose.” — American writer, editor, literary critic, and teacher William Zinsser

How do you write succinctly?

Here are three ways to polish your writing.

1. Delete

Your readers will appreciate it if you delete unnecessary words. If a word or phrase is essential, keep it. Don't force it.

Many readers dislike bloated sentences. Ask yourself if cutting a word or phrase will change the meaning or dilute your message.

For example, you could say, “It’s absolutely essential that I attend this meeting today, so I know the final outcome.” It’s better to say, “It’s critical I attend the meeting today, so I know the results.”

Key takeaway

Delete actually, completely, just, full, kind of, really, and totally. Keep the necessary words, cut the rest.

2. Just Do It

Don't tell readers your plans. Your readers don't need to know your plans. Who are you?

Don't say, "I want to highlight our marketing's problems." Our marketing issues are A, B, and C. This cuts 5–7 words per sentence.

Keep your reader's attention on the essentials, not the fluff. What are you doing? You won't lose readers because you get to the point quickly and don't build up.

Key takeaway

Delete words that don't add to your message. Do something, don't tell readers you will.

3. Cut Overlap

You probably repeat yourself unintentionally. You may add redundant sentences when brainstorming. Read aloud to detect overlap.

Remove repetition from your writing. It's important to edit our writing and thinking to avoid repetition.

Key Takeaway

If you're repeating yourself, combine sentences to avoid overlap.

4. Simplify

Write as you would to family or friends. Communicate clearly. Don't use jargon. These words confuse readers.

Readers want specifics, not jargon. Write simply. Done.

Most adults read at 8th-grade level. Jargon and buzzwords make speech fluffy. This confuses readers who want simple language.

Key takeaway

Ensure all audiences can understand you. USA Today's 5th-grade reading level is intentional. They want everyone to understand.

5. Active voice

Subjects perform actions in active voice. When you write in passive voice, the subject receives the action.

For example, “the board of directors decided to vote on the topic” is an active voice, while “a decision to vote on the topic was made by the board of directors” is a passive voice.

Key takeaway

Active voice clarifies sentences. Active voice is simple and concise.

Bringing It All Together

Five tips help you write clearly. Delete, just do it, cut overlap, use simple language, and write in an active voice.

Clear writing is effective. It's okay to occasionally use unnecessary words or phrases. Realizing it is key. Check your writing.

Adding words costs.

Write more concisely. People will appreciate it and read your future articles, emails, and messages. Spending extra time will increase trust and influence.

“Not that the story need be long, but it will take a long while to make it short.” — Naturalist, essayist, poet, and philosopher Henry David Thoreau

Elon Musk, Matt Taibbi, the 'Twitter Files,' and Hunter Biden's laptop: what gives?

Explaining Twitter Files

Matt Taibbi released "The Twitter Files," a batch of emails sent by Twitter executives discussing the company's decision to stop an October 2020 New York Post story online.

What's on Twitter? New York Post and Fox News call them "bombshell" documents. Or, as a Post columnist admitted, are they "not the smoking gun"? Onward!

What started this?

The New York Post published an exclusive, potentially explosive story in October 2020: Biden's Secret Emails: Ukrainian executive thanks Hunter Biden for'meeting' veep dad. The story purported to report the contents of a laptop brought to the tabloid by a Delaware computer repair shop owner who said it belonged to President Biden's second son, Hunter Biden. Emails and files on the laptop allegedly showed how Hunter peddled influence with Ukranian businessmen and included a "raunchy 12-minute video" of Hunter smoking crack and having sex.

Twitter banned links to the Post story after it was published, calling it "hacked material." The Post's Twitter account was suspended for multiple days.

Why? Yoel Roth, Twitter's former head of trust and safety, said the company couldn't verify the story, implying they didn't trust the Post.

Twitter's stated purpose rarely includes verifying news stories. This seemed like intentional political interference. This story was hard to verify because the people who claimed to have found the laptop wouldn't give it to other newspapers. (Much of the story, including Hunter's business dealings in Ukraine and China, was later confirmed.)

Roth: "It looked like a hack and leak."

So what are the “Twitter Files?”

Twitter's decision to bury the story became a political scandal, and new CEO Elon Musk promised an explanation. The Twitter Files, named after Facebook leaks.

Musk promised exclusive details of "what really happened" with Hunter Biden late Friday afternoon. The tweet was punctuated with a popcorn emoji.

Explaining Twitter Files

Three hours later, journalist Matt Taibbi tweeted more than three dozen tweets based on internal Twitter documents that revealed "a Frankensteinian tale of a human-built mechanism grown out of its designer's control."

Musk sees this release as a way to shape Twitter's public perception and internal culture in his image. We don't know if the CEO gave Taibbi the documents. Musk hyped the document dump before and during publication, but Taibbi cited "internal sources."

Taibbi shares email screenshots showing Twitter execs discussing the Post story and blocking its distribution. Taibbi says the emails show Twitter's "extraordinary steps" to bury the story.

Twitter communications chief Brandon Borrman has the most damning quote in the Files. Can we say this is policy? The story seemed unbelievable. It seemed like a hack... or not? Could Twitter, which ex-CEO Dick Costolo called "the free speech wing of the free speech party," censor a news story?

Many on the right say the Twitter Files prove the company acted at the behest of Democrats. Both parties had these tools, writes Taibbi. In 2020, both the Trump White House and Biden campaign made requests. He says the system for reporting tweets for deletion is unbalanced because Twitter employees' political donations favor Democrats. Perhaps. These donations may have helped Democrats connect with Twitter staff, but it's also possible they didn't. No emails in Taibbi's cache show these alleged illicit relations or any actions Twitter employees took as a result.

Even Musk's supporters were surprised by the drop. Miranda Devine of the New York Post told Tucker Carlson the documents weren't "the smoking gun we'd hoped for." Sebastian Gorka said on Truth Social, "So far, I'm deeply underwhelmed." DC Democrats collude with Palo Alto Democrats. Whoop!” The Washington Free Beacon's Joe Simonson said the Twitter files are "underwhelming." Twitter was staffed by Democrats who did their bidding. (Why?)

If "The Twitter Files" matter, why?

These emails led Twitter to suppress the Hunter Biden laptop story has real news value. It's rare for a large and valuable company like Twitter to address wrongdoing so thoroughly. Emails resemble FOIA documents. They describe internal drama at a company with government-level power. Katie Notopoulos tweeted, "Any news outlet would've loved this scoop!" It's not a'scandal' as teased."

Twitter's new owner calls it "the de facto public town square," implying public accountability. Like a government agency. Though it's exciting to receive once-hidden documents in response to a FOIA, they may be boring and tell you nothing new. Like Twitter files. We learned how Twitter blocked the Post's story, but not why. Before these documents were released, we knew Twitter had suppressed the story and who was involved.

These people were disciplined and left Twitter. Musk fired Vijaya Gadde, the former CLO who reportedly played a "key role" in the decision. Roth quit over Musk's "dictatorship." Musk arrived after Borrman left. Jack Dorsey, then-CEO, has left. Did those who digitally quarantined the Post's story favor Joe Biden and the Democrats? Republican Party opposition and Trump hatred? New York Post distaste? According to our documents, no. Was there political and press interference? True. We knew.

Taibbi interviewed anonymous ex-Twitter employees about the decision; all expressed shock and outrage. One source said, "Everyone knew this was fucked." Since Taibbi doesn't quote that expletive, we can assume the leaked emails contained few or no sensational quotes. These executives said little to support nefarious claims.

Outlets more invested in the Hunter Biden story than Gizmodo seem vexed by the release and muted headlines. The New York Post, which has never shied away from a blaring headline in its 221-year history, owns the story of Hunter Biden's laptop. Two Friday-night Post alerts about Musk's actions were restrained. Elon Musk will drop Twitter files on NY Post-Hunter Biden laptop censorship today. Elon Musk's Twitter dropped Post censorship details from Biden's laptop. Fox News' Apple News push alert read, "Elon Musk drops Twitter censorship documents."

Bombshell, bombshell, bombshell… what, exactly, is the bombshell? Maybe we've heard this story too much and are missing the big picture. Maybe these documents detail a well-documented decision.

The Post explains why on its website. "Hunter Biden laptop bombshell: Twitter invented reason to censor Post's reporting," its headline says.

Twitter's ad hoc decision to moderate a tabloid's content is not surprising. The social network had done this for years as it battled toxic users—violent white nationalists, virulent transphobes, harassers and bullies of all political stripes, etc. No matter how much Musk crows, the company never had content moderation under control. Buzzfeed's 2016 investigation showed how Twitter has struggled with abusive posters since 2006. Jack Dorsey and his executives improvised, like Musk.

Did the US government interfere with the ex-social VP's media company? That's shocking, a bombshell. Musk said Friday, "Twitter suppressing free speech by itself is not a 1st amendment violation, but acting under government orders with no judicial review is." Indeed! Taibbi believed this. August 2022: "The laptop is secondary." Zeynep Tufecki, a Columbia professor and New York Times columnist, says the FBI is cutting true story distribution. Taibbi retracted the claim Friday night: "I've seen no evidence of government involvement in the laptop story."

What’s the bottom line?

I'm still not sure what's at stake in the Hunter Biden scandal after dozens of New York Post articles, hundreds of hours of Fox News airtime, and thousands of tweets. Briefly: Joe Biden's son left his laptop with a questionable repairman. FBI confiscated it? The repairman made a copy and gave it to Rudy Giuliani's lawyer. The Post got it from Steve Bannon. On that laptop were videos of Hunter Biden smoking crack, cavorting with prostitutes, and emails about introducing his father to a Ukrainian businessman for $50,000 a month. Joe Biden urged Ukraine to fire a prosecutor investigating the company. What? The story seems to be about Biden family business dealings, right?

The discussion has moved past that point anyway. Now, the story is the censorship of it. Adrienne Rich wrote in "Diving Into the Wreck" that she came for "the wreck and not the story of the wreck" No matter how far we go, Hunter Biden's laptop is done. Now, the crash's story matters.

I'm dizzy. Katherine Miller of BuzzFeed wrote, "I know who I believe, and you probably do, too. To believe one is to disbelieve the other, which implicates us in the decision; we're stuck." I'm stuck. Hunter Biden's laptop is a political fabrication. You choose. I've decided.

This could change. Twitter Files drama continues. Taibbi said, "Much more to come." I'm dizzy.