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.

The crypto market crashed in June 2022. Bitcoin and other cryptocurrencies hit their lowest prices in over a year, causing market panic. Some believe this crash will benefit future investors.

Before I discuss how this crash might help investors, let's examine why it happened. Inflation in the U.S. reached a 30-year high in 2022 after Russia invaded Ukraine. In response, the U.S. Federal Reserve raised interest rates by 0.5%, the most in almost 20 years. This hurts cryptocurrencies like Bitcoin. Higher interest rates make people less likely to invest in volatile assets like crypto, so many investors sold quickly.

The crypto market collapsed. Bitcoin, Ethereum, and Binance dropped 40%. Other cryptos crashed so hard they were delisted from almost every exchange. Bitcoin peaked in April 2022 at $41,000, but after the May interest rate hike, it crashed to $28,000. Bitcoin investors were worried. Even in bad times, this crash is unprecedented.

Bitcoin wasn't "doomed." Before the crash, LUNA was one of the top 5 cryptos by market cap. LUNA was trading around $80 at the start of May 2022, but after the rate hike?

Less than 1 cent. LUNA lost 99.99% of its value in days and was removed from every crypto exchange. Bitcoin's "crash" isn't as devastating when compared to LUNA.

Many people said Bitcoin is "due" for a LUNA-like crash and that the only reason it hasn't crashed is because it's bigger. Still false. If so, Bitcoin should be worth zero by now. We didn't. Instead, Bitcoin reached 28,000, then 29k, 30k, and 31k before falling to 18k. That's not the world's greatest recovery, but it shows Bitcoin's safety.

Bitcoin isn't falling constantly. It fell because of the initial shock of interest rates, but not further. Now, Bitcoin's value is more likely to rise than fall. Bitcoin's low price also attracts investors. They know what prices Bitcoin can reach with enough hype, and they want to capitalize on low prices before it's too late.

Bitcoin's crash was bad, but in a way it wasn't. To understand, consider 2021. In March 2021, Bitcoin surpassed $60k for the first time. Elon Musk's announcement in May that he would no longer support Bitcoin caused a massive crash in the crypto market. In May 2017, Bitcoin's price hit $29,000. Elon Musk's statement isn't worth more than the Fed raising rates. Many expected this big announcement to kill Bitcoin.

Not so. Bitcoin crashed from $58k to $31k in 2021. Bitcoin fell from $41k to $28k in 2022. This crash is smaller. Bitcoin's price held up despite tensions and stress, proving investors still believe in it. What happened after the initial crash in the past?

Bitcoin fell until mid-July. This is also something we’re not seeing today. After a week, Bitcoin began to improve daily. Bitcoin's price rose after mid-July. Bitcoin's price fluctuated throughout the rest of 2021, but it topped $67k in November. Despite no major changes, the peak occurred after the crash. Elon Musk seemed uninterested in crypto and wasn't likely to change his mind soon. What triggered this peak? Nothing, really. What really happened is that people got over the initial statement. They forgot.

Internet users have goldfish-like attention spans. People quickly forgot the crash's cause and were back investing in crypto months later. Despite the market's setbacks, more crypto investors emerged by the end of 2017. Who gained from these peaks? Bitcoin investors who bought low. Bitcoin not only recovered but also doubled its ROI. It was like a movie, and it shows us what to expect from Bitcoin in the coming months.

The current Bitcoin crash isn't as bad as the last one. LUNA is causing market panic. LUNA and Bitcoin are different cryptocurrencies. LUNA crashed because Terra wasn’t able to keep its peg with the USD. Bitcoin is unanchored. It's one of the most decentralized investments available. LUNA's distrust affected crypto prices, including Bitcoin, but it won't last forever.

This is why Bitcoin will likely rebound in the coming months. In 2022, people will get over the rise in interest rates and the crash of LUNA, just as they did with Elon Musk's crypto stance in 2021. When the world moves on to the next big controversy, Bitcoin's price will soar.

Bitcoin may recover for another reason. Like controversy, interest rates fluctuate. The Russian invasion caused this inflation. World markets will stabilize, prices will fall, and interest rates will drop.

Next, lower interest rates could boost Bitcoin's price. Eventually, it will happen. The U.S. economy can't sustain such high interest rates. Investors will put every last dollar into Bitcoin if interest rates fall again.

Bitcoin has proven to be a stable investment. This boosts its investment reputation. Even if Ethereum dethrones Bitcoin as crypto king one day (or any other crypto, for that matter). Bitcoin may stay on top of the crypto ladder for a while. We'll have to wait a few months to see if any of this is true.

This post is a summary. Read the full article here.

WTF is wrong with you people?

When I was laid off as editor of my town's daily newspaper, I went silent on social media. I knew it was coming and had been quietly removing personal items each day, but the pain was intense.

I posted a day later. I didn't bad-mouth GateHouse Media but expressed my sadness at leaving the newspaper industry, pride in my accomplishments, and hope for success in another industry.

Normal job-loss response.

What do you recognize as abnormal?

The bullshit I’ve been reading from laid-off folks on LinkedIn.

If you're there, you know. Many Twitter or Facebook/Meta employees recently lost their jobs.

Well, many of them did not “lose their job,” actually. They were “impacted by the layoffs” at their former employer. I keep seeing that phrase.

Why don’t they want to actually say it? Why the euphemism?

Many are excited about the opportunities ahead. The jobless deny being sad.

They're ecstatic! They have big plans.

Hope so. Sincerely! Being laid off stinks, especially if, like me, your skills are obsolete. It's worse if, like me, you're too old to start a new career. Ageism exists despite denials.

Nowadays, professionalism seems to demand psychotic levels of fake optimism.

Why? Life is unpredictable. That's indisputable. You shouldn't constantly complain or cry in public, but you also shouldn't pretend everything's great.

It makes you look psychotic, not positive. It's like saying at work:

“I was impacted by the death of my spouse of 20 years this week, and many of you have reached out to me, expressing your sympathy. However, I’m choosing to remember the amazing things we shared. I feel confident that there is another marriage out there for me, and after taking a quiet weekend trip to reset myself, I’ll be out there looking for the next great marital adventure! #staypositive #available #opentolove

Also:

“Now looking for our next #dreamhome after our entire neighborhood was demolished by a wildfire last night. We feel so lucky to have lived near so many amazing and inspirational neighbors, all of whom we will miss as we go on our next housing adventure. The best house for us is yet to come! If you have a great neighborhood you’d recommend, please feel free to reach out and touch base with us! #newhouse #newneighborhood #newlife

Admit it. That’s creepy.

The constant optimism makes me feel sick to my stomach.

Viscerally.

I hate fakes.

Imagine a fake wood grain desk. Wouldn't it be better if the designer accepted that it's plastic and went with that?

Real is better but not always nice. When something isn't nice, you don't have to go into detail, but you also shouldn't pretend it's great.

How to announce your job loss to the world.

Do not pretend to be happy, but don't cry and drink vodka all afternoon.

Say you loved your job, and that you're looking for new opportunities.

Yes, if you'll miss your coworkers. Otherwise, don't badmouth. No bridge-burning!

Please specify the job you want. You may want to pivot.

Alternatively, try this.

You could always flame out.

If you've pushed yourself too far into toxic positivity, you may be ready to burn it all down. If so, make it worthwhile by writing something like this:

Well, I was shitcanned by the losers at #Acme today. That bitch Linda in HR threw me under the bus just because she saw that one of my “friends” tagged me in some beach pics on social media after I called in sick with Covid. The good thing is I will no longer have to watch my ass around that #asspincher Ron in accounting, but I’m sad that I will no longer have a cushy job with high pay or access to the primo office supplies I’ve been sneaking home for the last five years. (Those gel pens were the best!) I am going to be taking some time off to enjoy my unemployment and hammer down shots of Jägermeister but in about five months I’ll be looking for anything easy with high pay and great benefits. Reach out if you can help! #officesupplies #unemploymentrocks #drinkinglikeagirlboss #acmesucks

It beats the fake positivity.

Technical books aren't like novels. We need a new approach to technical texts. I've spent years looking for a decent reading method. I tried numerous ways before finding one that worked. This post explains how I read technical books efficiently.

What Do I Mean When I Say Effective?

Effectiveness depends on the book. Effective implies I know where to find answers after reading a reference book. Effective implies I learned the book's knowledge after reading it.

I use reference books as tools in my toolkit. I won't carry all my tools; I'll merely need them. Non-reference books teach me techniques. I never have to make an effort to use them since I always have them.

Reference books I like:

• Design Patterns: Elements of Reusable Object-Oriented Software

• Refactoring: Improving the Design of Existing Code 
  You can also check My Top Takeaways from Refactoring here.

• Practical Object-Oriented Design in Ruby: An Agile Primer

Non-reference books I like:

• The Pragmatic Programmer: your journey to mastery

• Clean Code: A Handbook of Agile Software Craftsmans

The Approach

Technical books might be overwhelming to read in one sitting. Especially when you have no idea what is coming next as you read. When you don't know how deep the rabbit hole goes, you feel lost as you read. This is my years-long method for overcoming this difficulty.

Whether you follow the step-by-step guide or not, remember these:

• Understand the terminology. Make sure you get the meaning of any terms you come across more than once. The likelihood that a term will be significant increases as you encounter it more frequently.

• Know when to stop. I've always believed that in order to truly comprehend something, I must delve as deeply as possible into it. That, however, is not usually very effective. There are moments when you have to draw the line and start putting theory into practice (if applicable).

• Look over your notes. When reading technical books or documents, taking notes is a crucial habit to develop. Additionally, you must regularly examine your notes if you want to get the most out of them. This will assist you in internalizing the lessons you acquired from the book. And you'll see that the urge to review reduces with time.

Let's talk about how I read a technical book step by step.

0. Read the Foreword/Preface

These sections are crucial in technical books. They answer Who should read it, What each chapter discusses, and sometimes How to Read? This is helpful before reading the book. Who could know the ideal way to read the book better than the author, right?

1. Scanning

I scan the chapter. Fast scanning is needed.

• I review the headings.

• I scan the pictures quickly.

• I assess the chapter's length to determine whether I might divide it into more manageable sections.

2. Skimming

Skimming is faster than reading but slower than scanning.

• I focus more on the captions and subtitles for the photographs.

• I read each paragraph's opening and closing sentences.

• I examined the code samples.

• I attempt to grasp each section's basic points without getting bogged down in the specifics.

• Throughout the entire reading period, I make an effort to make mental notes of what may require additional attention and what may not. Because I don't want to spend time taking physical notes, kindly notice that I am using the term "mental" here. It is much simpler to recall. You may think that this is more significant than typing or writing “Pay attention to X.”

• I move on quickly. This is something I considered crucial because, when trying to skim, it is simple to start reading the entire thing.

3. Complete reading

Previous steps pay off.

• I finished reading the chapter.

• I concentrate on the passages that I mentally underlined when skimming.

• I put the book away and make my own notes. It is typically more difficult than it seems for me. But it's important to speak in your own words. You must choose the right words to adequately summarize what you have read. How do those words make you feel? Additionally, you must be able to summarize your notes while you are taking them. Sometimes as I'm writing my notes, I realize I have no words to convey what I'm thinking or, even worse, I start to doubt what I'm writing down. This is a good indication that I haven't internalized that idea thoroughly enough.

• I jot my inquiries down. Normally, I read on while compiling my questions in the hopes that I will learn the answers as I read. I'll explore those issues more if I wasn't able to find the answers to my inquiries while reading the book.

Bonus!

Best part: If you take lovely notes like I do, you can publish them as a blog post with a few tweaks.

Conclusion

This is my learning journey. I wanted to show you. This post may help someone with a similar learning style. You can alter the principles above for any technical material.