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.

It’s only a loss if you sell, right?

Nayib Bukele proclaimed himself “the world’s coolest dictator”.

His jokes aren't clear.

El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.

His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.

He's popular, though.

Bukele won 53% of the vote by fighting violent crime and opposition party corruption.

El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.

They were powerless.

Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.

He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.

Transfer and deposit costs.

This makes logical sense when the president’s theatrics don’t blind you.

El Salvador's Bukele revealed plans to make bitcoin legal tender.

Remittances total $5.9 billion (23%) of the country's expenses.

Anything that reduces costs could boost the economy.

The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).

According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.

In 2021, 36% of El Salvadorans had bank accounts.

Large rural countries like Kenya seem to have resolved their unbanked dilemma.

An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.

Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.

96% of Kenyans utilize M-PESA, which doesn't require a bank account.

The software involves human agents who hang out with cash and a phone.

These people are like ATMs.

You offer them cash to deposit money in your mobile money account or withdraw cash.

In a country with a faulty banking system, cash availability and a safe place to deposit it are important.

William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.

2016 Science paper

Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.

The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.

Currently, El Salvador has 2,301 Bitcoin.

At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.

Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.

It's still falling.

This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.

The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.

You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.

Weekend project: limitless design service. It was fun to see whether I could start a business quickly.

I use no-code apps to save time and resources.

TL;DR I started a business utilizing EditorX for my website, Notion for client project management, and a few favors to finish my portfolio.

First step: research (Day 1)

I got this concept from a Kimp Instagram ad. The Minimalist Hustler Daily newsletter mentioned a similar and cheaper service (Graphically).

I Googled other unlimited design companies. Many provide different costs and services. Some supplied solely graphic design, web development, or copywriting.

Step 2: Brainstorming (Day 1)

I did something simple.

• What benefits and services to provide

• Price to charge

Since it's a one-person performance (for now), I'm focusing on graphic design. I can charge less.

So I don't overwhelm myself and can accommodate budget-conscious clientele.

Step 3: Construction (Day 1 & 2)

This project includes a management tool, a website, and a team procedure.

I built a project management tool and flow first. Once I had the flow and a Notion board, I tested it with design volunteers. They fake-designed while I built the website.

Tool for Project Management

I modified a Notion template. My goal is to keep clients and designers happy.

Team Approach

My sister, my partner, and I kept this business lean. I tweaked the Notion board to make the process smooth. By the end of Sunday, I’d say it’s perfect!

Website

I created the website after they finished the fake design demands. EditorX's drag-and-drop builder attracted me. I didn't need to learn code, and there are templates.

I used a template wireframe.

This project's hardest aspect is developing the site. It's my first time using EditorX and I'm no developer.

People answer all your inquiries in a large community forum.

As a first-time user developing a site in two days, I think I performed OK. Here's the site for feedback.

4th step: testing (Day 2)

Testing is frustrating because it works or doesn't. My testing day was split in two.

• testing the workflow from payment to onboarding to the website

• the demand being tested

It's working so far. If someone gets the trial, they can request design work.

I've gotten a couple of inquiries about demand. I’ll be working with them as a start.

Completion

Finally! I built my side project in one weekend. It's too early to tell if this is successful. I liked that I didn't squander months of resources testing out an idea.

Jobs' secret emails became public due to a litigation with Samsung.

Steve Jobs sent Phil Schiller an email at the end of 2010. Top 100 A was the codename for Apple's annual Top 100 executive meetings. The 2011 one was scheduled.

Everything about this gathering is secret, even attendance. The location is hidden, and attendees can't even drive themselves. Instead, buses transport them to a 2-3 day retreat.

Due to a litigation with Samsung, this Top 100 meeting's agenda was made public in 2014. This was a critical milestone in Apple's history, not a Top 100 meeting. Apple had many obstacles in the 2010s to remain a technological leader. Apple made more money with non-PC goods than with its best-selling Macintosh series. This was the last Top 100 gathering Steve Jobs would attend before passing, and he wanted to make sure his messages carried on before handing over his firm to Tim Cook.

In this post, we'll discuss lessons from Jobs' meeting agenda. Two sorts of entrepreneurs can use these tips:

1. Those who manage a team in a business and must ensure that everyone is working toward the same goals, upholding the same principles, and being inspired by the same future.

2. Those who are sole proprietors or independent contractors and who must maintain strict self-discipline in order to stay innovative in their industry and adhere to their own growth strategy.

Here's Steve Jobs's email outlining the annual meeting agenda. It's an 11-part summary of the company's shape and strategy.

Steve Jobs outlines Apple's 2011 strategy, 10/24/10

1. Correct your data

Business leaders must comprehend their company's metrics. Jobs either mentions critical information he already knows or demands slides showing the numbers he wants. These numbers fall under 2 categories:

Metrics for growth and strategy

• As we will see, this was a crucial statistic for Apple since it signaled the beginning of the Post PC era and required them to make significant strategic changes in order to stay ahead of the curve. Post PC products now account for 66% of our revenues.

• Within six months, iPad outsold Mac, another sign of the Post-PC age. As we will see, Jobs thought the iPad would be the next big thing, and item number four on the agenda is one of the most thorough references to the iPad.

• Geographical analysis: Here, Jobs emphasizes China, where the corporation has a slower start than anticipated. China was dominating Apple's sales growth with 16% of revenue one year after this meeting.

Metrics for people & culture

• The individuals that make up a firm are more significant to its success than its headcount or average age. That holds true regardless of size, from a 5-person startup to a Fortune 500 firm. Jobs was aware of this, which is why his suggested agenda begins by emphasizing demographic data.

• Along with the senior advancements in the previous year's requested statistic, it's crucial to demonstrate that if the business is growing, the employees who make it successful must also grow.

2. Recognize the vulnerabilities and strengths of your rivals

Steve Jobs was known for attacking his competition in interviews and in his strategies and roadmaps. This agenda mentions 18 competitors, including:

• Google 7 times

• Android 3 times

• Samsung 2 times

Jobs' agenda email was issued 6 days after Apple's Q4 results call (2010). On the call, Jobs trashed Google and Android. His 5-minute intervention included:

• Google has acknowledged that the present iteration of Android is not tablet-optimized.

• Future Android tablets will not work (Dead On Arrival)

• While Google Play only has 90,000 apps, the Apple App Store has 300,000.

• Android is extremely fragmented and is continuing to do so.

• The App Store for iPad contains over 35,000 applications. The market share of the latest generation of tablets (which debuted in 2011) will be close to nil.

Jobs' aim in blasting the competition on that call was to reassure investors about the upcoming flood of new tablets. Jobs often criticized Google, Samsung, and Microsoft, but he also acknowledged when they did a better job. He was great at detecting his competitors' advantages and devising ways to catch up.

• Jobs doesn't hold back when he says in bullet 1 of his agenda: "We further lock customers into our ecosystem while Google and Microsoft are further along on the technology, but haven't quite figured it out yet tie all of our goods together."

• The plan outlined in bullet point 5 is immediately clear: catch up to Android where we are falling behind (notifications, tethering, and speech), and surpass them (Siri,). It's important to note that Siri frequently let users down and never quite lived up to expectations.

• Regarding MobileMe, see Bullet 6 Jobs admits that when it comes to cloud services like contacts, calendars, and mail, Google is far ahead of Apple.

3. Adapt or perish

Steve Jobs was a visionary businessman. He knew personal computers were the future when he worked on the first Macintosh in the 1980s.

Jobs acknowledged the Post-PC age in his 2010 D8 interview.

Will the tablet replace the laptop, Walt Mossberg questioned Jobs? Jobs' response:

“You know, when we were an agrarian nation, all cars were trucks, because that’s what you needed on the farm. As vehicles started to be used in the urban centers and America started to move into those urban and suburban centers, cars got more popular and innovations like automatic transmission and things that you didn’t care about in a truck as much started to become paramount in cars. And now, maybe 1 out of every 25 vehicles is a truck, where it used to be 100%. PCs are going to be like trucks. They’re still going to be around, still going to have a lot of value, but they’re going to be used by one out of X people.”

Imagine how forward-thinking that was in 2010, especially for the Macintosh creator. You have to be willing to recognize that things were changing and that it was time to start over and focus on the next big thing.

Post-PC is priority number 8 in his 2010 agenda's 2011 Strategy section. Jobs says Apple is the first firm to get here and that Post PC items account about 66% of our income. The iPad outsold the Mac in 6 months, and the Post-PC age means increased mobility (smaller, thinner, lighter). Samsung had just introduced its first tablet, while Apple was working on the iPad 3. (as mentioned in bullet 4).

4. Plan ahead (and different)

Jobs' agenda warns that Apple risks clinging to outmoded paradigms. Clayton Christensen explains in The Innovators Dilemma that huge firms neglect disruptive technologies until they become profitable. Samsung's Galaxy tab, released too late, never caught up to Apple.

Apple faces a similar dilemma with the iPhone, its cash cow for over a decade. It doesn't sell as much because consumers aren't as excited about new iPhone launches and because technology is developing and cell phones may need to be upgraded.

Large companies' established consumer base typically hinders innovation. Clayton Christensen emphasizes that loyal customers from established brands anticipate better versions of current products rather than something altogether fresh and new technologies.

Apple's marketing is smart. Apple's ecosystem is trusted by customers, and its products integrate smoothly. So much so that Apple can afford to be a disruptor by doing something no one has ever done before, something the world's largest corporation shouldn't be the first to try. Apple can test the waters and produce a tremendous innovation tsunami, something few corporations can do.

In March 2011, Jobs appeared at an Apple event. During his address, Steve reminded us about Apple's brand:

“It’s in Apple’s DNA, that technology alone is not enough. That it’s technology married with liberal arts, married with the humanities that yields us the results that make our hearts sink. And nowhere is that more true that in these Post-PC devices.“

More than a decade later, Apple remains one of the most innovative and trailblazing companies in the Post-PC world (industry-disrupting products like Airpods or the Apple Watch came out after that 2011 strategy meeting), and it has reinvented how we use laptops with its M1-powered line of laptops offering unprecedented performance.

A decade after Jobs' death, Apple remains the world's largest firm, and its former CEO had a crucial part in its expansion. If you can do 1% of what Jobs did, you may be 1% as successful.

Not bad.