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.

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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.

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Sometimes doing everything correctly isn't enough.

In 2020, I could fix my life.

After failing to start a business, I owed $40,000 and had no work.

A $1.2 billion startup on the cusp of going public pulled me up.

Ironically, it was getting ready for an epic fall — with the world watching.

Life sometimes helps. Without a base, even the strongest fall. A corporation that did everything right failed 3 months after going public.

First-row view.

Apple is the creator of Adore.

Out of respect, I've altered the company and employees' names in this account, despite their failure.

Although being a publicly traded company, it may become obvious.

We’ll call it “Adore” — a revolutionary concept in retail shopping.

Two Apple execs established Adore in 2014 with a focus on people-first purchasing.

Jon and Tim:

• The concept for the stylish Apple retail locations you see today was developed by retail expert Jon Swanson, who collaborated closely with Steve Jobs.

• Tim Cruiter is a graphic designer who produced the recognizable bouncing lamp video that appears at the start of every Pixar film.

The dynamic duo realized their vision.

“What if you could combine the convenience of online shopping with the confidence of the conventional brick-and-mortar store experience.”

Adore's mobile store concept combined traditional retail with online shopping.

Adore brought joy to 70+ cities and 4 countries over 7 years, including the US, Canada, and the UK.

Being employed on the ground floor, with world dominance and IPO on the horizon, was exciting.

I started as an Adore Expert.

I delivered cell phones, helped consumers set them up, and sold add-ons.

As the company grew, I became a Virtual Learning Facilitator and trained new employees across North America using Zoom.

In this capacity, I gained corporate insider knowledge. I worked with the creative team and Jon and Tim.

It's where I saw company foundation fissures. Despite appearances, investors were concerned.

The business strategy was ground-breaking.

Even after seeing my employee stocks fall from a home down payment to $0 (when Adore filed for bankruptcy), it's hard to pinpoint what went wrong.

Solid business model, well-executed.

Jon and Tim's chase for public funding ended in glory.

Here’s the business model in a nutshell:

Buying cell phones is cumbersome. You have two choices:

1. Online purchase: not knowing what plan you require or how to operate your device.

2. Enter a store, which can be troublesome and stressful.

Apple, AT&T, and Rogers offered Adore as a free delivery add-on. Customers could:

• Have their phone delivered by UPS or Canada Post in 1-2 weeks.

• Alternately, arrange for a person to visit them the same day (or sometimes even the same hour) to assist them set up their phone and demonstrate how to use it (transferring contacts, switching the SIM card, etc.).

Each Adore Expert brought a van with extra devices and accessories to customers.

Happy customers.

Here’s how Adore and its partners made money:

Adores partners appreciated sending Experts to consumers' homes since they improved customer satisfaction, average sale, and gadget returns.

**Telecom enterprises have low customer satisfaction. The average NPS is 30/100. Adore's global NPS was 80.

Adore made money by:

• a set cost for each delivery

• commission on sold warranties and extras

Consumer product applications seemed infinite.

A proprietary scheduling system (“The Adore App”), allowed for same-day, even same-hour deliveries.

It differentiates Adore.

They treated staff generously by:

• Options on stock

• health advantages

• sales enticements

• high rates per hour

Four-day workweeks were set by experts.

Being hired early felt like joining Uber, Netflix, or Tesla. We hoped the company's stocks would rise.

Exciting times.

I smiled as I greeted more than 1,000 new staff.

I spent a decade in retail before joining Adore. I needed a change.

After a leap of faith, I needed a lifeline. So, I applied for retail sales jobs in the spring of 2019.

The universe typically offers you what you want after you accept what you need. I needed a job to settle my debt and reach $0 again.

And the universe listened.

After being hired as an Adore Expert, I became a Virtual Learning Facilitator. Enough said.

After weeks of economic damage from the pandemic.

This employment let me work from home during the pandemic. It taught me excellent business skills.

I was active in brainstorming, onboarding new personnel, and expanding communication as we grew.

This job gave me vital skills and a regular paycheck during the pandemic.

It wasn’t until January of 2022 that I left on my own accord to try to work for myself again — this time, it’s going much better.

Adore was perfect. We valued:

• Connection

• Discovery

• Empathy

Everything we did centered on compassion, and we held frequent Justice Calls to discuss diversity and work culture.

The last day of onboarding typically ended in tears as employees felt like they'd found a home, as I had.

Like all nice things, the wonderful vibes ended.

First indication of distress

My first day at the workplace was great.

Fun, intuitive, and they wanted creative individuals, not salesman.

While sales were important, the company's vision was more important.

“To deliver joy through life-changing mobile retail experiences.”

Thorough, forward-thinking training. We had a module on intuition. It gave us role ownership.

We were flown cross-country for training, gave feedback, and felt like we made a difference. Multiple contacts responded immediately and enthusiastically.

The atmosphere was genuine.

Making money was secondary, though. Incredible service was a priority.

Jon and Tim answered new hires' questions during Zoom calls during onboarding. CEOs seldom meet new hires this way, but they seemed to enjoy it.

All appeared well.

But in late 2021, things started changing.

Adore's leadership changed after its IPO. From basic values to sales maximization. We lost communication and were forced to fend for ourselves.

Removed the training wheels.

It got tougher to gain instructions from those above me, and new employees told me their roles weren't as advertised.

External money-focused managers were hired.

Instead of creative types, we hired salespeople.

With a new focus on numbers, Adore's uniqueness began to crumble.

Via Zoom, hundreds of workers were let go.

So.

Early in 2022, mass Zoom firings were trending. A CEO firing 900 workers over Zoom went viral.

Adore was special to me, but it became a headline.

30 June 2022, Vice Motherboard published Watch as Adore's CEO Fires Hundreds.

It described a leaked video of Jon Swanson laying off all staff in Canada and the UK.

They called it a “notice of redundancy”.

The corporation couldn't pay its employees.

I loved Adore's underlying ideals, among other things. We called clients Adorers and sold solutions, not add-ons.

But, like anything, a company is only as strong as its weakest link. And obviously, the people-first focus wasn’t making enough money.

There were signs. The expansion was presumably a race against time and money.

Adore finally declared bankruptcy.

Adore declared bankruptcy 3 months after going public. It happened in waves, like any large-scale fall.

• Initial key players to leave were

• Then, communication deteriorated.

• Lastly, the corporate culture disintegrated.

6 months after leaving Adore, I received a letter in the mail from a Law firm — it was about my stocks.

Adore filed Chapter 11. I had to sue to collect my worthless investments.

I hoped those stocks will be valuable someday. Nope. Nope.

Sad, I sighed.

$1.2 billion firm gone.

I left the workplace 3 months before starting a writing business. Despite being mediocre, I'm doing fine.

I got up as Adore fell.

Finally, can we scale kindness?

I trust my gut. Changes at Adore made me leave before it sank.

Adores' unceremonious slide from a top startup to bankruptcy is astonishing to me.

The company did everything perfectly, in my opinion.

• first to market,

• provided excellent service

• paid their staff handsomely.

• was responsible and attentive to criticism

The company wasn't led by an egotistical eccentric. The crew had centuries of cumulative space experience.

I'm optimistic about the future of work culture, but is compassion scalable?

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

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

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

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

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

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

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