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.

No, it’s not smartphones, overeating, or sugar.

Everyday practices affect brain health. Good brain practices increase memory and cognition.

Bad behaviors increase stress, which destroys brain cells.

Bad behaviors can reverse evolution and diminish the brain. So, avoid these practices for brain health.

1. The silent assassin

Introverts appreciated quarantine.

Before the pandemic, they needed excuses to remain home; thereafter, they had enough.

I am an introvert, and I didn’t hate quarantine. There are billions of people like me who avoid people.

Social relationships are important for brain health. Social anxiety harms your brain.

Antisocial behavior changes brains. It lowers IQ and increases drug abuse risk.

What you can do is as follows:

• Make a daily commitment to engage in conversation with a stranger. Who knows, you might turn out to be your lone mate.

• Get outside for at least 30 minutes each day.

• Shop for food locally rather than online.

• Make a call to a friend you haven't spoken to in a while.

2. Try not to rush things.

People love hustle culture. This economy requires a side gig to save money.

Long hours reduce brain health. A side gig is great until you burn out.

Work ages your wallet and intellect. Overworked brains age faster and lose cognitive function.

Working longer hours can help you make extra money, but it can harm your brain.

Side hustle but don't overwork.

What you can do is as follows:

• Decide what hour you are not permitted to work after.

• Three hours prior to night, turn off your laptop.

• Put down your phone and work.

• Assign due dates to each task.

3. Location is everything!

The environment may cause brain fog. High pollution can cause brain damage.

Air pollution raises Alzheimer's risk. Air pollution causes cognitive and behavioral abnormalities.

Polluted air can trigger early development of incurable brain illnesses, not simply lung harm.

Your city's air quality is uncontrollable. You may take steps to improve air quality.

In Delhi, schools and colleges are closed to protect pupils from polluted air. So I've adapted.

What you can do is as follows:

• To keep your mind healthy and young, make an investment in a high-quality air purifier.

• Enclose your windows during the day.

• Use a N95 mask every day.

4. Don't skip this meal.

Fasting intermittently is trendy. Delaying breakfast to finish fasting is frequent.

Some skip breakfast and have a hefty lunch instead.

Skipping breakfast might affect memory and focus. Skipping breakfast causes low cognition, delayed responsiveness, and irritation.

Breakfast affects mood and productivity.

Intermittent fasting doesn't prevent healthy breakfasts.

What you can do is as follows:

• Try to fast for 14 hours, then break it with a nutritious breakfast.

• So that you can have breakfast in the morning, eat dinner early.

• Make sure your breakfast is heavy in fiber and protein.

5. The quickest way to damage the health of your brain

Brain health requires water. 1% dehydration can reduce cognitive ability by 5%.

Cerebral fog and mental clarity might result from 2% brain dehydration. Dehydration shrinks brain cells.

Dehydration causes midday slumps and unproductivity. Water improves work performance.

Dehydration can harm your brain, so drink water throughout the day.

What you can do is as follows:

• Always keep a water bottle at your desk.

• Enjoy some tasty herbal teas.

• With a big glass of water, begin your day.

• Bring your own water bottle when you travel.

Conclusion

Bad habits can harm brain health. Low cognition reduces focus and productivity.

Unproductive work leads to procrastination, failure, and low self-esteem.

Avoid these harmful habits to optimize brain health and function.

Unlimited everything for $19 monthly!?

Even with pretty words, no one wants to read an ugly email.

• Not Gen Z

• Not Millennials

• Not Gen X

• Not Boomers

I am a minimalist.

I like Mozart. I like avos. I love Apple.

When I hear seamlessly, effortlessly, or Apple's new adverb fluidly, my toes curl.

No email marketing tool gave me that feeling.

As a marketing consultant helping high-growth brands create marketing that doesn't feel like marketing, I've worked with every email marketing platform imaginable, including that naughty monkey and the expensive platform whose sales teams don't stop calling.

Most email marketing platforms are flawed.

1. They are overpriced.

2. They use dreadful templates.

3. They employ a poor visual designer.

4. The user experience there is awful.

5. Too many useless buttons are present. (Similar to the TV remote!)

I may have finally found the perfect email marketing tool. It creates strong flows. It helps me focus on storytelling.

It’s called Flodesk.

It’s effortless. It’s seamless. It’s fluid.

Here’s why it excites me.

Unlimited everything for $19 per month

Sends unlimited. Emails unlimited. Signups unlimited.

Most email platforms penalize success.

Pay for performance?

• $87 for 10k contacts

• $605 for 100K contacts

• $1,300+ for 200K contacts

In the 1990s, this made sense, but not now. It reminds me of when ISPs capped internet usage at 5 GB per month.

Flodesk made unlimited email for a low price a reality. Affordable, attractive email marketing isn't just for big companies.

Flodesk doesn't penalize you for growing your list. Price stays the same as lists grow.

Flodesk plans cost $38 per month, but I'll give you a 30-day trial for $19.

Amazingly strong flows

Foster different people's flows.

Email marketing isn't one-size-fits-all.

Different times require different emails.

People don't open emails because they're irrelevant, in my experience. A colder audience needs a nurturing sequence.

Flodesk automates your email funnels so top-funnel prospects fall in love with your brand and values before mid- and bottom-funnel email flows nudge them to take action.

I wish I could save more custom audience fields to further customize the experience.

Dynamic editor

Easy. Effortless.

Flodesk's editor is Apple-like.

You understand how it works almost instantly.

Like many Apple products, it's intentionally limited. No distractions. You can focus on emotional email writing.

Flodesk's inability to add inline HTML to emails is my biggest issue with larger projects. I wish I could upload HTML emails.

Simple sign-up procedures

Dream up joining.

I like how easy it is to create conversion-focused landing pages. Linkly lets you easily create 5 landing pages and A/B test messaging.

I like that you can use signup forms to ask people what they're interested in so they get relevant emails instead of mindless mass emails nobody opens.

I love how easy it is to embed in-line on a website.

Wonderful designer templates

Beautiful, connecting emails.

Flodesk has calm email templates. My designer's eye felt at rest when I received plain text emails with big impacts.

As a typography nerd, I love Flodesk's handpicked designer fonts. It gives emails a designer feel that is hard to replicate on other platforms without coding and custom font licenses.

Small adjustments can have a big impact

Details matter.

Flodesk remembers your brand colors. Flodesk automatically adds your logo and social handles to emails after signup.

Flodesk uses Zapier. This lets you send emails based on a user's action.

A bad live chat can trigger a series of emails to win back a customer.

Flodesk isn't for everyone.

Flodesk is great for Apple users like me.

The Top 8 Growth Hacking Techniques for Startups

These startups, and how they used growth-hack marketing to flourish, are some of the more ethical ones, while others are less so.

Before the 1970 World Cup began, Puma paid footballer Pele $120,000 to tie his shoes. The cameras naturally focused on Pele and his Pumas, causing people to realize that Puma was the top football brand in the world.

Early workers of Uber canceled over 5,000 taxi orders made on competing applications in an effort to financially hurt any of their rivals.

PayPal developed a bot that advertised cheap goods on eBay, purchased them, and paid for them with PayPal, fooling eBay into believing that customers preferred this payment option. Naturally, Paypal became eBay's primary method of payment.

Anyone renting a space on Craigslist had their emails collected by AirBnB, who then urged them to use their service instead. A one-click interface was also created to list immediately on AirBnB from Craigslist.

To entice potential single people looking for love, Tinder developed hundreds of bogus accounts of attractive people. Additionally, for at least a year, users were "accidentally" linked.

Reddit initially created a huge number of phony accounts and forced them all to communicate with one another. It eventually attracted actual users—the real meaning of "fake it 'til you make it"! Additionally, this gave Reddit control over the tone of voice they wanted for their site, which is still present today.

To disrupt the conferences of their main rival, Salesforce recruited fictitious protestors. The founder then took over all of the event's taxis and gave a 45-minute pitch for his startup. No place to hide!

When a wholesaler required a minimum purchase of 10, Amazon CEO Jeff Bezos wanted a way to purchase only one book from them. A wholesaler would deliver the one book he ordered along with an apology for the other eight books after he discovered a loophole and bought the one book before ordering nine books about lichens. On Amazon, he increased this across all of the users.

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