Prioritize big-ticket purchases

I've made several spending blunders. I get sick thinking about how much money I spent.

My financial mental model was poor back then.

Stoicism and mindfulness keep me from attaching to those feelings. It still hurts.

Until four or five years ago, I bought a new winter jacket every year.

Ten years ago, I spent twice as much. Now that I have a fantastic, warm winter parka, I don't even consider acquiring another one. No more spending. I'm not looking for jackets either.

Saving time and money by spending well is my thinking paradigm.

The philosophy is expressed in most languages. Cheap is expensive in the Netherlands. This applies beyond shopping.

In this essay, I will offer three examples of how this mental paradigm transformed my financial life.

Publishing books

In 2015, I presented and positioned my first book poorly.

I called the book Huge Life Success and made a funny Canva cover in 30 minutes. This:

That looks nothing like my present books. No logo or style. The book felt amateurish.

The book started bothering me a few weeks after publication. The advice was good, but it didn't appear professional. I studied the book business extensively.

I created a style for all my designs. Branding. Win Your Inner Wars was reissued a year later.

Title, cover, and description changed. Rearranging the chapters improved readability.

Seven years later, the book sells hundreds of copies a month. That taught me a lot.

Rushing to finish a project is enticing. Send it and move forward.

Avoid rushing everything. Relax. Develop your projects. Perform well. Perform the job well.

My first novel was underfunded and underworked. A bad book arrived. I then invested time and money in writing the greatest book I could.

That book still sells.

Traveling

I hate travel. Airports, flights, trains, and lines irritate me.

But, I enjoy traveling to beautiful areas.

I do it strangely. I make up travel rules. I never go to airports in summer. I hate being near airports on holidays. Unworthy.

No vacation packages for me. Those airline packages with a flight, shuttle, and hotel. I've had enough.

I try to avoid crowds and popular spots. July Paris? Nuts and bolts, please. Christmas in NYC? No, please keep me sane.

I fly business class behind. I accept upgrades upon check-in. I prefer driving. I drove from the Netherlands to southern Spain.

Thankfully, no lines. What if travel costs more? Thus? I enjoy it from the start. I start traveling then.

I rarely travel since I'm so difficult. One great excursion beats several average ones.

Personal effectiveness

New apps, tools, and strategies intrigue most productivity professionals.

No.

I researched years ago. I spent years investigating productivity in university.

I bought books, courses, applications, and tools. It was expensive and time-consuming.

Im finished. Productivity no longer costs me time or money. OK. I worked on it once and now follow my strategy.

I avoid new programs and systems. My stuff works. Why change winners?

Spending wisely saves time and money.

Spending wisely means spending once. Many people ignore productivity. It's understudied. No classes.

Some assume reading a few articles or a book is enough. Productivity is personal. You need a personal system.

Time invested is one-time. You can trust your system for life once you find it.

Concentrate on the expensive choices.

Life's short. Saving money quickly is enticing.

Spend less on groceries today. True. That won't fix your finances.

Adopt a lifestyle that makes you affluent over time. Consider major choices.

Are they causing long-term poverty? Are you richer?

Leasing cars comes to mind. The automobile costs a fortune today. The premium could accomplish a million nice things.

Focusing on important decisions makes life easier. Consider your future. You want to improve next year.

Ray Dalio's super-effective roadmap 

Ray Dalio's $22 billion net worth ranks him 36th globally. From 1975 to 2011, he built the world's most successful hedge fund, never losing more than 4% from 1991 to 2020. (and only doing so during 3 calendar years). 

Dalio describes a 5-step process in his best-selling book Principles. It's the playbook he's used to build his hedge fund, beat the markets, and face personal challenges. 

This 5-step system is so valuable and well-explained that I didn't edit or change anything; I only added my own insights in the parts I found most relevant and/or relatable as a young entrepreneur. The system's overview: 

1. Have clear goals 

2. Identify and don’t tolerate problems 

3. Diagnose problems to get at their root causes 

4. Design plans that will get you around those problems 

5. Do what is necessary to push through the plans to get results 

If you follow these 5 steps in a virtuous loop, you'll almost always see results. Repeat the process for each goal you have. 

1. Have clear goals 

a) Prioritize: You can have almost anything, but not everything. 

I started and never launched dozens of projects for 10 years because I was scattered. I opened a t-shirt store, traded algorithms, sold art on Instagram, painted skateboards, and tinkered with electronics. I decided to try blogging for 6 months to see where it took me. Still going after 3 years. 

b) Don’t confuse goals with desires. 

A goal inspires you to act. Unreasonable desires prevent you from achieving your goals. 

c) Reconcile your goals and desires to decide what you want. 

d) Don't confuse success with its trappings. 

e) Never dismiss a goal as unattainable. 

Always one path is best. Your perception of what's possible depends on what you know now. I never thought I'd make money writing online so quickly, and now I see a whole new horizon of business opportunities I didn't know about before. 

f) Expectations create abilities. 

Don't limit your abilities. More you strive, the more you'll achieve. 

g) Flexibility and self-accountability can almost guarantee success. 

Flexible people accept what reality or others teach them. Self-accountability is the ability to recognize your mistakes and be more creative, flexible, and determined. 

h) Handling setbacks well is as important as moving forward. 

Learn when to minimize losses and when to let go and move on. 

2. Don't ignore problems 

a) See painful problems as improvement opportunities. 

Every problem, painful situation, and challenge is an opportunity. Read The Art of Happiness for more. 

b) Don't avoid problems because of harsh realities. 

Recognizing your weaknesses isn't the same as giving in. It's the first step in overcoming them. 

c) Specify your issues. 

There is no "one-size-fits-all" solution. 

d) Don’t mistake a cause of a problem with the real problem. 

"I can't sleep" is a cause, not a problem. "I'm underperforming" could be a problem. 

e) Separate big from small problems. 

You have limited time and energy, so focus on the biggest problems. 

f) Don't ignore a problem. 

Identifying a problem and tolerating it is like not identifying it. 

3. Identify problems' root causes 

a) Decide "what to do" after assessing "what is." 

"A good diagnosis takes 15 to 60 minutes, depending on its accuracy and complexity. [...] Like principles, root causes recur in different situations. 

b) Separate proximate and root causes. 

"You can only solve problems by removing their root causes, and to do that, you must distinguish symptoms from disease." 

c) Knowing someone's (or your own) personality can help you predict their behavior. 

4. Design plans that will get you around the problems 

a) Retrace your steps. 

Analyze your past to determine your future. 

b) Consider your problem a machine's output. 

Consider how to improve your machine. It's a game then. 

c) There are many ways to reach your goals. 

Find a solution. 

d) Visualize who will do what in your plan like a movie script. 

Consider your movie's actors and script's turning points, then act accordingly. The game continues. 

e) Document your plan so others can judge your progress. 

Accountability boosts success. 

f) Know that a good plan doesn't take much time. 

The execution is usually the hardest part, but most people either don't have a plan or keep changing it. Don't drive while building the car. Build it first, because it'll be bumpy. 

5. Do what is necessary to push through the plans to get results 

a) Great planners without execution fail. 

Life is won with more than just planning. Similarly, practice without talent beats talent without practice. 

b) Work ethic is undervalued. 

Hyper-productivity is praised in corporate America, even if it leads nowhere. To get things done, use checklists, fewer emails, and more desk time. 

c) Set clear metrics to ensure plan adherence. 

I've written about the OKR strategy for organizations with multiple people here. If you're on your own, I recommend the Wheel of Life approach. Both systems start with goals and tasks to achieve them. Then start executing on a realistic timeline. 

If you find solutions, weaknesses don't matter. 

Everyone's weak. You, me, Gates, Dalio, even Musk. Nobody will be great at all 5 steps of the system because no one can think in all the ways required. Some are good at analyzing and diagnosing but bad at executing. Some are good planners but poor communicators. Others lack self-discipline. 

Stay humble and ask for help when needed. Nobody has ever succeeded 100% on their own, without anyone else's help. That's the paradox of individual success: teamwork is the only way to get there. 

Most people won't have the skills to execute even the best plan. You can get missing skills in two ways: 

1. Self-taught (time-consuming) 

2. Others' (requires humility) light

On knowing what to do with your life 

“Some people have good mental maps and know what to do on their own. Maybe they learned them or were blessed with common sense. They have more answers than others. Others are more humble and open-minded. […] Open-mindedness and mental maps are most powerful.” — Ray Dalio 

I've always known what I wanted to do, so I'm lucky. I'm almost 30 and have always had trouble executing. Good thing I never stopped experimenting, but I never committed to anything long-term. I jumped between projects. I decided 3 years ago to stick to one project for at least 6 months and haven't looked back. 

Maybe you're good at staying focused and executing, but you don't know what to do. Maybe you have none of these because you haven't found your purpose. Always try new projects and talk to as many people as possible. It will give you inspiration and ideas and set you up for success. 

There is almost always a way to achieve a crazy goal or idea. 

Enjoy the journey, whichever path you take.

These quotes will change your perspective.

I try to reflect on these quotes daily. Reading it in the morning can affect your day, decisions, and priorities. Let's start.

1. Friedrich Nietzsche once said, "He who has a why to live for can bear almost any how."

What's your life goal?

80% of people don't know why they live or what they want to accomplish in life if you ask them randomly.

Even those with answers may not pursue their why. Without a purpose, life can be dull.

Your why can guide you through difficult times.

Create a life goal. Growing may change your goal. Having a purpose in life prevents feeling lost.

2. Seneca said, "He who fears death will never do anything fit for a man in life."

FAILURE STINKS Yes.

This quote is great if you're afraid to try because of failure. What if I'm not made for it? What will they think if I fail?

This wastes most of our lives. Many people prefer not failing over trying something with a better chance of success, according to studies.

Failure stinks in the short term, but it can transform our lives over time.

3. Two men peered through the bars of their cell windows; one saw mud, the other saw stars. — Dale Carnegie

It’s not what you look at that matters; it’s what you see.

The glass-full-or-empty meme is everywhere. It's hard to be positive when facing adversity.

This is a skill. Positive thinking can change our future.

We should stop complaining about our life and how easy success is for others.

Seductive pessimism. Realize this and start from first principles.

4. “Smart people learn from everything and everyone, average people from their experiences, and stupid people already have all the answers.” — Socrates.

Knowing we're ignorant can be helpful.

Every person and situation teaches you something. You can learn from others' experiences so you don't have to. Analyzing your and others' actions and applying what you learn can be beneficial.

Reading (especially non-fiction or biographies) is a good use of time. Walter Issacson wrote Benjamin Franklin's biography. Ben Franklin's early mistakes and successes helped me in some ways.

Knowing everything leads to disaster. Every incident offers lessons.

5. “We must all suffer one of two things: the pain of discipline or the pain of regret or disappointment.“ — James Rohn

My favorite Jim Rohn quote.

Exercise hurts. Healthy eating can be painful. But they're needed to get in shape. Avoiding pain can ruin our lives.

Always choose progress over hopelessness. Myth: overnight success Everyone who has mastered a craft knows that mastery comes from overcoming laziness.

Turn off your inner critic and start working. Try Can't Hurt Me by David Goggins.

6. “A champion is defined not by their wins, but by how they can recover when they fail.“ — Serena Williams

Have you heard of Traf-o-Data?

Gates and Allen founded Traf-O-Data. After some success, it failed. Traf-o-Data's failure led to Microsoft.

Allen said Traf-O-Data's setback was important for Microsoft's first product a few years later. Traf-O-Data was a business failure, but it helped them understand microprocessors, he wrote in 2017.

“The obstacle in the path becomes the path. Never forget, within every obstacle is an opportunity to improve our condition.” — Ryan Holiday.

Bonus Quotes

More helpful quotes:

“Those who cannot change their minds cannot change anything.” — George Bernard Shaw.

“Do something every day that you don’t want to do; this is the golden rule for acquiring the habit of doing your duty without pain.” — Mark Twain.

“Never give up on a dream just because of the time it will take to accomplish it. The time will pass anyway.” — Earl Nightingale.

“A life spent making mistakes is not only more honorable, but more useful than a life spent doing nothing.” — George Bernard Shaw.

“We don’t stop playing because we grow old; we grow old because we stop playing.” — George Bernard Shaw.

Conclusion

Words are powerful. Utilize it. Reading these inspirational quotes will help you.

Email, landing pages, and digital content

Today's most significant skill:

Copywriting.

Unfortunately, most people don't know how to write successful copy because they weren't taught in school.

I've been obsessed with copywriting for two years. I've read 15 books, completed 3 courses, and studied internet's best digital entrepreneurs.

Here are 8 copywriting frameworks that educate more than a four-year degree.

1. Feature — Advantage — Benefit (F.A.B)

This is the most basic copywriting foundation. Email marketing, landing page copy, and digital video ads can use it.

F.A.B says:

• How it works (feature)

• which is helpful (advantage)

• What's at stake (benefit)

The Hustle uses this framework on their landing page to convince people to sign up:

2. P. A. S. T. O. R.

This framework is for longer-form copywriting. PASTOR uses stories to engage with prospects. It explains why people should buy this offer.

PASTOR means:

• Problem

• Amplify

• Story

• Testimonial

• Offer

• Response

Dan Koe's landing page is a great example. It shows PASTOR frame-by-frame.

3. Before — After — Bridge

Before-after-bridge is a copywriting framework that draws attention and shows value quickly.

This framework highlights:

• where you are

• where you want to be

• how to get there

Works great for: Email threads/landing pages

Zain Kahn utilizes this framework to write viral threads.

4. Q.U.E.S.T

QUEST is about empathetic writing. You know their issues, obstacles, and headaches. This allows coverups.

QUEST:

• Qualifies

• Understands

• Educates

• Stimulates

• Transitions

Tom Hirst's landing page uses the QUEST framework.

5. The 4P’s model

The 4P’s approach pushes your prospect to action. It educates and persuades quickly.

4Ps:

• The problem the visitor is dealing with

• The promise that will help them

• The proof the promise works

• A push towards action

Mark Manson is a bestselling author, digital creator, and pop-philosopher. He's also a great copywriter, and his membership offer uses the 4P’s framework.

6. Problem — Agitate — Solution (P.A.S)

Up-and-coming marketers should understand problem-agitate-solution copywriting. Once you understand one structure, others are easier. It drives passion and presents a clear solution.

PAS outlines:

• The issue the visitor is having

• It then intensifies this issue through emotion.

• finally offers an answer to that issue (the offer)

The customer's story loops. Nicolas Cole and Dickie Bush use PAS to promote Ship 30 for 30.

7. Star — Story — Solution (S.S.S)

PASTOR + PAS = star-solution-story. Like PAS, it employs stories to persuade.

S.S.S. is effective storytelling:

• Star: (Person had a problem)

• Story: (until they had a breakthrough)

• Solution: (That created a transformation)

Ali Abdaal is a YouTuber with a great S.S.S copy.

8. Attention — Interest — Desire — Action

AIDA is another classic. This copywriting framework is great for fast-paced environments (think all digital content on Linkedin, Twitter, Medium, etc.).

It works with:

• Page landings

• writing on thread

• Email

It's a good structure since it's concise, attention-grabbing, and action-oriented.

Shane Martin, Twitter's creator, uses this approach to create viral content.

TL;DR

8 copywriting frameworks that teach marketing better than a four-year degree

• Feature-advantage-benefit

• Before-after-bridge

• Star-story-solution

• P.A.S.T.O.R

• Q.U.E.S.T

• A.I.D.A

• P.A.S

• 4P’s

Peter Thiel studied philosophy as an undergraduate.

Peter Thiel has $7.36 billion.

Peter is a world-ranked chess player, has a legal degree, and has written profitable novels.

In 1999, he co-founded PayPal with Max Levchin, which merged with X.com.

Peter Thiel made $55 million after selling the company to eBay for $1.5 billion in 2002.

You may be wondering…

How did Peter turn $55 million into his now multi-billion dollar net worth?

One amazing investment?

Facebook.

Thiel was Facebook's first external investor. He bought 10% of the company for $500,000 in 2004.

This investment returned 159% annually, 200x in 8 years.

By 2012, Thiel sold almost all his Facebook shares, becoming a billionaire.

What was the investment thesis of Peter?

This investment appeared ridiculous. Facebook was an innovative startup.

Thiel's $500,000 contribution transformed Facebook.

Harvard students have access to Facebook's 8 features and 1 photo per profile.

How did Peter determine that this would be a wise investment, then?

Facebook is a mimetic desire machine.

Social media's popularity is odd. Why peek at strangers' images on a computer?

Peter Thiel studied under French thinker Rene Girard at Stanford.

Mimetic Desire explains social media's success.

Mimetic Desire is the idea that humans desire things simply because other people do.

If nobody wanted it, would you?

Would you desire a family, a luxury car, or expensive clothes if no one else did? Girard says no.

People we admire affect our aspirations because we're social animals. Every person has a role model.

Our nonreligious culture implies role models are increasingly other humans, not God.

The idea explains why social media influencers are so powerful.

Why would Andrew Tate or Kim Kardashian matter if people weren't mimetic?

Humanity is fundamentally motivated by social comparison.

Facebook takes advantage of this need for social comparison, and puts it on a global scale.

It aggregates photographs and updates from millions of individuals.

Facebook mobile allows 24/7 social comparison.

Thiel studied mimetic desire with Girard and realized Facebook exploits the urge for social comparison to gain money.

Social media is more significant and influential than ever, despite Facebook's decline.

Thiel and Girard show that applied philosophy (particularly in business) can be immensely profitable.

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.