His success was unforeseeable.

2014 saw Facebook's $2.3 billion purchase of Oculus VR.

19-year-old Palmer Luckey founded Oculus. He quit journalism school. His parents worried about his college dropout.

Facebook bought Oculus VR in less than 2 years.

Palmer Luckey started Anduril Industries. Palmer has raised $385 million with Anduril.

The Oculus journey began in a trailer

Palmer Luckey, 19, owned the trailer.

Luckey had his trailer customized. The trailer had all six of Luckey's screens. In the trailer's remaining area, Luckey conducted hardware tests.

At 16, he became obsessed with virtual reality. Virtual reality was rare at the time.

Luckey didn't know about VR when he started.

Previously, he liked "portabilizing" mods. Hacking ancient game consoles into handhelds.

In his city, fewer portabilizers actively traded.

Luckey started "ModRetro" for other portabilizers. Luckey was exposed to VR headsets online.

Luckey:

“Man, ModRetro days were the best.”

Palmer Luckey used VR headsets for three years. His design had 50 prototypes.

Luckey used to work at the Long Beach Sailing Center for minimum salary, servicing diesel engines and cleaning boats.

Luckey worked in a USC Institute for Creative Technologies mixed reality lab in July 2011. (ICT).

Luckey cleaned the lab, did reports, and helped other students with VR projects.

Luckey's lab job was dull.

Luckey chose to work in the lab because he wanted to engage with like-minded folks.

By 2012, Luckey had a prototype he hoped to share globally. He made cheaper headsets than others.

Luckey wanted to sell an easy-to-assemble virtual reality kit on Kickstarter.

He realized he needed a corporation to do these sales legally. He started looking for names. "Virtuality," "virtual," and "VR" are all taken.

Hence, Oculus.

If Luckey sold a hundred prototypes, he would be thrilled since it would boost his future possibilities.

John Carmack, legendary game designer

Carmack has liked sci-fi and fantasy since infancy.

Carmack loved imagining intricate gaming worlds.

His interest in programming and computer science grew with age.

He liked graphics. He liked how mismatching 0 and 1 might create new colors and visuals.

Carmack played computer games as a teen. He created Shadowforge in high school.

He founded Id software in 1991. When Carmack created id software, console games were the best-sellers.

Old computer games have weak graphics. John Carmack and id software developed "adaptive tile refresh."

This technique smoothed PC game scrolling. id software launched 3-D, Quake, and Doom using "adaptive tile refresh."

These games made John Carmack a gaming star. Later, he sold Id software to ZeniMax Media.

How Palmer Luckey met Carmack

In 2011, Carmack was thinking a lot about 3-D space and virtual reality.

He was underwhelmed by the greatest HMD on the market. Because of their flimsiness and latency.

His disappointment was partly due to the view (FOV). Best HMD had 40-degree field of view.

Poor. The best VR headset is useless with a 40-degree FOV.

Carmack intended to show the press Doom 3 in VR. He explored VR headsets and internet groups for this reason.

Carmack identified a VR enthusiast in the comments section of "LEEP on the Cheap." "PalmerTech" was the name.

Carmack approached PalmerTech about his prototype. He told Luckey about his VR demos, so he wanted to see his prototype.

Carmack got a Rift prototype. Here's his May 17 tweet.

John Carmack tweeted an evaluation of the Luckey prototype.

Dan Newell, a Valve engineer, and Mick Hocking, a Sony senior director, pre-ordered Oculus Rift prototypes with Carmack's help.

Everyone praised Luckey after Carmack demoed Rift.

Palmer Luckey received a job offer from Sony.

• It was a full-time position at Sony Computer Europe.

• He would run Sony’s R&D lab.

• The salary would be $70k.

Who is Brendan Iribe?

Brendan Iribe started early with Startups. In 2004, he and Mike Antonov founded Scaleform.

Scaleform created high-performance middleware. This package allows 3D Flash games.

In 2011, Iribe sold Scaleform to Autodesk for $36 million.

How Brendan Iribe discovered Palmer Luckey.

Brendan Iribe's friend Laurent Scallie.

Laurent told Iribe about a potential opportunity.

Laurent promised Iribe VR will work this time. Laurent introduced Iribe to Luckey.

Iribe was doubtful after hearing Laurent's statements. He doubted Laurent's VR claims.

But since Laurent took the name John Carmack, Iribe thought he should look at Luckey Innovation. Iribe was hooked on virtual reality after reading Palmer Luckey stories.

He asked Scallie about Palmer Luckey.

Iribe convinced Luckey to start Oculus with him

First meeting between Palmer Luckey and Iribe.

The Iribe team wanted Luckey to feel comfortable.

Iribe sought to convince Luckey that launching a company was easy. Iribe told Luckey anyone could start a business.

Luckey told Iribe's staff he was homeschooled from childhood. Luckey took self-study courses.

Luckey had planned to launch a Kickstarter campaign and sell kits for his prototype. Many companies offered him jobs, nevertheless.

He's considering Sony's offer.

Iribe advised Luckey to stay independent and not join a firm. Iribe asked Luckey how he could raise his child better. No one sees your baby like you do?

Iribe's team pushed Luckey to stay independent and establish a software ecosystem around his device.

After conversing with Iribe, Luckey rejected every job offer and merger option.

Iribe convinced Luckey to provide an SDK for Oculus developers.

After a few months. Brendan Iribe co-founded Oculus with Palmer Luckey. Luckey trusted Iribe and his crew, so he started a corporation with him.

Crowdfunding

Brendan Iribe and Palmer Luckey launched a Kickstarter.

Gabe Newell endorsed Palmer's Kickstarter video.

Gabe Newell wants folks to trust Palmer Luckey since he's doing something fascinating and answering tough questions.

Mark Bolas and David Helgason backed Palmer Luckey's VR Kickstarter video.

Luckey introduced Oculus Rift during the Kickstarter campaign. He introduced virtual reality during press conferences.

Oculus' Kickstarter effort was a success. Palmer Luckey felt he could raise $250,000.

Oculus raised $2.4 million through Kickstarter. Palmer Luckey's virtual reality vision was well-received.

Mark Zuckerberg's Oculus discovery

Brendan Iribe and Palmer Luckey hired the right personnel after a successful Kickstarter campaign.

Oculus needs a lot of money for engineers and hardware. They needed investors' money.

Series A raised $16M.

Next, Andreessen Horowitz partner Brain Cho approached Iribe.

Cho told Iribe that Andreessen Horowitz could invest in Oculus Series B if the company solved motion sickness.

Mark Andreessen was Iribe's dream client.

Marc Andreessen and his partners gave Oculus $75 million.

Andreessen introduced Iribe to Zukerberg. Iribe and Zukerberg discussed the future of games and virtual reality by phone.

Facebook's Oculus demo

Iribe showed Zuckerberg Oculus.

Mark was hooked after using Oculus. The headset impressed him.

The whole Facebook crew who saw the demo said only one thing.

“Holy Crap!”

This surprised them all.

Mark Zuckerberg was impressed by the team's response. Mark Zuckerberg met the Oculus team five days after the demo.

First meeting Palmer Luckey.

Palmer Luckey is one of Mark's biggest supporters and loves Facebook.

Oculus Acquisition

Zuckerberg wanted Oculus.

Brendan Iribe had requested for $4 billion, but Mark wasn't interested.

Facebook bought Oculus for $2.3 billion after months of drama.

After selling his company, how does Palmer view money?

Palmer loves the freedom money gives him. Money frees him from small worries.

Money has allowed him to pursue things he wouldn't have otherwise.

“If I didn’t have money I wouldn’t have a collection of vintage military vehicles…You can have nice hobbies that keep you relaxed when you have money.”

He didn't start Oculus to generate money. His virtual reality passion spanned years.

He didn't have to lie about how virtual reality will transform everything until he needed funding.

The company's success was an unexpected bonus. He was merely passionate about a good cause.

After Oculus' $2.3 billion exit, what changed?

Palmer didn't mind being rich. He did similar things.

After Facebook bought Oculus, he moved to Silicon Valley and lived in a 12-person shared house due to high rents.

Palmer might have afforded a big mansion, but he prefers stability and doing things because he wants to, not because he has to.

“Taco Bell is never tasted so good as when you know you could afford to never eat taco bell again.”

Palmer's leadership shifted.

Palmer changed his leadership after selling Oculus.

When he launched his second company, he couldn't work on his passions.

“When you start a tech company you do it because you want to work on a technology, that is why you are interested in that space in the first place. As the company has grown, he has realized that if he is still doing optical design in the company it’s because he is being negligent about the hiring process.”

Once his startup grows, the founder's responsibilities shift. He must recruit better firm managers.

Recruiting talented people becomes the top priority. The founder must convince others of their influence.

A book that helped me write this:

The History of the Future: Oculus, Facebook, and the Revolution That Swept Virtual Reality — Blake Harris

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

I pitched 10 brands per workday for a total of 200.

How did I do?

It was difficult.

I've never pitched so much.

What did this challenge teach me?

• the superiority of quality over quantity

• When you need help, outsource

• Don't disregard burnout in order to complete a challenge because it exists.

First, pitching brands for brand deals requires quality. Find firms that align with your brand to expose to your audience.

If you associate with any company, you'll lose audience loyalty. I didn't lose sight of that, but I couldn't resist finishing the task.

Outsourcing.

Delegating work to teammates is effective.

I wish I'd done it.

Three people can pitch 200 companies a month significantly faster than one.

One person does research, one to two do outreach, and one to two do follow-up and negotiating.

Simple.

In 2022, I'll outsource everything.

Burnout.

I felt this, so I slowed down at the end of the month.

Thanksgiving week in November was slow.

I was buying and decorating for Christmas. First time putting up outdoor holiday lights was fun.

Much was happening.

I'm not perfect.

I'm being honest.

The Outcomes

Less than 50 brands pitched.

Result: A deal with 3 brands.

I hoped for 4 brands with reaching out to 200 companies, so three with under 50 is wonderful.

That’s a 6% conversion rate!

Whoo-hoo!

I needed 2%.

Here's a screenshot from one of the deals I booked.

These companies fit my company well. Each campaign is different, but I've booked $2,450 in brand work with a couple of pending transactions for December and January.

$2,450 in brand work booked!

How did I do? You tell me.

Is this something you’d try yourself?

A top-down company plan facilitates decision-making and impresses investors.

A startup business plan starts with the product, the target customers, how to reach them, and how to grow the business.

Bottom-up is terrific unless venture investors fund it.

If it can prove how it can exceed $100M in sales, investors will invest. If not, the business may be wonderful, but it's not venture capital-investable.

As a rule, venture investors only fund firms that expect to reach $100M within 5 years.

Investors get nothing until an acquisition or IPO. To make up for 90% of failed investments and still generate 20% annual returns, portfolio successes must exit with a 25x return. A $20M-valued company must be acquired for $500M or more.

This requires $100M in sales (or being on a nearly vertical trajectory to get there). The company has 5 years to attain that milestone and create the requisite ROI.

This motivates venture investors (venture funds and angel investors) to hunt for $100M firms within 5 years. When you pitch investors, you outline how you'll achieve that aim.

I'm wary of pitches after seeing a million hockey sticks predicting $5M to $100M in year 5 that never materialized. Doubtful.

Startups fail because they don't have enough clients, not because they don't produce a great product. That jump from $5M to $100M never happens. The company reaches $5M or $10M, growing at 10% or 20% per year.  That's great, but not enough for a $500 million deal.

Once it becomes clear the company won’t reach orbit, investors write it off as a loss. When a corporation runs out of money, it's shut down or sold in a fire sale. The company can survive if expenses are trimmed to match revenues, but investors lose everything.

When I hear a pitch, I'm not looking for bright income projections but a viable plan to achieve them. Answer these questions in your pitch.

• Is the market size sufficient to generate $100 million in revenue?

• Will the initial beachhead market serve as a springboard to the larger market or as quicksand that hinders progress?

• What marketing plan will bring in $100 million in revenue? Is the market diffuse and will cost millions of dollars in advertising, or is it one, focused market that can be tackled with a team of salespeople?

• Will the business be able to bridge the gap from a small but fervent set of early adopters to a larger user base and avoid lock-in with their current solution?

• Will the team be able to manage a $100 million company with hundreds of people, or will hypergrowth force the organization to collapse into chaos?

• Once the company starts stealing market share from the industry giants, how will it deter copycats?

The requirement to reach $100M may be onerous, but it provides a context for difficult decisions: What should the product be? Where should we concentrate? who should we hire? Every strategic choice must consider how to reach $100M in 5 years.

Focusing on $100M streamlines investor pitches. Instead of explaining everything, focus on how you'll attain $100M.

As an investor, I know I'll lose my money if the startup doesn't reach this milestone, so the revenue prediction is the first thing I look at in a pitch deck.

Reaching the $100M goal needs to be the first thing the entrepreneur thinks about when putting together the business plan, the central story of the pitch, and the criteria for every important decision the company makes.

Taking great financial advice is key to financial success.

A wealthy man told me to INVEST MY MONEY when I was young.

As I entered Starbucks, an older man was leaving. I noticed his watch and expensive-looking shirt, not like the guy in the photo, but one made of fine fabric like vicuna wool, which can only be shorn every two to three years. His Bentley confirmed my suspicions about his wealth.

This guy looked like James Bond, so I asked him how to get rich like him.

"Drug dealer?" he laughed.

Whether he was telling the truth, I'll never know, and I didn't want to be an accessory, but he quickly added, "Kid, invest your money; it will do wonders." He left.

When he told me to invest, he didn't say what. Later, I realized the investment game has so many levels that even if he drew me a blueprint, I wouldn't understand it.

The best advice I received was to invest my earnings. I must decide where to invest.

I'll preface by saying I'm not a financial advisor or Your financial advisor, but I'll share what I've learned from books, links, and sources. The rest is up to you.

Basically:

Invest your Money

Money is money, whether you call it cake, dough, moolah, benjamins, paper, bread, etc.

If you're lucky, you can buy one of the gold shirts in the photo.

Investing your money today means putting it towards anything that could be profitable.

According to the website Investopedia:
“Investing is allocating money to generate income or profit.”

You can invest in a business, real estate, or a skill that will pay off later.

Everyone has different goals and wants at different stages of life, so investing varies.

He was probably a sugar daddy with his Bentley, nice shirt, and Rolex.

In my twenties, I started making "good" money; now, in my forties, with a family and three kids, I'm building a legacy for my grandkids.

“It’s not how much money you make, but how much money you keep, how hard it works for you, and how many generations you keep it for.” — Robert Kiyosaki.

Money isn't evil, but lack of it is.

Financial stress is a major source of problems, according to studies. 

Being broke hurts, especially if you want to provide for your family or do things.

“An investment in knowledge pays the best interest.” — Benjamin Franklin.

Investing in knowledge is invaluable. Before investing, do your homework.

You probably didn't learn about investing when you were young, like I didn't. My parents were in survival mode, making investing difficult.

In my 20s, I worked in banking to better understand money.

So, why invest?

Growth requires investment.

Investing puts money to work and can build wealth. Your money may outpace inflation with smart investing. Compounding and the risk-return tradeoff boost investment growth.

Investing your money means you won't have to work forever — unless you want to.

Two common ways to make money are;

-working hard,

and

-interest or capital gains from investments.

Capital gains can help you invest.

“How many millionaires do you know who have become wealthy by investing in savings accounts? I rest my case.” — Robert G. Allen

If you keep your money in a savings account, you'll earn less than 2% interest at best; the bank makes money by loaning it out.

Savings accounts are a safe bet, but the low-interest rates limit your gains.

Don't skip it. An emergency fund should be in a savings account, not the market.

Other reasons to invest:

Investing can generate regular income.

If you own rental properties, the tenant's rent will add to your cash flow.

Daily, weekly, or monthly rentals (think Airbnb) generate higher returns year-round.

Capital gains are taxed less than earned income if you own dividend-paying or appreciating stock.

Time is on your side

“Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn’t — pays it.” — Albert Einstein

Historical data shows that young investors outperform older investors. So you can use compound interest over decades instead of investing at 45 and having less time to earn.

If I had taken that man's advice and invested in my twenties, I would have made a decent return by my thirties. (Depending on my investments)

So for those who live a YOLO (you only live once) life, investing can't hurt.

Investing increases your knowledge.

Lessons are clearer when you're invested. Each win boosts confidence and draws attention to losses. Losing money prompts you to investigate.

Before investing, I read many financial books, but I didn't understand them until I invested.

Now what?

What do you invest in? Equities, mutual funds, ETFs, retirement accounts, savings, business, real estate, cryptocurrencies, marijuana, insurance, etc.

The key is to start somewhere. Know you don't know everything. You must care.

“A journey of a thousand miles must begin with a single step.” — Lao Tzu.

Start simple because there's so much information. My first investment book was:

Robert Kiyosaki's "Rich Dad, Poor Dad"

This easy-to-read book made me hungry for more. This book is about the money lessons rich parents teach their children, which poor and middle-class parents neglect. The poor and middle-class work for money, while the rich let their assets work for them, says Kiyosaki.

There is so much to learn, but you gotta start somewhere.

More books:

***Wisdom

I hope I'm not suggesting that investing makes everything rosy. Remember three rules:

1. Losing money is possible.

2. Losing money is possible.

3. Losing money is possible.

You can lose money, so be careful.

Read, research, invest.

Golden rules for Investing your money

• Never invest money you can't lose.

• Financial freedom is possible regardless of income.

• "Courage taught me that any sound investment will pay off, no matter how bad a crisis gets." Helu Carlos

• "I'll tell you Wall Street's secret to wealth. When others are afraid, you're greedy. You're afraid when others are greedy. Buffett

• Buy low, sell high, and have an exit strategy.

• Ask experts or wealthy people for advice.

• "With a good understanding of history, we can have a clear vision of the future." Helu Carlos

• "It's not whether you're right or wrong, but how much money you make when you're right." Soros

• "The individual investor should act as an investor, not a speculator." Graham

• "It's different this time" is the most dangerous investment phrase. Templeton

Lastly,

• Avoid quick-money schemes. Building wealth takes years, not months.

Start small and work your way up.

Thanks for reading!

This post is a summary. Read the full article here

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.