New ventures are exciting. It's fun to imagine yourself rich, successful, and famous (if that's your thing). How you'll help others and make your family proud. This excitement can pull you forward for years, even when you intuitively realize that the path you're on may not lead to your desired success.

Know when to change course. Switching course can mean pivoting or changing direction.

In this not-so-short blog, I'll describe the journey of building your dream. And how the journey might look when you think you're building your dream, but fall short of that vision. Both can feel similar in the beginning, but there are subtle differences.

Let’s dive in.

How an exciting journey to a dead end looks and feels.

You want to help many people. You're business-minded, creative, and ambitious. You jump into entrepreneurship. You're excited, free, and in control.

I'll use tech as an example because that's what I know best, but this applies to any entrepreneurial endeavor.

So you start learning the basics of your field, say coding/software development. You read books, take courses, and may even join a bootcamp. You start practicing, and the journey begins. Once you reach a certain level of skill (which can take months, usually 12-24), you gain the confidence to speak with others in the field and find common ground. You might attract a co-founder this way with time. You and this person embark on a journey (Tip: the idea you start with is rarely the idea you end with).

Amateur mistake #1: You spend months building a product before speaking to customers.

Building something pulls you forward blindly. You make mistakes, avoid customers, and build with your co-founder or small team in the dark for months, usually 6-12 months.

You're excited when the product launches. We'll be billionaires! The market won't believe it. This excites you and the team. Launch.

….

Nothing happens.

Some people may sign up out of pity, only to never use the product or service again.

You and the team are confused, discouraged and in denial. They don't get what we've built yet. We need to market it better, we need to talk to more investors, someone will understand our vision.

This is a hopeless path, and your denial could last another 6 months. If you're lucky, while talking to consumers and investors (which you should have done from the start), someone who has been there before would pity you and give you an idea to pivot into that can create income.

Suppose you get this idea and pivot your business. Again, you've just pivoted into something limited by what you've already built. It may be a revenue-generating idea, but it's rarely new. Now you're playing catch-up, doing something others are doing but you can do better. (Tip #2: Don't be late.) Your chances of winning are slim, and you'll likely never catch up.

You're finally seeing revenue and feel successful. You can compete, but if you're not a first mover, you won't earn enough over time. You'll get by or work harder than ever to earn what a skilled trade could provide. You didn't go into business to stress out and make $100,000 or $200,000 a year. When you can make the same amount by becoming a great software developer, electrician, etc.

You become stuck. Either your firm continues this way for years until you realize there isn't enough growth to recruit a strong team and remove yourself from day-to-day operations due to competition. Or a catastrophic economic event forces you to admit that what you were building wasn't new and unique and wouldn't get you where you wanted to be.

This realization could take 6-10 years. No kidding.

The good news is, you’ve learned a lot along the way and this information can be used towards your next venture (if you have the energy).

Key Lesson: Don’t build something if you aren’t one of the first in the space building it just for the sake of building something.

-

Let's discuss what it's like to build something that can make your dream come true.

Case 2: Building something the market loves is difficult but rewarding.

It starts with a problem that hasn't been adequately solved for a long time but is now solvable due to technology. Or a new problem due to a change in how things are done.

Let's examine each example.

Example #1: Mass communication. The problem is now solvable due to some technological breakthrough.

Twitter — One of the first web 2 companies that became successful with the rise of smart mobile computing.

People can share their real-time activities via mobile device with friends, family, and strangers. Web 2 and smartphones made it easy and fun.

Example #2: A new problem has emerged due to some change in the way things are conducted.

Zoom- A web-conferencing company that reached massive success due to the movement towards “work from home”, remote/hybrid work forces.

Online web conferencing allows for face-to-face communication.

-

These two examples show how to build a unicorn-type company. It's a mix of solving the right problem at the right time, either through a technological breakthrough that opens up new opportunities or by fundamentally changing how people do things.

Let's find these opportunities.

Start by examining problems, such as how the world has changed and how we can help it adapt. It can also be both. Start team brainstorming. Research technologies, current world-trends, use common sense, and make a list. Then, choose the top 3 that you're most excited about and seem most workable based on your skillsets, values, and passion.

Once you have this list, create the simplest MVP you can and test it with customers. The prototype can be as simple as a picture or diagram of user flow and end-user value. No coding required. Market-test. Twitter's version 1 was simple. It was a web form that asked, "What are you doing?" Then publish it from your phone. A global status update, wherever you are. Currently, this company has a $50 billion market cap.

Here's their MVP screenshot.

Small things grow. Tiny. Simplify.

Remember Frequency and Value when brainstorming. Your product is high frequency (Twitter, Instagram, Snapchat, TikTok) or high value (Airbnb for renting travel accommodations), or both (Gmail).

Once you've identified product ideas that meet the above criteria, they're simple, have a high frequency of use, or provide deep value. You then bring it to market in the simplest, most cost-effective way. You can sell a half-working prototype with imagination and sales skills. You need just enough of a prototype to convey your vision to a user or customer.

With this, you can approach real people. This will do one of three things: give you a green light to continue on your vision as is, show you that there is no opportunity and people won't use it, or point you in a direction that is a blend of what you've come up with and what the customer / user really wants, and you update the prototype and go back to the maze. Repeat until you have enough yeses and conviction to build an MVP.

I recently wrote about getting VCs excited about Marketplace start-ups. SaaS founders became envious!

Understanding how people wire tens of millions is the only Series A hack I recommend.

Few people understand the intellectual process behind investing.

VC is risk management.

Series A-focused VCs must cover two risks.

1. Market risk

You need a large market to cross a threshold beyond which you can build defensibilities. Series A VCs underwrite market risk.

They must see you have reached product-market fit (PMF) in a large total addressable market (TAM).

2. Execution risk

When evaluating your growth engine's blitzscaling ability, execution risk arises.

When investors remove operational uncertainty, they profit.

Series A VCs like businesses with derisked revenue streams. Don't raise unless you have a predictable model, pipeline, and growth.

Please beat these 3 metrics before Series A:

Achieve $1.5m ARR in 12-24 months (Market risk)

Above 100% Net Dollar Retention. (Market danger)

Lead Velocity Rate supporting $10m ARR in 2–4 years (Execution risk)

Hit the 3 and you'll raise $10M in 4 months. Discussing 2/3 may take 6–7 months.

If none, don't bother raising and focus on becoming a capital-efficient business (Topics for other posts).

Let's examine these 3 metrics for the brave ones.

1. Lead Velocity Rate supporting €$10m ARR in 2 to 4 years

Last because it's the least discussed. LVR is the most reliable data when evaluating a growth engine, in my opinion.

SaaS allows you to see the future.

Monthly Sales and Sales Pipelines, two predictive KPIs, have poor data quality. Both are lagging indicators, and minor changes can cause huge modeling differences.

Analysts and Associates will trash your forecasts if they're based only on Monthly Sales and Sales Pipeline.

LVR, defined as month-over-month growth in qualified leads, is rock-solid. There's no lag. You can See The Future if you use Qualified Leads and a consistent formula and process to qualify them.

With this metric in your hand, scaling your company turns into an execution play on which VCs are able to perform calculations risk.

2. Above-100% Net Dollar Retention.

Net Dollar Retention is a better-known SaaS health metric than LVR.

Net Dollar Retention measures a SaaS company's ability to retain and upsell customers. Ask what $1 of net new customer spend will be worth in years n+1, n+2, etc.

Depending on the business model, SaaS businesses can increase their share of customers' wallets by increasing users, selling them more products in SaaS-enabled marketplaces, other add-ons, and renewing them at higher price tiers.

If a SaaS company's annualized Net Dollar Retention is less than 75%, there's a problem with the business.

Slack's ARR chart (below) shows how powerful Net Retention is. Layer chart shows how existing customer revenue grows. Slack's S1 shows 171% Net Dollar Retention for 2017–2019.

Slack S-1

3. $1.5m ARR in the last 12-24 months.

According to Point 9, $0.5m-4m in ARR is needed to raise a $5–12m Series A round.

Target at least what you raised in Pre-Seed/Seed. If you've raised $1.5m since launch, don't raise before $1.5m ARR.

Capital efficiency has returned since Covid19. After raising $2m since inception, it's harder to raise $1m in ARR.

P9's 2016-2021 SaaS Funding Napkin

In summary, less than 1% of companies VCs meet get funded. These metrics can help you win.

If there’s demand for it, I’ll do one on direct-to-consumer.

Cheers!

Are you an idiot at fundraising?

Humans undervalue what they don't grasp. Consider NASCAR. How is that a sport? ask uneducated observers. Circular traffic. Driving near a car's physical limits is different from daily driving. When driving at 200 mph, seemingly simple things like changing gas weight or asphalt temperature might be life-or-death.

Venture investors do something similar in entrepreneurship. Most entrepreneurs don't realize how complex venture finance is.

In my early startup days, I didn't comprehend venture capital's intricacy. I thought VCs were rich folks looking for the next Mark Zuckerberg. I was meant to be a sleek, enthusiastic young entrepreneur who could razzle-dazzle investors.

Finally, one of the VCs I was trying to woo set me straight. He insulted me.

How I learned that I was approaching the wrong investor

I was constructing a consumer-facing, pre-revenue marketplace firm. I looked for investors in my old university's alumni database. My city had one. After some research, I learned he was a partner at a growth-stage, energy-focused VC company with billions under management.

Billions? I thought. Surely he can write a million-dollar cheque. He'd hardly notice.

I emailed the VC about our shared alumni status, explaining that I was building a startup in the area and wanted advice. When he agreed to meet the next week, I prepared my pitch deck.

First error.

The meeting seemed like a funding request. Imagine the awkwardness.

His assistant walked me to the firm's conference room and told me her boss was running late. While waiting, I prepared my pitch. I connected my computer to the projector, queued up my PowerPoint slides, and waited for the VC.

He didn't say hello or apologize when he entered a few minutes later. What are you doing?

Hi! I said, Confused but confident. Dinin Aaron. My startup's pitch.

Who? Suspicious, he replied. Your email says otherwise. You wanted help.

I said, "Isn't that a euphemism for contacting investors?" Fundraising I figured I should pitch you.

As he sat down, he smiled and said, "Put away your computer." You need to study venture capital.

Recognizing the business aspects of venture capital

The VC taught me venture capital in an hour. Young entrepreneur me needed this lesson. I assume you need it, so I'm sharing it.

Most people view venture money from an entrepreneur's perspective, he said. They envision a world where venture capital serves entrepreneurs and startups.

As my VC indicated, VCs perceive their work differently. Venture investors don't serve entrepreneurs. Instead, they run businesses. Their product doesn't look like most products. Instead, the VCs you're proposing have recognized an undervalued market segment. By investing in undervalued companies, they hope to profit. It's their investment thesis.

Your company doesn't fit my investment thesis, the venture capitalist told me. Your pitch won't beat my investing theory. I invest in multimillion-dollar clean energy companies. Asking me to invest in you is like ordering a breakfast burrito at a fancy steakhouse. They could, but why? They don't do that.

Yeah, I’m not a fine steak yet, I laughed, feeling like a fool for pitching a growth-stage VC used to looking at energy businesses with millions in revenues on my pre-revenue, consumer startup.

He stressed that it's not necessary. There are investors targeting your company. Not me. Find investors and pitch them.

Remember this when fundraising. Your investors aren't philanthropists who want to help entrepreneurs realize their company goals. Venture capital is a sophisticated investment strategy, and VC firm managers are industry experts. They're looking for companies that meet their investment criteria. As a young entrepreneur, I didn't grasp this, which is why I struggled to raise money. In retrospect, I probably seemed like an idiot. Hopefully, you won't after reading this.

SpaceX's Starship.

Launched last week.

Four minutes in:

SpaceX will succeed. When it does, its massiveness will matter.

Its payload will revolutionize space economics.

Civilization will shift.

We don't yet understand how this will affect space and Earth culture. Grab it.

The Cost of Space Transportation Has Decreased Exponentially

Space launches have increased dramatically in recent years.

We mostly send items to LEO, the green area below:

SpaceX's reusable rockets can send these things to LEO. Each may launch dozens of payloads into space.

With all these launches, we're sending more than simply things to space. Volume and mass. Since the 1980s, launching a kilogram of payload to LEO has become cheaper:

One kilogram in a large rocket cost over $75,000 in the 1980s. Carrying one astronaut cost nearly $5M! Falcon Heavy's $1,500/kg price is 50 times lower. SpaceX's larger, reusable rockets are amazing.

SpaceX's Starship rocket will continue. It can carry over 100 tons to LEO, 50% more than the current Falcon heavy. Thousands of launches per year. Elon Musk predicts Falcon Heavy's $1,500/kg cost will plummet to $100 in 23 years.

In context:

People underestimate this.

2. The Benefits of Affordable Transportation

Compare Earth's transportation costs:

It's no surprise that the US and Northern Europe are the wealthiest and have the most navigable interior waterways.

So what? since sea transportation is cheaper than land. Inland waterways are even better than sea transportation since weather is less of an issue, currents can be controlled, and rivers serve two banks instead of one for coastal transportation.

In France, because population density follows river systems, rivers are valuable. Cheap transportation brought people and money to rivers, especially their confluences.

How come? Why were humans surrounding rivers?

Imagine selling meat for $10 per kilogram. Transporting one kg one kilometer costs $1. Your margin decreases $1 each kilometer. You can only ship 10 kilometers. For example, you can only trade with four cities:

If instead, your cost of transportation is half, what happens? It costs you $0.5 per km. You now have higher margins with each city you traded with. More importantly, you can reach 20-km markets.

However, 2x distance 4x surface! You can now trade with sixteen cities instead of four! Metcalfe's law states that a network's value increases with its nodes squared. Since now sixteen cities can connect to yours. Each city now has sixteen connections! They get affluent and can afford more meat.

Rivers lower travel costs, connecting many cities, which can trade more, get wealthy, and buy more.

The right network is worth at least an order of magnitude more than the left! The cheaper the transport, the more trade at a lower cost, the more income generated, the more that wealth can be reinvested in better canals, bridges, and roads, and the wealth grows even more.

Throughout history. Rome was established around cheap Mediterranean transit and preoccupied with cutting overland transportation costs with their famous roadways. Communications restricted their empire.

The Egyptians lived around the Nile, the Vikings around the North Sea, early Japan around the Seto Inland Sea, and China started canals in the 5th century BC.

Transportation costs shaped empires.Starship is lowering new-world transit expenses. What's possible?

3. Change Organizations, Change Companies, Change the World

Starship is a conveyor belt to LEO. A new world of opportunity opens up as transportation prices drop 100x in a decade.

Satellite engineers have spent decades shedding milligrams. Weight influenced every decision: pricing structure, volumes to be sent, material selections, power sources, thermal protection, guiding, navigation, and control software. Weight was everything in the mission. To pack as much science into every millimeter, NASA missions had to be miniaturized. Engineers were indoctrinated against mass.

No way.

Starship is not constrained by any space mission, robotic or crewed.

Starship obliterates the mass constraint and every last vestige of cultural baggage it has gouged into the minds of spacecraft designers. A dollar spent on mass optimization no longer buys a dollar saved on launch cost. It buys nothing. It is time to raise the scope of our ambition and think much bigger. — Casey Handmer, Starship is still not understood

A Tesla Roadster in space makes more sense.

It went beyond bad PR. It told the industry: Did you care about every microgram? No more. My rockets are big enough to send a Tesla without noticing. Industry watchers should have noticed.

Most didn’t. Artemis is a global mission to send astronauts to the Moon and build a base. Artemis uses disposable Space Launch System rockets. Instead of sending two or three dinky 10-ton crew habitats over the next decade, Starship might deliver 100x as much cargo and create a base for 1,000 astronauts in a year or two. Why not? Because Artemis remains in a pre-Starship paradigm where each kilogram costs a million dollars and we must aggressively descope our objective.

Space agencies can deliver 100x more payload to space for the same budget with 100x lower costs and 100x higher transportation volumes. How can space economy saturate this new supply?

Before Starship, NASA supplied heavy equipment for Moon base construction. After Starship, Caterpillar and Deere may space-qualify their products with little alterations. Instead than waiting decades for NASA engineers to catch up, we could send people to build a space outpost with John Deere equipment in a few years.

History is littered with the wreckage of former industrial titans that underestimated the impact of new technology and overestimated their ability to adapt: Blockbuster, Motorola, Kodak, Nokia, RIM, Xerox, Yahoo, IBM, Atari, Sears, Hitachi, Polaroid, Toshiba, HP, Palm, Sony, PanAm, Sega, Netscape, Compaq, GM… — Casey Handmer, Starship is still not understood

Everyone saw it coming, but senior management failed to realize that adaption would involve moving beyond their established business practice. Others will if they don't.

4. The Starship Possibilities

It's Starlink.

SpaceX invented affordable cargo space and grasped its implications first. How can we use all this inexpensive cargo nobody knows how to use?

Satellite communications seemed like the best way to capitalize on it. They tried. Starlink, designed by SpaceX, provides fast, dependable Internet worldwide. Beaming information down is often cheaper than cable. Already profitable.

Starlink is one use for all this cheap cargo space. Many more. The longer firms ignore the opportunity, the more SpaceX will acquire.

What are these chances?

Satellite imagery is outdated and lacks detail. We can improve greatly. Synthetic aperture radar can take beautiful shots like this:

Have you ever used Google Maps and thought, "I want to see this in more detail"? What if I could view Earth live? What if we could livestream an infrared image of Earth?

We could launch hundreds of satellites with such mind-blowing visual precision of the Earth that we would dramatically improve the accuracy of our meteorological models; our agriculture; where crime is happening; where poachers are operating in the savannah; climate change; and who is moving military personnel where. Is that useful?

What if we could see Earth in real time? That affects businesses? That changes society?

Alex Mashinsky, CEO of Celsius, speaks at Web Summit 2021 in Lisbon. 

Celsius Network filed for Chapter 11 bankruptcy a month after freezing customer withdrawals, joining other crypto casualties.

Celsius took the step to stabilize its business and restructure for all stakeholders. The filing was done in the Southern District of New York.

The company, which amassed more than $20 billion by offering 18% interest on cryptocurrency deposits, paused withdrawals and other functions in mid-June, citing "extreme market conditions."

As the Fed raises interest rates aggressively, it hurts risk sentiment and squeezes funding costs. Voyager Digital Ltd. filed for Chapter 11 bankruptcy this month, and Three Arrows Capital has called in liquidators.

Celsius called the pause "difficult but necessary." Without the halt, "the acceleration of withdrawals would have allowed certain customers to be paid in full while leaving others to wait for Celsius to harvest value from illiquid or longer-term asset deployment activities," it said.

Celsius declined to comment. CEO Alex Mashinsky said the move will strengthen the company's future.

The company wants to keep operating. It's not requesting permission to allow customer withdrawals right now; Chapter 11 will handle customer claims. The filing estimates assets and liabilities between $1 billion and $10 billion.

Celsius is advised by Kirkland & Ellis, Centerview Partners, and Alvarez & Marsal.

Yield-promises

Celsius promised 18% returns on crypto loans. It lent those coins to institutional investors and participated in decentralized-finance apps.

When TerraUSD (UST) and Luna collapsed in May, Celsius pulled its funds from Terra's Anchor Protocol, which offered 20% returns on UST deposits. Recently, another large holding, staked ETH, or stETH, which is tied to Ether, became illiquid and discounted to Ether.

The lender is one of many crypto companies hurt by risky bets in the bear market. Also, Babel halted withdrawals. Voyager Digital filed for bankruptcy, and crypto hedge fund Three Arrows Capital filed for Chapter 15 bankruptcy.

According to blockchain data and tracker Zapper, Celsius repaid all of its debt in Aave, Compound, and MakerDAO last month.

Celsius charged Symbolic Capital Partners Ltd. 2,000 Ether as collateral for a cash loan on June 13. According to company filings, Symbolic was charged 2,545.25 Ether on June 11.

In July 6 filings, it said it reshuffled its board, appointing two new members and firing others.

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.