Building A Second Brain helps us remember connections, ideas, inspirations, and insights. Using contemporary technologies and networks increases our intelligence.

This approach makes and preserves concepts. It's a straightforward, practical way to construct a second brain—a remote, centralized digital store for your knowledge and its sources.

How to build ‘The Second Brain’

Have you forgotten any brilliant ideas? What insights have you ignored?

We're pressured to read, listen, and watch informative content. Where did the data go? What happened?

Our brains can store few thoughts at once. Our brains aren't idea banks.

Building a Second Brain helps us remember thoughts, connections, and insights. Using digital technologies and networks expands our minds.

Ten Rules for Creating a Second Brain

1. Creative Stealing

Instead of starting from scratch, integrate other people's ideas with your own.

This way, you won't waste hours starting from scratch and can focus on achieving your goals.

Users of Notion can utilize and customize each other's templates.

2. The Habit of Capture

We must record every idea, concept, or piece of information that catches our attention since our minds are fragile.

When reading a book, listening to a podcast, or engaging in any other topic-related activity, save and use anything that resonates with you.

3. Recycle Your Ideas

Reusing our own ideas across projects might be advantageous since it helps us tie new information to what we already know and avoids us from starting a project with no ideas.

4. Projects Outside of Category

Instead of saving an idea in a folder, group it with documents for a project or activity.

If you want to be more productive, gather suggestions.

5. Burns Slowly

Even if you could finish a job, work, or activity if you focused on it, you shouldn't.

You'll get tired and can't advance many projects. It's easier to divide your routine into daily tasks.

Few hours of daily study is more productive and healthier than entire nights.

6. Begin with a surplus

Instead of starting with a blank sheet when tackling a new subject, utilise previous articles and research.

You may have read or saved related material.

7. Intermediate Packets

A bunch of essay facts.

You can utilize it as a document's section or paragraph for different tasks.

Memorize useful information so you can use it later.

8. You only know what you make

We can see, hear, and read about anything.

What matters is what we do with the information, whether that's summarizing it or writing about it.

9. Make it simpler for yourself in the future.

Create documents or files that your future self can easily understand. Use your own words, mind maps, or explanations.

10. Keep your thoughts flowing.

If you don't employ the knowledge in your second brain, it's useless.

Few people exercise despite knowing its benefits.

Conclusion:

• You may continually move your activities and goals closer to completion by organizing and applying your information in a way that is results-focused.

• Profit from the information economy's explosive growth by turning your specialized knowledge into cash.

• Make up original patterns and linkages between topics.

• You may reduce stress and information overload by appropriately curating and managing your personal information stream.

• Learn how to apply your significant experience and specific knowledge to a new job, business, or profession.

• Without having to adhere to tight, time-consuming constraints, accumulate a body of relevant knowledge and concepts over time.

• Take advantage of all the learning materials that are at your disposal, including podcasts, online courses, webinars, books, and articles.

I started smoking in 2002 at IIT BHU. Most of us thought it was enjoyable at first. I didn't realize the cost later.

In 2005, during my final semester, I lost my father. Suddenly, I felt more accountable for my mother and myself.

I quit before starting my first job in Bangalore. I didn't see any smoking friends in my hometown for 2 months before moving to Bangalore.

For the next 5-6 years, I had no regimen and smoked only when drinking.

Due to personal concerns, I started smoking again after my 2011 marriage. Now smoking was a constant guilty pleasure.

I smoked 3-4 cigarettes a day, but never in front of my family or on weekends. I used to excuse this with pride! First office ritual: smoking. Even with guilt, I couldn't stop this time because of personal concerns.

After 8-9 years, in mid 2019, a personal development program solved all my problems. I felt complete in myself. After this, I just needed one cigarette each day.

The hardest thing was leaving this final cigarette behind, even though I didn't want it.

James Clear's Atomic Habits was published last year. I'd only read 2-3 non-tech books before reading this one in August 2021. I knew everything but couldn't use it.

In April 2022, I realized the compounding effect of a bad habit thanks to my subconscious mind. 1 cigarette per day (excluding weekends) equals 240 = 24 packs per year, which is a lot. No matter how much I did, it felt negative.

Then I applied the 2nd principle of this book, identifying the trigger. I tried to identify all the major triggers of smoking. I found social drinking is one of them & If I am able to control it during that time, I can easily control it in other situations as well. Going further whenever I drank, I was pre-determined to ignore the craving at any cost. Believe me, it was very hard initially but gradually this craving started fading away even with drinks.

I've been smoke-free for 3 months. Now I know a bad habit's effects. After realizing the power of habits, I'm developing other good habits which I ignored all my life.

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

Know why it's important.

The entrepreneur I was meeting with said, "I should be doing crypto, or maybe AI? Aren't those the hot spots? I should look there for a startup idea.”

I shook my head. Yes, they're exciting, but that doesn't mean they're best for you and your business.

“There are different types of entrepreneurs?” he asked.

I said "obviously." Two types, actually. Knowing what type of entrepreneur you are helps you build the right startup.

The two types of businesspeople

The best way for me to describe the two types of entrepreneurs is to start by telling you exactly the kinds of entrepreneurial opportunities I never get excited about: future opportunities.

In the early 1990s, my older brother showed me the World Wide Web and urged me to use it. Unimpressed, I returned to my Super Nintendo.

My roommate tried to get me to join Facebook as a senior in college. I remember thinking, This is dumb. Who'll use it?

In 2011, my best friend tried to convince me to buy bitcoin and I laughed.

Heck, a couple of years ago I had to buy a new car, and I never even considered buying something that didn’t require fossilized dinosaur bones.

I'm no visionary. I don't anticipate the future. I focus on the present.

This tendency makes me a problem-solving entrepreneur. I identify entrepreneurial opportunities by spotting flaws and/or inefficiencies in the world and devising solutions.

There are other ways to find business opportunities. Visionary entrepreneurs also exist. I don't mean visionary in the hyperbolic sense that implies world-changing impact. I mean visionary as an entrepreneur who identifies future technological shifts that will change how people work and live and create new markets.

Problem-solving and visionary entrepreneurs are equally good. But the two approaches to building companies are very different. Knowing the type of entrepreneur you are will help you build a startup that fits your worldview.

What is the distinction?

Let's use some simple hypotheticals to compare problem-solving and visionary entrepreneurship.

Imagine a city office building without nearby restaurants. Those office workers love to eat. Sometimes they'd rather eat out than pack a lunch. As an entrepreneur, you can solve the lack of nearby restaurants. You'd open a restaurant near that office, say a pizza parlor, and get customers because you solved the lack of nearby restaurants. Problem-solving entrepreneurship.

Imagine a new office building in a developing area with no residents or workers. In this scenario, a large office building is coming. The workers will need to eat then. As a visionary entrepreneur, you're excited about the new market and decide to open a pizzeria near the construction to meet demand.

Both possibilities involve the same product. You opened a pizzeria. How you launched that pizza restaurant and what will affect its success are different.

Why is the distinction important?

Let's say you opened a pizzeria near an office. You'll probably get customers. Because people are nearby and demand isn't being met, someone from a nearby building will stop in within the first few days of your pizzeria's grand opening. This makes solving the problem relatively risk-free. You'll get customers unless you're a fool.

The market you're targeting existed before you entered it, so you're not guaranteed success. This means people in that market solved the lack of nearby restaurants. Those office workers are used to bringing their own lunches. Why should your restaurant change their habits? Even when they eat out, they're used to traveling far. They've likely developed pizza preferences.

To be successful with your problem-solving startup, you must convince consumers to change their behavior, which is difficult.

Unlike opening a pizza restaurant near a construction site. Once the building opens, workers won't have many preferences or standardized food-getting practices. Your pizza restaurant can become the incumbent quickly. You'll be the first restaurant in the area, so you'll gain a devoted following that makes your food a routine.

Great, right? It's easier than changing people's behavior. The benefit comes with a risk. Opening a pizza restaurant near a construction site increases future risk. What if builders run out of money? No one moves in? What if the building's occupants are the National Association of Pizza Haters? Then you've opened a pizza restaurant next to pizza haters.

Which kind of businessperson are you?

This isn't to say one type of entrepreneur is better than another. Each type of entrepreneurship requires different skills.

As my simple examples show, a problem-solving entrepreneur must operate in markets with established behaviors and habits. To be successful, you must be able to teach a market a new way of doing things.

Conversely, the challenge of being a visionary entrepreneur is that you have to be good at predicting the future and getting in front of that future before other people.

Both are difficult in different ways. So, smart entrepreneurs don't just chase opportunities. Smart entrepreneurs pursue opportunities that match their skill sets.

No other calendar is simpler, smaller, and reusable year after year. It works and is used here.

Most of us discard and replace our calendars annually. Each month, we move our calendar ahead another page, thus if we need to know which day of the week corresponds to a given day/month combination, we have to calculate it or flip forward/backward to the corresponding month. Questions like:

• What day does this year's American Thanksgiving fall on?

• Which months contain a Friday the thirteenth?

• When is July 4th? What day of the week?

• Alternatively, what day of the week is Christmas?

They're hard to figure out until you switch to the right month or look up all the months.

However, mathematically, the answers to these questions or any question that requires matching the day of the week with the day/month combination in a year are predictable, basic, and easy to work out. If you use this one-page calendar instead of a 12-month calendar, it lasts the whole year and is easy to alter for future years. Let me explain.

The 2023 one-page calendar is above. The days of the month are on the lower left, which works for all months if you know that:

• There are 31 days in January, March, May, July, August, October, and December.

• All of the months of April, June, September, and November have 30 days.

• And depending on the year, February has either 28 days (in non-leap years) or 29 days (in leap years).

If you know this, this calendar makes it easy to match the day/month of the year to the weekday.

Here are some instances. American Thanksgiving is always on the fourth Thursday of November. You'll always know the month and day of the week, but the date—the day in November—changes each year.

On any other calendar, you'd have to flip to November to see when the fourth Thursday is. This one-page calendar only requires:

• pick the month of November in the top-right corner to begin.

• drag your finger down until Thursday appears,

• then turn left and follow the monthly calendar until you reach the fourth Thursday.

It's obvious: 2023 is the 23rd American Thanksgiving. For every month and day-of-the-week combination, start at the month, drag your finger down to the desired day, and then move to the left to see which dates match.

What if you knew the day of the week and the date of the month, but not the month(s)?

A different method using the same one-page calendar gives the answer. Which months have Friday the 13th this year? Just:

• begin on the 13th of the month, the day you know you desire,

• then swipe right with your finger till Friday appears.

• and then work your way up until you can determine which months the specific Friday the 13th falls under.

One Friday the 13th occurred in January 2023, and another will occur in October.

The most typical reason to consult a calendar is when you know the month/day combination but not the day of the week.

Compared to single-month calendars, the one-page calendar excels here. Take July 4th, for instance. Find the weekday here:

• beginning on the left on the fourth of the month, as you are aware,

• also begin with July, the month of the year you are most familiar with, at the upper right,

• you should move your two fingers in the opposite directions till they meet: on a Tuesday in 2023.

That's how you find your selected day/month combination's weekday.

Another example: Christmas. Christmas Day is always December 25th, however unless your conventional calendar is open to December of your particular year, a question like "what day of the week is Christmas?" difficult to answer.

Unlike the one-page calendar!

Remember the left-hand day of the month. Top-right, you see the month. Put two fingers, one from each hand, on the date (25th) and the month (December). Slide the day hand to the right and the month hand downwards until they touch.

They meet on Monday—December 25, 2023.

For 2023, that's fine, but what happens in 2024? Even worse, what if we want to know the day-of-the-week/day/month combo many years from now?

I think the one-page calendar shines here.

Except for the blue months in the upper-right corner of the one-page calendar, everything is the same year after year. The months also change in a consistent fashion.

Each non-leap year has 365 days—one more than a full 52 weeks (which is 364). Since January 1, 2023 began on a Sunday and 2023 has 365 days, we immediately know that December 31, 2023 will conclude on a Sunday (which you can confirm using the one-page calendar) and that January 1, 2024 will begin on a Monday. Then, reorder the months for 2024, taking in mind that February will have 29 days in a leap year.

Please note the differences between 2023 and 2024 month placement. In 2023:

• October and January began on the same day of the week.

• On the following Monday of the week, May began.

• August started on the next day,

• then the next weekday marked the start of February, March, and November, respectively.

• Unlike June, which starts the following weekday,

• While September and December start on the following day of the week,

• Lastly, April and July start one extra day later.

Since 2024 is a leap year, February has 29 days, disrupting the rhythm. Month placements change to:

• The first day of the week in January, April, and July is the same.

• October will begin the following day.

• Possibly starting the next weekday,

• February and August start on the next weekday,

• beginning on the following day of the week between March and November,

• beginning the following weekday in June,

• and commencing one more day of the week after that, September and December.

Due to the 366-day leap year, 2025 will start two days later than 2024 on January 1st.

Now, looking at the 2025 calendar, you can see that the 2023 pattern of which months start on which days is repeated! The sole variation is a shift of three days-of-the-week ahead because 2023 had one more day (365) than 52 full weeks (364), and 2024 had two more days (366). Again,

• On Wednesday this time, January and October begin on the same day of the week.

• Although May begins on Thursday,

• August begins this Friday.

• March, November, and February all begin on a Saturday.

• Beginning on a Sunday in June

• Beginning on Monday are September and December,

• and on Tuesday, April and July begin.

In 2026 and 2027, the year will commence on a Thursday and a Friday, respectively.

We must return to our leap year monthly arrangement in 2028. Yes, January 1, 2028 begins on a Saturday, but February, which begins on a Tuesday three days before January, will have 29 days. Thus:

• Start dates for January, April, and July are all Saturdays.

• Given that October began on Sunday,

• Although May starts on a Monday,

• beginning on a Tuesday in February and August,

• Beginning on a Wednesday in March and November,

• Beginning on Thursday, June

• and Friday marks the start of September and December.

This is great because there are only 14 calendar configurations: one for each of the seven non-leap years where January 1st begins on each of the seven days of the week, and one for each of the seven leap years where it begins on each day of the week.

The 2023 calendar will function in 2034, 2045, 2051, 2062, 2073, 2079, 2090, 2102, 2113, and 2119. Except when passing over a non-leap year that ends in 00, like 2100, the repeat time always extends to 12 years or shortens to an extra 6 years.

• The pattern is repeated in 2025's calendar in 2031, 2042, 2053, 2059, 2070, 2081, 2087, 2098, 2110, and 2121.

• The extra 6-year repeat at the end of the century on the calendar for 2026 will occur in the years 2037, 2043, 2054, 2065, 2071, 2082, 2093, 2099, 2105, and 2122.

• The 2027s calendar repeats in 2038, 2049, 2055, 2066, 2077, 2083, 2094, 2100, 2106, and 2117, almost exactly matching the 2026s pattern.

For leap years, the recurrence pattern is every 28 years when not passing a non-leap year ending in 00, or 12 or 40 years when we do. 2024's calendar repeats in 2052, 2080, 2120, 2148, 2176, and 2216; 2028's in 2056, 2084, 2124, 2152, 2180, and 2220.

Knowing January 1st and whether it's a leap year lets you construct a one-page calendar for any year. Try it—you might find it easier than any other alternative!

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.