As a marketer, I joined Twitter on August 31, 2022 to use it.

Growth has been volatile, causing up-and-down engagements. 500 followers in 11 days.

I met amazing content creators, marketers, and people.

Those who use Twitter may know that one-liners win the algorithm, especially if they're funny or humorous, but as a marketer I can't risk posting content that my audience won't like.

I researched, learned some strategies, and A/B tested; some worked, some didn't.

In this article, I share what worked for me so you can do the same.

Thanks for reading!

Let's check my Twitter stats.

• Tweets: how many tweets I sent in the first 28 days.

• A user may be presented with a Tweet in their timeline or in search results.

• In-person visits how many times my Twitter profile was viewed in the first 28 days.

• Mentions: the number of times a tweet has mentioned my name.

• Number of followers: People who were following me

Getting 500 Twitter followers isn't difficult.

Not easy, but doable.

Follow these steps to begin:

Determine your content pillars in step 1.

My formula is Growth = Content + Marketing + Community.

I discuss growth strategies.

My concept for growth is : 1. Content = creating / writing + sharing content in my niche. 2. Marketing = Marketing everything in business + I share my everyday learnings in business, marketing & entrepreneurship. 3. Community = Building community of like minded individuals (Also,I share how to’s) + supporting marketers to build & grow through community building.

Identify content pillars to create content for your audience.

2. Make your profile better

Create a profile picture. Your recognition factor is this.

Professional headshots are worthwhile.

This tool can help you create a free, eye-catching profile pic.

Use a niche-appropriate avatar if you don't want to show your face.

2. Create a bio that converts well mainly because first impressions count.

what you're sharing + why + +social proof what are you making

Be brief and precise. (155 characters)

3. Configure your banner

Banners complement profile pictures.

Use this space to explain what you do and how Twitter followers can benefit.

Canva's Twitter header maker is free.

Birdy can test multiple photo, bio, and banner combinations to optimize your profile.

• Versions A and B of your profile should be completed.

• Find the version that converts the best.

• Use the profile that converts the best.

4. Special handle

If your username/handle is related to your niche, it will help you build authority and presence among your audience. Mine on Twitter is @marketershruti.

5. Participate expertly

Proficiently engage while you'll have no audience at first. Borrow your dream audience for free.

Steps:

• Find a creator who has the audience you want.

• Activate their post notifications and follow them.

• Add a valuable comment first.

6. Create fantastic content

Use:

• Medium (Read articles about your topic.)

• Podcasts (Listen to experts on your topics)

• YouTube (Follow channels in your niche)

Tweet what?

• Listicle ( Hacks, Books, Tools, Podcasts)

• Lessons (Teach your audience how to do 1 thing)

• Inspirational (Inspire people to take action)

Consistent writing?

• You MUST plan ahead and schedule your Tweets.

• Use a scheduling tool that is effective for you; hypefury is mine.

Lastly, consistency is everything that attracts growth. After optimizing your profile, stay active to gain followers, engagements, and clients.

If you found this helpful, please like and comment below.

This resulted to obimy's new audience.

Hi! obimy's official account. Here, we'll teach app developers and marketers. In 2022, our downloads increased dramatically, so we'll share what we learned.

obimy is what we call a ‘senseger’. It's a new method to communicate digitally. Instead of text, obimy users connect through senses and moods. Feeling playful? Flirt with your partner, pat a pal, or dump water on a classmate. Each feeling is an interactive animation with vibration. It's a wordless app. App Store and Google Play have obimy.

We had 20,000 users in 2022. Two to five thousand of them opened the app monthly. Our DAU metric was 500.

We have 6 million users after 6 months. 500,000 individuals use obimy daily. obimy was the top lifestyle app this week in the U.S.

And TikTok helped.

TikTok fuels obimys' growth. It's why our app exploded. How and what did we learn? Our Head of Marketing, Anastasia Avramenko, knows.

our actions prior to TikTok

We wanted to achieve product-market fit through organic expansion. Quora, Reddit, Facebook Groups, Facebook Ads, Google Ads, Apple Search Ads, and social media activity were tested. Nothing worked. Our CPI was sometimes $4, so unit economics didn't work.

We studied our markets and made audience hypotheses. We promoted our goods and studied our audience through social media quizzes. Our target demographic was Americans in long-distance relationships. I designed quizzes like Test the Strength of Your Relationship to better understand the user base. After each quiz, we encouraged users to download the app to enhance their connection and bridge the distance.

We got 1,000 responses for $50. This helped us comprehend the audience's grief and coping strategies (aka our rivals). I based action items on answers given. If you can't embrace a loved one, use obimy.

We also tried Facebook and Google ads. From the start, we knew it wouldn't work.

We were desperate to discover a free way to get more users.

Our journey to TikTok

TikTok is a great venue for emerging creators. It also helped reach people. Before obimy, my TikTok videos garnered 12 million views without sponsored promotion.

We had to act. TikTok was required.

I wasn't a TikTok user before obimy. Initially, I uploaded promotional content. Call-to-actions appear strange next to dancing challenges and my money don't jiggle jiggle. I learned TikTok. Watch TikTok for an hour was on my to-do list. What a dream job!

Our most popular movies presented the app alongside text outlining what it does. We started promoting them in Europe and the U.S. and got a 16% CTR and $1 CPI, an improvement over our previous efforts.

Somehow, we were expanding. So we came up with new hypotheses, calls to action, and content.

Four months passed, yet we saw no organic growth.

Russia attacked Ukraine.

Our app aimed to be helpful. For now, we're focusing on our Ukrainian audience. I posted sloppy TikToks illustrating how obimy can help during shelling or air raids.

In two hours, Kostia sent me our visitor count. Our servers crashed.

Initially, we had several thousand daily users. Over 200,000 users joined obimy in a week. They posted obimy videos on TikTok, drawing additional users. We've also resumed U.S. video promotion.

We gained 2,000,000 new members with less than $100 in ads, primarily in the U.S. and U.K.

TikTok helped.

The figures

We were confident we'd chosen the ideal tool for organic growth.

• Over 45 million people have viewed our own videos plus a ton of user-generated content with the hashtag #obimy.

• About 375 thousand people have liked all of our individual videos.

• The number of downloads and the virality of videos are directly correlated.

Where are we now?

TikTok fuels our organic growth. We post 56 videos every week and pay to promote viral content.

We use UGC and influencers. We worked with Universal Music Italy on Eurovision. They offered to promote us through their million-follower TikTok influencers. We thought their followers would improve our audience, but it didn't matter. Integration didn't help us. Users that share obimy videos with their followers can reach several million views, which affects our download rate.

After the dust settled, we determined our key audience was 13-18-year-olds. They want to express themselves, but it's sometimes difficult. We're searching for methods to better engage with our users. We opened a Discord server to discuss anime and video games and gather app and content feedback.

TikTok helps us test product updates and hypotheses. Example: I once thought we might raise MAU by prompting users to add strangers as friends. Instead of asking our team to construct it, I made a TikTok urging users to share invite URLs. Users share links under every video we upload, embracing people worldwide.

Key lessons

Don't direct-sell. TikTok isn't for Instagram, Facebook, or YouTube promo videos. Conventional advertisements don't fit. Most users will swipe up and watch humorous doggos.

More product videos are better. Finally. So what?

Encourage interaction. Tagging friends in comments or making videos with the app promotes it more than any marketing spend.

Be odd and risqué. A user mistakenly sent a French kiss to their mom in one of our most popular videos.

TikTok helps test hypotheses and build your user base. It also helps develop apps. In our upcoming blog, we'll guide you through obimy's design revisions based on TikTok. Follow us on Twitter, Instagram, and TikTok.

A J.Crew collaboration for fashion label Eveliina Vintage is not a paradox; it is a solution.

Eveliina Vintage's capsule collection debuted yesterday at J.Crew. This J.Crew partnership stopped me in my tracks.

Eveliina Vintage sells vintage goods. Eeva Musacchia founded the shop in Finland in the 1970s. It's recognized for its one-of-a-kind slip dresses from the 1930s and 1940s.

I wondered why a vintage brand would partner with a mass shop. Fast fashion against vintage shopping? Will Eveliina Vintages customers be turned off?

But Eveliina Vintages customers don't care about sustainability. They want Eveliina's Instagram look. Eveliina Vintage collaborated with J.Crew to give customers what they wanted: more Eveliina at a lower price.

Vintage: A Fashion Option That Is Eco-Conscious

Secondhand shopping is a trendy response to quick fashion. J.Crew releases hundreds of styles annually. Waste and environmental damage have been criticized. A pair of jeans requires 1,800 gallons of water. J.Crew's limited-time deals promote more purchases. J.Crew items are likely among those Americans wear 7 times before discarding.

Consumers and designers have emphasized sustainability in recent years. Stella McCartney and Eileen Fisher are popular eco-friendly brands. They've also flocked to ThredUp and similar sites.

Gap, Levis, and Allbirds have listened to consumer requests. They promote recycling, ethical sourcing, and secondhand shopping.

Secondhand shoppers feel good about reusing and recycling clothing that might have ended up in a landfill.

Eco-conscious fashionistas shop vintage. These shoppers enjoy the thrill of the hunt (that limited-edition Chanel bag!) and showing off a unique piece (nobody will have my look!). They also reduce their environmental impact.

Is Eveliina Vintage capitalizing on an aesthetic or is it a sustainable brand?

Eveliina Vintage emphasizes environmental responsibility. Vogue's Amanda Musacchia emphasized sustainability. Amanda, founder Eeva's daughter, is a company leader.

But Eveliina's press message doesn't address sustainability, unlike Instagram. Scarcity and fame rule.

Eveliina Vintages Instagram has see-through dresses and lace-trimmed slip dresses. Celebrities and influencers are often photographed in Eveliina's apparel, which has 53,000+ followers. Vogue appreciates Eveliina's style. Multiple publications discuss Alexa Chung's Eveliina dress.

Eveliina Vintage markets its one-of-a-kind goods. It teases future content, encouraging visitors to return. Scarcity drives demand and raises clothing prices. One dress is $1,600+, but most are $500-$1,000.

The catch: Eveliina can't monetize its expanding popularity due to exorbitant prices and limited quantity. Why?

1. Most people struggle to pay for their clothing. But Eveliina Vintage lacks those more affordable entry-level products, in contrast to other luxury labels that sell accessories or perfume.

2. Many people have trouble fitting into their clothing. The bodies of most women in the past were different from those for which vintage clothing was designed. Each Eveliina dress's specific measurements are mentioned alongside it. Be careful, you can fall in love with an ill-fitting dress.

3. No matter how many people can afford it and fit into it, there is only one item to sell. To get the item before someone else does, those people must be on the Eveliina Vintage website as soon as it becomes available.

A Way for Eveliina Vintage to Make Money (and Expand) with J.Crew Its following

Eveliina Vintages' cooperation with J.Crew makes commercial sense.

This partnership spreads Eveliina's style. Slightly better pricing The $390 outfits have multicolored slips and gauzy cotton gowns. Sizes range from 00 to 24, which is wider than vintage racks.

Eveliina Vintage customers like the combination. Excited comments flood the brand's Instagram launch post. Nobody is mocking the 50-year-old vintage brand's fast-fashion partnership.

Vintage may be a sustainable fashion trend, but that's not why Eveliina's clients love the brand. They only care about the old look.

And that is a tale as old as fashion.

Don't overcomplicate to-do lists. DWTS (Done, Waiting, Top 3, Soon) organizes your to-dos.

How Are You Going to Manage Everything?

Modern America is busy. Work involves meetings. Anytime, Slack communications arrive. Many software solutions offer a @-mention notification capability. Emails.

Work obligations continue. At home, there are friends, family, bills, chores, and fun things.

How are you going to keep track of it all? Enter the todo list. It’s been around forever. It’s likely to stay forever in some way, shape, or form.

Everybody has their own system. You probably modified something from middle school. Post-its? Maybe it’s an app? Maybe both, another system, or none.

I suggest a format that has worked for me in 15 years of professional and personal life.

Try it out and see if it works for you. If not, no worries. You do you! Hopefully though you can learn a thing or two, and I from you too.

It is merely a Google Doc, yes.

It's a giant list. One task per line. Indent subtasks on a new line. Add or move new tasks as needed.

I recommend using Google Docs. It's easy to use and flexible for structuring.

Prioritizing these tasks is key. I organize them using DWTS (Done, Waiting, Top 3, Soon). Chronologically is good because it implicitly provides both a priority (high, medium, low) and an ETA (now, soon, later).

Yes, I recognize the similarities to DWTS (Dancing With The Stars) TV Show. Although I'm not a fan, it's entertaining. The acronym is easy to remember and adds fun to something dull.

What each section contains

Done

All tasks' endpoint. Finish here. Don't worry about it again.

Waiting

You're blocked and can't continue. Blocked tasks usually need someone. Write Person Task so you know who's waiting.

Blocking tasks shouldn't last long. After a while, remind them kindly. If people don't help you out of kindness, they will if you're persistent.

Top 3

Mental focus areas. These can be short- to mid-term goals or recent accomplishments. 2 to 5 is a good number to stay focused.

Top 3 reminds us to prioritize. If they don't fit your Top 3 goals, delay them.

Every 1:1 at work is a project update. Another chance to list your top 3. You should know your Top 3 well and be able to discuss them confidently.

Soon

Here's your short-term to-do list. Rank them from highest to lowest.

I usually subdivide it with empty lines. First is what I have to do today, then week, then month. Subsections can be arranged however you like.

Inventories by Concept

Tasks that aren’t in your short or medium future go into the backlog. 
Eventually you’ll complete these tasks, assign them to someone else, or mark them as “wont’ do” (like done but in another sense).

Backlog tasks don't need to be organized chronologically because their timing and priority may change. Theme-organize them. When planning/strategic, you can choose themes to focus on, so future top 3 topics.

More Tips on Todos

Decide Upon a Morning Goal

Morning routines are universal. Coffee and Wordle. My to-do list is next. Two things:

• As needed, update the to-do list: based on the events of yesterday and any fresh priorities.

• Pick a few jobs to complete today: Pick a few goals that you know you can complete today. Push the remainder below and move them to the top of the Soon section. I typically select a few tasks I am confident I can complete along with one stretch task that might extend into tomorrow.

Finally. By setting and achieving small goals every day, you feel accomplished and make steady progress on medium and long-term goals.

Tech companies call this a daily standup. Everyone shares what they did yesterday, what they're doing today, and any blockers. The name comes from a tradition of holding meetings while standing up to keep them short. Even though it's virtual, everyone still wants a quick meeting.

Your team may or may not need daily standups. Make a daily review a habit with your coffee.

Review Backwards & Forwards on a regular basis

While you're updating your to-do list daily, take time to review it.

Review your Done list. Remember things you're proud of and things that could have gone better. Your Done list can be long. Archive it so your main to-do list isn't overwhelming.

Future-gaze. What you considered important may no longer be. Reorder tasks. Backlog grooming is a workplace term.

Backwards-and-forwards reviews aren't required often. Every 3-6 months is fine. They help you see the forest as often as the trees.

Final Remarks

Keep your list simple. Done, Waiting, Top 3, Soon. These are the necessary sections. If you like, add more subsections; otherwise, keep it simple.

I recommend a morning review. By having clear goals and an action-oriented attitude, you'll be successful.

11 science-backed ways to improve health, happiness, healing, relaxation, and mental clarity.

Vagus nerve is the main parasympathetic nervous system component.

It helps us rest and digest by slowing and stabilizing a resting heart rate, slowing and stabilizing the breath, promoting digestion, improving recovery and healing times, producing saliva, releasing endorphins and hormones like dopamine, oxytocin, and serotonin, and boosting the immune, digestive, and cardiovascular systems.

The vagus nerve sends anti-inflammatory signals to other parts of the body and is located behind the tongue, in the throat, neck, heart, lungs, abdomen, and brainstem.

Vagus means wandering in Latin. So, it's bold.

Here are 11 proven ways to boost health, happiness, and the vagus nerve.

1. Extend

“Yoga stimulates different nerves in your body, especially the vagus nerve that carries information from the brain to most of the body’s major organs, slows everything down and allows self-regulation. It’s the nerve that is associated with the parasympathetic system and emotions like love, joy, and compassion.” — Deepak Chopra

Stretching doesn't require a yoga background.

Listen to your body and ease into simple poses. This connects the mind and body.

If you're new to yoga or don't have access to an in-person class, try Yoga with Adrienne. Over 600 YouTube videos give her plenty of material.

2. Inhale

Because inhaling and exhaling activate the autonomic nervous system, we can breathe to relax.

Exhaling activates the parasympathetic nervous system (rest and digest). One inhales stress, the other exhales it.

So, faster or more intense breathing increases stress. Slower breathing relaxes us.

Breathe slowly, smoothly, and less.

Rhythmic breathing helps me relax.

What to do is as follows:

1. Take 4 smooth, forceless nose breaths.

2. Exhale smoothly and forcefully for 4 seconds

3. Don't pause at the inhale or exhale.

4. Continue for 5 minutes/40 breaths

5. Hold your breath as long as comfortable.

6. Breathe normally.

If four seconds is too long, try breathing in and out for two seconds, or in and out for three seconds, until your breath naturally relaxes. Once calmer, extend your breath.

Any consistent rhythm without force is good. Your heart will follow your lead and become coherent.

3. Chant/Hum

Singing, chanting, or humming activate the vagus nerve through the back of the throat.

Humming emits nitric oxide.

Nitric oxide improves blood circulation, blood flow, heart health, and blood pressure.

Antiviral, antibacterial, anti-inflammatory, antioxidant, and antimicrobial properties kill viruses and bacteria in the nose and throat.

Gargling water stimulates the vagus nerve.

Simple ways to heal, boost energy, and boost mood are often the healthiest. They're free and can be done anywhere.

4. Have more fun

Laughing stimulates the throat muscles, activating the vagus nerve. What's not to like? It releases dopamine.

Take time to enjoy life. Maybe it's a book, podcast, movie, socializing with friends, or laughing yoga.

Follow your bliss, as Joseph Campbell says.

Laugh at yourself

Actually. Really.

Gagging activates vagus nerve-connected muscles. Some doctors use the gag reflex to test the vagus nerve.

Grossness isn't required. While brushing, gag quickly. My girlfriend's brother always does it.

I'm done brushing when I gag, he says.

6. Take in the outdoors

Nature relaxes body and mind. Better if you can walk barefoot.

Earthing is associated with hippies dancing in daisies.

Science now supports hippies.

7. Enter some chilly water.

The diving reflex activates the vagus nerve when exposed to cold water.

The diving reflex involves holding your breath in cold water. Cold showers work best.

Within minutes of being in cold water, parasympathetic nervous system activity, which calms the body, increases.

8. Workout

Exercise increases dopamine, blood circulation, and breathing. So we feel energized, calm, and well-rested.

After resting, the parasympathetic nervous system engages.

It's worth waiting for, though.

9. Play music with brainwaves

Brainwave music harmonizes brainwave activity, boosts productivity and mental clarity, and promotes peace and relaxation by stimulating the vagus nerve.

Simply play a song.

My favorite.

10. Make gentle eyes

Eyes, like breath, often reflect inner state. Sharp, dilated, focused eyes indicate alertness.

Soft, open eyes reflect relaxation and ease. Soft eyes relax the nervous system.

This practice reduces stress, anxiety, and body tension. It's a quick and effective way to enter a calm, peaceful state.

Wild animals can be hunted one minute and graze the next.

Put it into action:

Relax while seated.

Gaze at a distant object

Use peripheral vision while looking straight ahead

Without moving your eyes, look up and down. Connect side spaces to your vision.

Focus on everything as your eyes soften.

Keep breathing

Stay as long as you like

11. Be intimate

We kiss, moan, and breathe deeper during love. We get dopamine, oxytocin, serotonin, and vagus nerve stimulation.

Why not?

To sum up

Here are 11 vagus nerve resets:

1. Stretch

2. Breathe

3. Hum/Chant

4. More humor

5. Amuse yourself

6. Spend time outdoors

7. Leap into chilly water

8. Exercise

9. Play music with brainwaves.

10. Make gentle eyes.

11. Be intimate

If these words have inspired you, try my favorite breathwork technique. Combining breathing, chanting, and brainwave music. Win-win-win :)

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.