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.

OpenAI's ChatGPT is an amazing AI-powered chatbot. It can write poetry, fiction, and marketing ideas. I was impressed how well ChatGPT followed the interview, albeit there were a few classics “I’m afraid I can’t do that”.

Full interview below.

Me: I'm your interviewer. We'll discuss crypto. Thanks for being interviewed. What do you think about cryptocurrency?

CGPT: Cryptocurrency is an innovative technology that could transform money and financial activities. It might boost the efficiency and security of financial institutions and let consumers deal directly without banks. Cryptocurrency is still a new and untested technology with many unknowns and hurdles before it can be broadly accepted.

Me: Over 22,000 cryptocurrencies exist, and many have little use. We need how much cryptocurrencies?

CGPT: The appropriate quantity will rely on a multitude of factors, including each cryptocurrency's use cases, adoption rate, and market rivalry. Only a few cryptocurrencies will likely receive widespread adoption and be successful. Many existing cryptocurrencies may not survive since they can't deliver new or valued solutions or compete with more established cryptocurrencies.

Me: Will cryptocurrencies grow? What's next for worldwide adoption?

CGPT: It's hard to forecast the future of cryptocurrencies because the acceptance of new technologies is unpredictable. I believe cryptocurrencies will expand and acquire popularity in the future years. Regulatory ambiguity, security concerns, and scalability issues must be overcome for this to happen. Cryptocurrency companies and organizations must engage together with governments and authorities to address these obstacles and establish a growth-friendly climate. Continued innovation and improvement of the technology, including public education and awareness, will drive acceptance.

Me: Do bear markets affect bitcoin adoption?

CGPT: On the one hand, a bear market might induce uncertainty and hesitation among potential consumers. Bear markets can present opportunities for long-term investors who can acquire cryptocurrencies at cheaper prices and are more likely to hang onto their assets. A bear market can also test the tenacity and power of the underlying technology and community, which can boost long-term confidence and adoption. A bear market might have profound and nuanced effects on bitcoin adoption.

Me: What are some likely outcomes if Bitcoin reaches $100,000 by next year?

CGPT: If Bitcoin saw rapid growth, it would undoubtedly have repercussions for the cryptocurrency market and economy. Possible scenarios include increased public interest and awareness of cryptocurrency, a surge in demand for Bitcoin and other cryptocurrencies, and a growth in the cryptocurrency industry's market value. By shifting their focus and cash to the cryptocurrency market, investors and traders may also affect other asset classes, such as traditional equities and commodities. Governments and financial organizations may focus more on the cryptocurrency sector as they try to grasp Bitcoin's rapid rise.

Me: Will other countries accept crypto as currency?

CGPT: It's hard to say if additional governments will adopt cryptocurrencies as legal tender. Some countries have recognized the promise of bitcoin and are exploring its use, while others are cautious. Adoption of bitcoin as legal cash may vary by country, based on financial infrastructure, rules, and political atmosphere. As cryptocurrency technology evolves and becomes more accepted, more governments may embrace it as legal tender.

Me: Final question: will the recent cryptocurrency hacks affect long term adoption?

CGPT: Hacks might tarnish cryptocurrency's reputation and public faith in its security, the technology underpinning it is continually improving. Many cryptocurrency projects have developed strong security mechanisms to prevent hacks and other dangers, and these safeguards will undoubtedly improve over time. The decentralized nature of cryptocurrency makes it more resistant to attacks. Hacks and other security breaches are a concern, but they may not affect cryptocurrency adoption.

New social media producers' biggest error

The Treadmill of Social Media Content

New creators focus on the wrong platforms.

They post to Instagram, Twitter, TikTok, etc.

They create daily material, but it's never enough for social media algorithms.

Creators recognize they're on a content creation treadmill.

They have to keep publishing content daily just to stay on the algorithm’s good side and avoid losing the audience they’ve built on the platform.

This is exhausting and unsustainable, causing creator burnout.

They focus on short-lived platforms, which is an issue.

Comparing low- and high-return social media platforms

Social media networks are great for reaching new audiences.

Their algorithm is meant to viralize material.

Social media can use you for their aims if you're not careful.

To master social media, focus on the right platforms.

To do this, we must differentiate low-ROI and high-ROI platforms:

Low ROI platforms are ones where content has a short lifespan. High ROI platforms are ones where content has a longer lifespan.

A tweet may be shown for 12 days. If you write an article or blog post, it could get visitors for 23 years.

ROI is drastically different.

New creators have limited time and high learning curves.

Nothing is possible.

First create content for high-return platforms.

ROI for social media platforms

Here are high-return platforms:

1. Your Blog - A single blog article can rank and attract a ton of targeted traffic for a very long time thanks to the power of SEO.

2. YouTube - YouTube has a reputation for showing search results or sidebar recommendations for videos uploaded 23 years ago. A superb video you make may receive views for a number of years.

3. Medium - A platform dedicated to excellent writing is called Medium. When you write an article about a subject that never goes out of style, you're building a digital asset that can drive visitors indefinitely.

These high ROI platforms let you generate content once and get visitors for years.

This contrasts with low ROI platforms:

1. Twitter

2. Instagram

3. TikTok

4. LinkedIn

5. Facebook

The posts you publish on these networks have a 23-day lifetime. Instagram Reels and TikToks are exceptions since viral content can last months.

If you want to make content creation sustainable and enjoyable, you must focus the majority of your efforts on creating high ROI content first. You can then use the magic of repurposing content to publish content to the lower ROI platforms to increase your reach and exposure.

How To Use Your Content Again

So, you’ve decided to focus on the high ROI platforms.

Great!

You've published an article or a YouTube video.

You worked hard on it.

Now you have fresh stuff.

What now?

If you are not repurposing each piece of content for multiple platforms, you are throwing away your time and efforts.

You've created fantastic material, so why not distribute it across platforms?

Repurposing Content Step-by-Step

For me, it's writing a blog article, but you might start with a video or podcast.

The premise is the same regardless of the medium.

Start by creating content for a high ROI platform (YouTube, Blog Post, Medium). Then, repurpose, edit, and repost it to the lower ROI platforms.

Here's how to repurpose pillar material for other platforms:

1. Post the article on your blog.

2. Put your piece on Medium (use the canonical link to point to your blog as the source for SEO)

3. Create a video and upload it to YouTube using the talking points from the article.

4. Rewrite the piece a little, then post it to LinkedIn.

5. Change the article's format to a Thread and share it on Twitter.

6. Find a few quick quotes throughout the article, then use them in tweets or Instagram quote posts.

7. Create a carousel for Instagram and LinkedIn using screenshots from the Twitter Thread.

8. Go through your film and select a few valuable 30-second segments. Share them on LinkedIn, Facebook, Twitter, TikTok, YouTube Shorts, and Instagram Reels.

9. Your video's audio can be taken out and uploaded as a podcast episode.

If you (or your team) achieve all this, you'll have 20-30 pieces of social media content.

If you're just starting, I wouldn't advocate doing all of this at once.

Instead, focus on a few platforms with this method.

You can outsource this as your company expands. (If you'd want to learn more about content repurposing, contact me.)

You may focus on relevant work while someone else grows your social media on autopilot.

You develop high-ROI pillar content, and it's automatically chopped up and posted on social media.

This lets you use social media algorithms without getting sucked in.

Thanks for reading!

No Paid Ads Required

I had an online store. After a year of running the company alongside my 9-to-5, I made enough to resign.

That day was amazing.

This Instagram marketing plan helped the store succeed.

How did I increase my sales to five figures a month without using any paid advertising?

I used customer event marketing.

I'm not sure this term exists. I invented it to describe what I was doing.

Instagram word-of-mouth, fan engagement, and interaction drove sales.

If a customer liked or disliked a product, the buzz would drive attention to the store.

I used customer-based events to increase engagement and store sales.

Success!

Here are the weekly Instagram customer events I coordinated while running my business:

• Be the Buyer Days

• Flash sales

• Mystery boxes

Be the Buyer Days: How do they work?

Be the Buyer Days are exactly that.

You choose a day to share stock selections with social media followers.

This is an easy approach to engaging customers and getting fans enthusiastic about new releases.

First, pick a handful of items you’re considering ordering. I’d usually pick around 3 for Be the Buyer Day.

Then I'd poll the crowd on Instagram to vote on their favorites.

This was before Instagram stories, polls, and all the other cool features Instagram offers today. I think using these tools now would make this event even better.

I'd ask customers their favorite back then.

The growing comments excited customers.

Then I'd declare the winner, acquire the products, and start selling it.

How do flash sales work?

I mostly ran flash sales.

You choose a limited number of itemsdd for a few-hour sale.

We wanted most sales to result in sold-out items.

When an item sells out, it contributes to the sensation of scarcity and can inspire customers to visit your store to buy a comparable product, join your email list, become a fan, etc.

We hoped they'd act quickly.

I'd hold flash deals twice a week, which generated scarcity and boosted sales.

The store had a few thousand Instagram followers when I started flash deals.

Each flash sale item would make $400 to $600.

$400 x 3= $1,200

That's $1,200 on social media!

Twice a week, you'll make roughly $10K a month from Instagram.

$1,200/day x 8 events/month=$9,600

Flash sales did great.

We held weekly flash deals and sent social media and email reminders. That’s about it!

How are mystery boxes put together?

All you do is package a box of store products and sell it as a mystery box on TikTok or retail websites.

A $100 mystery box would cost $30.

You're discounting high-value boxes.

This is a clever approach to get rid of excess inventory and makes customers happy.

It worked!

Be the Buyer Days, flash deals, and mystery boxes helped build my company without paid advertisements.

All companies can use customer event marketing. Involving customers and providing an engaging environment can boost sales.

Try it!