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.

Yes, salary details ought to be mentioned in job postings. Recruiters and candidates both agree, so why doesn't it happen?

The short answer is “Unfortunately, it’s not the Recruiter’s decision”. The longer answer is well… A LOT.

Starting in November 2022, NYC employers must include salary ranges in job postings. It should have started in May, but companies balked.

I'm thrilled about salary transparency. This decision will promote fair, inclusive, and equitable hiring practices, and I'm sure other states will follow suit. Good news!

Candidates, recruiters, and ED&I practitioners have advocated for pay transparency for years. Why the opposition?

Let's quickly review why companies have trouble sharing salary bands.

💰 Pay Parity

Many companies and leaders still oppose pay parity. Yes, even in 2022.

💰 Pay Equity

Many companies believe in pay parity and have reviewed their internal processes and systems to ensure equality.

However, Pay Equity affects who gets roles/promotions/salary raises/bonuses and when. Enter the pay gap!

💰Pay Transparency and its impact on Talent Retention

Sharing salary bands with external candidates (and the world) means current employees will have access to that information, which is one of the main reasons companies don't share salary data.

If a company has Pay Parity and Pay Equity issues, they probably have a Pay Transparency policy as well.

Sharing salary information with external candidates without ensuring current employees understand their own salary bands and how promotions/raises are decided could impact talent retention strategies.

This information should help clarify recent conversations.

How I monetized and sold an abandoned side project for $20,000

The Origin Story

I've always wanted to be an entrepreneur but never succeeded. I often had business ideas, made a landing page, and told my buddies. Never got customers.

In April 2021, I decided to try again with a new strategy. I noticed that I had trouble acquiring an initial set of customers, so I wanted to start by acquiring a product that had a small user base that I could grow.

I found a SaaS marketplace called MicroAcquire.com where you could buy and sell SaaS products. I liked Shareit.video, an online Loom-like screen recorder.

Shareit.video didn't generate revenue, but 50 people visited daily to record screencasts.

Purchasing a Failed Side Project

I eventually bought Shareit.video for $12,000 from its owner.

$12,000 was probably too much for a website without revenue or registered users.

I thought time was most important. I could have recreated the website, but it would take months. $12,000 would give me an organized code base and a working product with a few users to monetize.

I considered buying a screen recording website and trying to grow it versus buying a new car or investing in crypto with the $12K.

Buying the website would make me a real entrepreneur, which I wanted more than anything.

Putting down so much money would force me to commit to the project and prevent me from quitting too soon.

A Year of Development

I rebranded the website to be called RecordJoy and worked on it with my cousin for about a year. Within a year, we made $5000 and had 3000 users.

We spent $3500 on ads, hosting, and software to run the business.

AppSumo promoted our $120 Life Time Deal in exchange for 30% of the revenue.

We put RecordJoy on maintenance mode after 6 months because we couldn't find a scalable user acquisition channel.

We improved SEO and redesigned our landing page, but nothing worked.

Despite not being able to grow RecordJoy any further, I had already learned so much from working on the project so I was fine with putting it on maintenance mode. RecordJoy still made $500 a month, which was great lunch money.

Getting Taken Over

One of our customers emailed me asking for some feature requests and I replied that we weren’t going to add any more features in the near future. They asked if we'd sell.

We got on a call with the customer and I asked if he would be interested in buying RecordJoy for 15k. The customer wanted around $8k but would consider it.

Since we were negotiating with one buyer, we put RecordJoy on MicroAcquire to see if there were other offers.

We quickly received 10+ offers. We got 18.5k. There was also about $1000 in AppSumo that we could not withdraw, so we agreed to transfer that over for $600 since about 40% of our sales on AppSumo usually end up being refunded.

Lessons Learned

First, create an acquisition channel

We couldn't discover a scalable acquisition route for RecordJoy. If I had to start another project, I'd develop a robust acquisition channel first. It might be LinkedIn, Medium, or YouTube.

Purchase Power of the Buyer Affects Acquisition Price

Some of the buyers we spoke to were individuals looking to buy side projects, as well as companies looking to launch a new product category. Individual buyers had less budgets than organizations.

Customers of AppSumo vary.

AppSumo customers value lifetime deals and low prices, which may not be a good way to build a business with recurring revenue. Designed for AppSumo users, your product may not connect with other users.

Try to increase acquisition trust

Acquisition often fails. The buyer can go cold feet, cease communicating, or run away with your stuff. Trusting the buyer ensures a smooth asset exchange. First acquisition meeting was unpleasant and price negotiation was tight. In later meetings, we spent the first few minutes trying to get to know the buyer’s motivations and background before jumping into the negotiation, which helped build trust.

Operating expenses can reduce your earnings.

Monitor operating costs. We were really happy when we withdrew the $5000 we made from AppSumo and Stripe until we realized that we had spent $3500 in operating fees. Spend money on software and consultants to help you understand what to build.

Don't overspend on advertising

We invested $1500 on Google Ads but made little money. For a side project, it’s better to focus on organic traffic from SEO rather than paid ads unless you know your ads are going to have a positive ROI.

Startup entrepreneurs assume what risks when pitching?

Last week I signed four NDAs.

Four!

NDA stands for non-disclosure agreement. A legal document given to someone receiving confidential information. By signing, the person pledges not to share the information for a certain time. If they do, they may be in breach of contract and face legal action.

Companies use NDAs to protect trade secrets and confidential internal information from employees and contractors. Appropriate. If you manage a huge, successful firm, you don't want your employees selling their information to your competitors. To be true, business NDAs don't always prevent corporate espionage, but they usually make employees and contractors think twice before sharing.

I understand employee and contractor NDAs, but I wasn't asked to sign one. I counsel entrepreneurs, thus the NDAs I signed last week were from startups that wanted my feedback on their concepts.

I’m not a startup investor. I give startup guidance online. Despite that, four entrepreneurs thought their company ideas were so important they wanted me to sign a generically written legal form they probably acquired from a shady, spam-filled legal templates website before we could chat.

False. One company tried to get me to sign their NDA a few days after our conversation. I gently rejected, but their tenacity encouraged me. I considered sending retroactive NDAs to everyone I've ever talked to about one of my startups in case they establish a successful company based on something I said.

Two of the other three NDAs were from nearly identical companies. Good thing I didn't sign an NDA for the first one, else they may have sued me for talking to the second one as though I control the firms people pitch me.

I wasn't talking to the fourth NDA company. Instead, I received an unsolicited email from someone who wanted comments on their fundraising pitch deck but required me to sign an NDA before sending it.

That's right, before I could read a random Internet stranger's unsolicited pitch deck, I had to sign his NDA, potentially limiting my ability to discuss what was in it.

You should understand. Advisors, mentors, investors, etc. talk to hundreds of businesses each year. They cannot manage all the companies they deal with, thus they cannot risk legal trouble by talking to someone. Well, if I signed NDAs for all the startups I spoke with, half of the 300+ articles I've written on Medium over the past several years could get me sued into the next century because I've undoubtedly addressed topics in my articles that I discussed with them.

The four NDAs I received last week are part of a recent trend of entrepreneurs sending out NDAs before meetings, despite the practical and legal issues. They act like asking someone to sign away their right to talk about all they see and hear in a day is as straightforward as asking for a glass of water.

Given this inflow of NDAs, I wanted to briefly remind entrepreneurs reading this blog about the merits and cons of requesting investors (or others in the startup ecosystem) to sign your NDA.

Benefits of having investors sign your NDA include:

None. Zero. Nothing.

Disadvantages of requesting investor NDAs:

• You'll come off as an amateur who has no idea what it takes to launch a successful firm.

• Investors won't trust you with their money since you appear to be a complete amateur.

• Printing NDAs will be a waste of paper because no genuine entrepreneur will ever sign one.

I apologize for missing any cons. Please leave your remarks.