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.

It’s only a loss if you sell, right?

Nayib Bukele proclaimed himself “the world’s coolest dictator”.

His jokes aren't clear.

El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.

His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.

He's popular, though.

Bukele won 53% of the vote by fighting violent crime and opposition party corruption.

El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.

They were powerless.

Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.

He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.

Transfer and deposit costs.

This makes logical sense when the president’s theatrics don’t blind you.

El Salvador's Bukele revealed plans to make bitcoin legal tender.

Remittances total $5.9 billion (23%) of the country's expenses.

Anything that reduces costs could boost the economy.

The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).

According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.

In 2021, 36% of El Salvadorans had bank accounts.

Large rural countries like Kenya seem to have resolved their unbanked dilemma.

An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.

Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.

96% of Kenyans utilize M-PESA, which doesn't require a bank account.

The software involves human agents who hang out with cash and a phone.

These people are like ATMs.

You offer them cash to deposit money in your mobile money account or withdraw cash.

In a country with a faulty banking system, cash availability and a safe place to deposit it are important.

William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.

2016 Science paper

Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.

The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.

Currently, El Salvador has 2,301 Bitcoin.

At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.

Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.

It's still falling.

This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.

The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.

You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.

We're opening up Maki through a series of Medium posts. We'll describe what Maki is building and how. We'll explain how we built a SaaS in 100 days. This isn't a step-by-step guide to starting a business, but a product philosophy to help you build quickly.

Focus on end-users.

This may seem obvious, but it's important to talk to users first. When we started thinking about Maki, we interviewed 100 HR directors from SMBs, Next40 scale-ups, and major Enterprises to understand their concerns. We initially thought about the future of employment, but most of their worries centered on Recruitment. We don't have a clear recruiting process, it's time-consuming, we recruit clones, we don't support diversity, etc. And as hiring managers, we couldn't help but agree.

Co-create your product with your end-users.

We went to the drawing board, read as many books as possible (here, here, and here), and when we started getting a sense for a solution, we questioned 100 more operational HR specialists to corroborate the idea and get a feel for our potential answer. This confirmed our direction to help hire more objectively and efficiently.

Back to the drawing board, we designed our first flows and screens. We organized sessions with certain survey respondents to show them our early work and get comments. We got great input that helped us build Maki, and we met some consumers. Obsess about users and execute alongside them.

Don’t shoot for the moon, yet. Make pragmatic choices first.

Once we were convinced, we began building. To launch a SaaS in 100 days, we needed an operating principle that allowed us to accelerate while still providing a reliable, secure, scalable experience. We focused on adding value and outsourced everything else. Example:

Concentrate on adding value. Reuse existing bricks.

When determining which technology to use, we looked at our strengths and the future to see what would last. Node.js for backend, React for frontend, both with typescript. We thought this technique would scale well since it would attract more talent and the surrounding mature ecosystem would help us go quicker.

We explored for ways to bootstrap services while setting down strong foundations that might support millions of users. We built our backend services on NestJS so we could extend into microservices later. Hasura, a GraphQL APIs engine, automates Postgres data exposing through a graphQL layer. MUI's ready-to-use components powered our design-system. We used well-maintained open-source projects to speed up certain tasks.

We outsourced important components of our platform (Auth0 for authentication, Stripe for billing, SendGrid for notifications) because, let's face it, we couldn't do better. We choose to host our complete infrastructure (SQL, Cloud run, Logs, Monitoring) on GCP to simplify our work between numerous providers.

Focus on your business, use existing bricks for the rest. For the curious, we'll shortly publish articles detailing each stage.

Most importantly, empower people and step back.

We couldn't have done this without the incredible people who have supported us from the start. Since Powership is one of our key values, we provided our staff the power to make autonomous decisions from day one. Because we believe our firm is its people, we hired smart builders and let them build.

Nicolas left Spendesk to create scalable interfaces using react-router, react-queries, and MUI. JD joined Swile and chose Hasura as our GraphQL engine. Jérôme chose NestJS to build our backend services. Since then, Justin, Ben, Anas, Yann, Benoit, and others have followed suit.

If you consider your team a collective brain, you should let them make decisions instead of directing them what to do. You'll make mistakes, but you'll go faster and learn faster overall.

Invest in great talent and develop a strong culture from the start. Here's how to establish a SaaS in 100 days.

Open to ideas or acquisitions

Failure is an unavoidable aspect of life, yet many recoil at the word.

I've worked on unrelated startup projects. This is a list of products I developed (often as the tech lead or co-founder) and why they failed to launch.

Chess Bet (Betting)

As a chess player who plays 5 games a day and has an ELO rating of 2100, I tried to design a chess engine to rival stockfish and Houdini.

While constructing my chess engine, my cofounder asked me about building a p2p chess betting app. Chess Bet. There couldn't be a better time.

Two people in different locations could play a staked game. The winner got 90% of the bet and we got 10%. The business strategy was clear, but our mini-launch was unusual.

People started employing the same cheat engines I mentioned, causing user churn and defaming our product.

It was the first programming problem I couldn't solve after building a cheat detection system based on player move strengths and prior games. Chess.com, the most famous online chess software, still suffers from this.

We decided to pivot because we needed an expensive betting license.

We relaunched as Chess MVP after deciding to focus on chess learning. A platform for teachers to create chess puzzles and teach content. Several chess students used our product, but the target market was too tiny.

We chose to quit rather than persevere or pivot.

BodaCare (Insure Tech)

‘BodaBoda’ in Swahili means Motorcycle. My Dad approached me in 2019 (when I was working for a health tech business) about establishing an Insurtech/fintech solution for motorbike riders to pay for insurance using SNPL.

We teamed up with an underwriter to market motorcycle insurance. Once they had enough premiums, they'd get an insurance sticker in the mail. We made it better by splitting the cover in two, making it more reasonable for motorcyclists struggling with lump-sum premiums.

Lack of capital and changing customer behavior forced us to close, with 100 motorcyclists paying 0.5 USD every day. Our unit econ didn't make sense, and CAC and retention capital only dug us deeper.

Circle (Social Networking)

Having learned from both product failures, I began to understand what worked and what didn't. While reading through Instagram, an idea struck me.

Suppose social media weren't virtual.

Imagine meeting someone on your way home. Like-minded person

People were excited about social occasions after covid restrictions were eased. Anything to escape. I just built a university student-popular experiences startup. Again, there couldn't be a better time.

I started the Android app. I launched it on Google Beta and oh my! 200 people joined in two days.

It works by signaling if people are in a given place and allowing users to IM in hopes of meeting up in near real-time. Playstore couldn't deploy the app despite its success in beta for unknown reasons. I appealed unsuccessfully.

My infrastructure quickly lost users because I lacked funding.

In conclusion

This essay contains many failures, some of which might have been avoided and others not, but they were crucial learning points in my startup path.

If you liked any idea, I have the source code on Github.

Happy reading until then!

SaaS spending is 40/40/20. 40% of operating expenses should be R&D, 40% sales and marketing, and 20% G&A. We wanted to see the statistics behind the rules of thumb. Since October 2017, 73 SaaS startups have gone public. Perhaps the rule of thumb should be 30/50/20. The data is below.

30/50/20. R&D accounts for 26% of opex, sales and marketing 48%, and G&A 22%. We think R&D/S&M/G&A should be 30/50/20.

There are outliers. There are exceptions to rules of thumb. Dropbox spent 45% on R&D whereas Zoom spent 13%. Zoom spent 73% on S&M, Dropbox 37%, and Bill.com 28%. Snowflake spent 130% of revenue on S&M, while their EBITDA margin is -192%.

G&A shouldn't stand out. Minimize G&A spending. Priorities should be product development and sales. Cloudflare, Sendgrid, Snowflake, and Palantir spend 36%, 34%, 37%, and 43% on G&A.

Another myth is that COGS is 20% of revenue. Median and averages are 29%.

Where is the profitability? Data-driven operating income calculations were simplified (Revenue COGS R&D S&M G&A). 20 of 73 IPO businesses reported operational income. Median and average operating income margins are -21% and -27%.

As long as you're growing fast, have outstanding retention, and marquee clients, you can burn cash since recurring income that doesn't churn is a valuable annuity.

The data was compelling overall. 30/50/20 is the new 40/40/20 for more established SaaS enterprises, unprofitability is alright as long as your business is expanding, and COGS can be somewhat more than 20% of revenue.