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.

Ethereum's blockchain rules NFTs. Many consider it the one-stop shop for NFTs, and it's become the most talked-about and trafficked blockchain in existence.

Other blockchains are becoming popular in NFTs. Crypto-artists and NFT enthusiasts have sought new places to mint and trade NFTs due to Ethereum's high transaction costs and environmental impact.

When choosing a blockchain to mint on, there are several factors to consider. Size, creator costs, consumer spending habits, security, and community input are important. We've created a high-level summary of blockchains for NFTs to help clarify the fast-paced world of web3 tech.

Ethereum

Ethereum currently has the most NFTs. It's decentralized and provides financial and legal services without intermediaries. It houses popular NFT marketplaces (OpenSea), projects (CryptoPunks and the Bored Ape Yacht Club), and artists (Pak and Beeple).

It's also expensive and energy-intensive. This is because Ethereum works using a Proof-of-Work (PoW) mechanism. PoW requires computers to solve puzzles to add blocks and transactions to the blockchain. Solving these puzzles requires a lot of computer power, resulting in astronomical energy loss.

You should consider this blockchain first due to its popularity, security, decentralization, and ease of use.

Solana

Solana is a fast programmable blockchain. Its proof-of-history and proof-of-stake (PoS) consensus mechanisms eliminate complex puzzles. Reduced validation times and fees result.

PoS users stake their cryptocurrency to become a block validator. Validators get SOL. This encourages and rewards users to become stakers. PoH works with PoS to cryptographically verify time between events. Solana blockchain ensures transactions are in order and found by the correct leader (validator).

Solana's PoS and PoH mechanisms keep transaction fees and times low. Solana isn't as popular as Ethereum, so there are fewer NFT marketplaces and blockchain traders.

Tezos

Tezos is a greener blockchain. Tezos rose in 2021. Hic et Nunc was hailed as an economic alternative to Ethereum-centric marketplaces until Nov. 14, 2021.

Similar to Solana, Tezos uses a PoS consensus mechanism and only a PoS mechanism to reduce computational work. This blockchain uses two million times less energy than Ethereum. It's cheaper than Ethereum (but does cost more than Solana).

Tezos is a good place to start minting NFTs in bulk. Objkt is the largest Tezos marketplace.

Flow

Flow is a high-performance blockchain for NFTs, games, and decentralized apps (dApps). Flow is built with scalability in mind, so billions of people could interact with NFTs on the blockchain.

Flow became the NBA's blockchain partner in 2019. Flow, a product of Dapper labs (the team behind CryptoKitties), launched and hosts NBA Top Shot, making the blockchain integral to the popularity of non-fungible tokens.

Flow uses PoS to verify transactions, like Tezos. Developers are working on a model to handle 10,000 transactions per second on the blockchain. Low transaction fees.

Flow NFTs are tradeable on Blocktobay, OpenSea, Rarible, Foundation, and other platforms. NBA, NFL, UFC, and others have launched NFT marketplaces on Flow. Flow isn't as popular as Ethereum, resulting in fewer NFT marketplaces and blockchain traders.

Asset Exchange (WAX)

WAX is king of virtual collectibles. WAX is popular for digitalized versions of legacy collectibles like trading cards, figurines, memorabilia, etc.

Wax uses a PoS mechanism, but also creates carbon offset NFTs and partners with Climate Care. Like Flow, WAX transaction fees are low, and network fees are redistributed to the WAX community as an incentive to collectors.

WAX marketplaces host Topps, NASCAR, Hot Wheels, and cult classic film franchises like Godzilla, The Princess Bride, and Spiderman.

Binance Smart Chain

BSC is another good option for balancing fees and performance. High-speed transactions and low fees hurt decentralization. BSC is most centralized.

Binance Smart Chain uses Proof of Staked Authority (PoSA) to support a short block time and low fees. The 21 validators needed to run the exchange switch every 24 hours. 11 of the 21 validators are directly connected to the Binance Crypto Exchange, according to reports.

While many in the crypto and NFT ecosystems dislike centralization, the BSC NFT market picked up speed in 2021. OpenBiSea, AirNFTs, JuggerWorld, and others are gaining popularity despite not having as robust an ecosystem as Ethereum.

University entrepreneurship is like a Willy Wonka Factory of ideas. Classes, roommates, discussions, and the cafeteria all inspire new ideas. I've seen people establish a business without knowing its roots.

Chicken or egg? On my mind: I've asked university founders around the world whether the problem or solution came first.

The Problem

One African team I met started with the “instant noodles” problem in their academic ecosystem. Many of us have had money issues in college, which may have led to poor nutritional choices.

Many university students in a war-torn country ate quick noodles or pasta for dinner.

Noodles required heat, water, and preparation in the boarding house. Unreliable power from one hot plate per blue moon. What's healthier, easier, and tastier than sodium-filled instant pots?

BOOM. They were fixing that. East African kids need affordable, nutritious food.

This is a real difficulty the founders faced every day with hundreds of comrades.

This sparked their serendipitous entrepreneurial journey and became their business's cornerstone.

The Solution

I asked a UK team about their company idea. They said the solution fascinated them.

The crew was fiddling with social media algorithms. Why are some people more popular? They were studying platforms and social networks, which offered a way for them.

Solving a problem? Yes. Long nights of university research lead them to it. Is this like world hunger? Social media influencers confront this difficulty regularly.

It made me ponder something. Is there a correct response?

In my heart, yes, but in my head…maybe?

I believe you should lead with empathy and embrace the problem, not the solution. Big or small, businesses should solve problems. This should be your focus. This is especially true when building a social company with an audience in mind.

Philosophically, invention and innovation are occasionally accidental. Also not penalized. Think about bugs and the creation of Velcro, or the inception of Teflon. They tackle difficulties we overlook. The route to the problem may look different, but there is a path there.

There's no golden ticket to the Chicken-Egg debate, but I'll keep looking this summer.

No wonder brands use Twitter to reach their audience. 53% of Twitter users buy new products first. 

Twitter growth does more than make your brand look popular. It helps clients trust your business. It boosts your industry standing. It shows clients, prospects, and even competitors you mean business.

How can you naturally gain Twitter followers?

• Share useful information

• Post visual content

• Tweet consistently

• Socialize

• Spread your @name everywhere.

• Use existing customers

• Promote followers

Share useful information

Twitter users join conversations and consume material. To build your followers, make sure your material appeals to them and gives value, whether it's sales, product lessons, or current events.

Use Twitter Analytics to learn what your audience likes.

Explore popular topics by utilizing relevant keywords and hashtags. Check out this post on how to use Twitter trends.

Post visual content

97% of Twitter users focus on images, so incorporating media can help your Tweets stand out. Visuals and videos make content more engaging and memorable.

Tweet often

Your audience should expect regular content updates. Plan your ideas and tweet during crucial seasons and events with a content calendar.

Socialize

Twitter connects people. Do more than tweet. Follow industry leaders. Retweet influencers, engage with thought leaders, and reply to mentions and customers to boost engagement.

Micro-influencers can promote your brand or items. They can help you gain new audiences' trust.

Spread your @name everywhere.

Maximize brand exposure. Add a follow button on your website, link to it in your email signature and newsletters, and promote it on business cards or menus.

Use existing customers

Emails can be used to find existing Twitter clients. Upload your email contacts and follow your customers on Twitter to start a dialogue.

Promote followers

Run a followers campaign to boost your organic growth. Followers campaigns promote your account to a particular demographic, and you only pay when someone follows you.

Consider short campaigns to enhance momentum or an always-on campaign to gain new followers.

Increasing your brand's Twitter followers takes effort and experimentation, but the payback is huge.

👋 Follow me on twitter

Ethereum is the second-largest cryptocurrency. The Merge, a mid-September event that will convert Ethereum's consensus process from proof-of-work to proof-of-stake if all goes according to plan, will be a game changer.

Why is Ethereum ditching proof-of-work? Because it can. We're talking about a fully functioning, open-source ecosystem with a capacity for evolution that other cryptocurrencies lack, a change that would allow it to scale up its performance from 15 transactions per second to 100,000 as its blockchain is used for more and more things. It would reduce its energy consumption by 99.95%. Vitalik Buterin, the system's founder, would play a less active role due to decentralization, and miners, who validated transactions through proof of work, would be far less important.

Why has this conversion taken so long and been so cautious? Because it involves modifying a core process while it's running to boost its performance. It requires running the new mechanism in test chains on an ever-increasing scale, assessing participant reactions, and checking for issues or restrictions. The last big test was in early June and was successful. All that's left is to converge the mechanism with the Ethereum blockchain to conclude the switch.

What's stopping Bitcoin, the leader in market capitalization and the cryptocurrency that began blockchain's appeal, from doing the same? Satoshi Nakamoto, whoever he or she is, departed from public life long ago, therefore there's no community leadership. Changing it takes a level of consensus that is impossible to achieve without strong leadership, which is why Bitcoin's evolution has been sluggish and conservative, with few modifications.

Secondly, The Merge will balance the consensus mechanism (proof-of-work or proof-of-stake) and the system decentralization or centralization. Proof-of-work prevents double-spending, thus validators must buy hardware. The system works, but it requires a lot of electricity and, as it scales up, tends to re-centralize as validators acquire more hardware and the entire network activity gets focused in a few nodes. Larger operations save more money, which increases profitability and market share. This evolution runs opposed to the concept of decentralization, and some anticipate that any system that uses proof of work as a consensus mechanism will evolve towards centralization, with fewer large firms able to invest in efficient network nodes.

Yet radical bitcoin enthusiasts share an opposite argument. In proof-of-stake, transaction validators put their funds at stake to attest that transactions are valid. The algorithm chooses who validates each transaction, giving more possibilities to nodes that put more coins at stake, which could open the door to centralization and government control.

In both cases, we're talking about long-term changes, but Bitcoin's proof-of-work has been evolving longer and seems to confirm those fears, while proof-of-stake is only employed in coins with a minuscule volume compared to Ethereum and has no predictive value.

As of mid-September, we will have two significant cryptocurrencies, each with a different consensus mechanisms and equally different characteristics: one is intrinsically conservative and used only for economic transactions, while the other has been evolving in open source mode, and can be used for other types of assets, smart contracts, or decentralized finance systems. Some even see it as the foundation of Web3.

Many things could change before September 15, but The Merge is likely to be a turning point. We'll have to follow this closely.