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.

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

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

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

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

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

Vintage: A Fashion Option That Is Eco-Conscious

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And that is a tale as old as fashion.

Product management wizardry.

Early adopters buy a product even if it hasn't hit the market or has flaws.

Who are the early adopters?

Early adopters try a new technology or product first. Early adopters are interested in trying or buying new technologies and products before others. They're risk-tolerant and can provide initial cash flow and product reviews. They help a company's new product or technology gain social proof.

Early adopters are most common in the technology industry, but they're in every industry. They don't follow the crowd. They seek innovation and report product flaws before mass production. If the product works well, the first users become loyal customers, and colleagues value their opinion.

What to do with early adopters?

They can be used to collect feedback and initial product promotion, first sales, and product value validation.

How to find early followers?

Start with your immediate environment and target audience. Communicate with them to see if they're interested in your value proposition.

1) Innovators (2.5% of the population) are risk-takers seeking novelty. These people are the first to buy new and trendy items and drive social innovation. However, these people are usually elite;

Early adopters (13.5%) are inclined to accept innovations but are more cautious than innovators; they start using novelties when innovators or famous people do;

3) The early majority (34%) is conservative; they start using new products when many people have mastered them. When the early majority accepted the innovation, it became ingrained in people's minds.

4) Attracting 34% of the population later means the novelty has become a mass-market product. Innovators are using newer products;

5) Laggards (16%) are the most conservative, usually elderly people who use the same products.

Stages of new information acceptance

1. The information is strange and rejected by most. Accepted only by innovators;

2. When early adopters join, more people believe it's not so bad; when a critical mass is reached, the novelty becomes fashionable and most people use it.

3. Fascination with a novelty peaks, then declines; the majority and laggards start using it later; novelty becomes obsolete; innovators master something new.

Problems with early implementation

Early adopter sales have disadvantages.

Higher risk of defects

Selling to first-time users increases the risk of defects. Early adopters are often influential, so this can affect the brand's and its products' long-term perception.

Not what was expected

First-time buyers may be disappointed by the product. Marketing messages can mislead consumers, and if the first users believe the company misrepresented the product, this will affect future sales.

Compatibility issues

Some technological advances cause compatibility issues. Consumers may be disappointed if new technology is incompatible with their electronics.

Method 5 WHY

Let's talk about 5 why, a good tool for finding project problems' root causes. This method is also known as the five why rule, method, or questions.

The 5 why technique came from Toyota's lean manufacturing and helps quickly determine a problem's root cause.

On one, two, and three, you simply do this:

1. We identify and frame the issue for which a solution is sought.

2. We frequently ponder this question. The first 2-3 responses are frequently very dull, making you want to give up on this pointless exercise. However, after that, things get interesting. And occasionally it's so fascinating that you question whether you really needed to know.

3. We consider the final response, ponder it, and choose a course of action.

Always do the 5 whys with the customer or team to have a reasonable discussion and better understand what's happening.

And the “five whys” is a wonderful and simplest tool for introspection. With the accumulated practice, it is used almost automatically in any situation like “I can’t force myself to work, the mood is bad in the morning” or “why did I decide that I have no life without this food processor for 20,000 rubles, which will take half of my rather big kitchen.”

An illustration of the five whys

A simple, but real example from my work practice that I think is very indicative, given the participants' low IT skills.  Anonymized, of course.

Users spend too long looking for tender documents.

Why? Because they must search through many company tender documents.

Why? Because the system can't filter department-specific bids.

Why? Because our contract management system requirements didn't include a department-tender link. That's it, right? We'll add a filter and be happy. but still…

why? Because we based the system's requirements on regulations for working with paper tender documents (when they still had envelopes and autopsies), not electronic ones, and there was no search mechanism.

Why? We didn't consider how our work would change when switching from paper to electronic tenders when drafting the requirements.

Now I know what to do in the future. We add a filter, enter department data, and teach users to use it. This is tactical, but strategically we review the same forgotten requirements to make all the necessary changes in a package, plus we include it in the checklist for the acceptance of final requirements for the future.

Errors when using 5 why

Five whys seems simple, but it can be misused.

Popular ones:

1. The accusation of everyone and everything is then introduced. After all, the 5 why method focuses on identifying the underlying causes rather than criticizing others. As a result, at the third step, it is not a good idea to conclude that the system is ineffective because users are stupid and that we can therefore do nothing about it.

2. to fight with all my might so that the outcome would be exactly 5 reasons, neither more nor less. 5 questions is a typical number (it sounds nice, yes), but there could be 3 or 7 in actuality.

3. Do not capture in-between responses. It is difficult to overestimate the power of the written or printed word, so the result is so-so when the focus is lost. That's it, I suppose. Simple, quick, and brilliant, like other project management tools.

Conclusion

Today we analyzed important study elements:

Early adopters and 5 WHY We've analyzed cases and live examples of how these methods help with product research and growth point identification. Next, consider the HADI cycle.