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.

Recent investigations have shown that generative AI can boost hackers and misinformation spreaders.

Since its inception in late November 2022, OpenAI's ChatGPT has entertained and assisted many online users in writing, coding, task automation, and linguistic translation. Given this versatility, it is maybe unsurprising but nonetheless regrettable that fraudsters and mis-, dis-, and malinformation (MDM) spreaders are also considering ChatGPT and related AI models to streamline and improve their operations.

Malign actors may benefit from ChatGPT, according to a WithSecure research. ChatGPT promises to elevate unlawful operations across many attack channels. ChatGPT can automate spear phishing attacks that deceive corporate victims into reading emails from trusted parties. Malware, extortion, and illicit fund transfers can result from such access.

ChatGPT's ability to simulate a desired writing style makes spear phishing emails look more genuine, especially for international actors who don't speak English (or other languages like Spanish and French).

This technique could let Russian, North Korean, and Iranian state-backed hackers conduct more convincing social engineering and election intervention in the US. ChatGPT can also create several campaigns and various phony online personas to promote them, making such attacks successful through volume or variation. Additionally, image-generating AI algorithms and other developing techniques can help these efforts deceive potential victims.

Hackers are discussing using ChatGPT to install malware and steal data, according to a Check Point research. Though ChatGPT's scripts are well-known in the cyber security business, they can assist amateur actors with little technical understanding into the field and possibly develop their hacking and social engineering skills through repeated use.

Additionally, ChatGPT's hacking suggestions may change. As a writer recently indicated, ChatGPT's ability to blend textual and code-based writing might be a game-changer, allowing the injection of innocent content that would subsequently turn out to be a malicious script into targeted systems. These new AI-powered writing- and code-generation abilities allow for unique cyber attacks, regardless of viability.

OpenAI fears ChatGPT usage. OpenAI, Georgetown University's Center for Security and Emerging Technology, and Stanford's Internet Observatory wrote a paper on how AI language models could enhance nation state-backed influence operations. As a last resort, the authors consider polluting the internet with radioactive or misleading data to ensure that AI language models produce outputs that other language models can identify as AI-generated. However, the authors of this paper seem unaware that their "solution" might cause much worse MDM difficulties.

Literally False News

The public argument about ChatGPTs content-generation has focused on originality, bias, and academic honesty, but broader global issues are at stake. ChatGPT can influence public opinion, troll individuals, and interfere in local and national elections by creating and automating enormous amounts of social media material for specified audiences.

ChatGPT's capacity to generate textual and code output is crucial. ChatGPT can write Python scripts for social media bots and give diverse content for repeated posts. The tool's sophistication makes it irrelevant to one's language skills, especially English, when writing MDM propaganda.

I ordered ChatGPT to write a news piece in the style of big US publications declaring that Ukraine is on the verge of defeat in its fight against Russia due to corruption, desertion, and exhaustion in its army. I also gave it a fake reporter's byline and an unidentified NATO source's remark. The outcome appears convincing:

Worse, terrible performers can modify this piece to make it more credible. They can edit the general's name or add facts about current wars. Furthermore, such actors can create many versions of this report in different forms and distribute them separately, boosting its impact.

In this example, ChatGPT produced a news story regarding (fictional) greater moviegoer fatality rates:

Editing this example makes it more plausible. Dr. Jane Smith, the putative author of the medical report, might be replaced with a real-life medical person or a real victim of this supposed medical hazard.

Can deceptive texts be found? Detecting AI text is behind AI advancements. Minor AI-generated text alterations can upset these technologies.

Some OpenAI individuals have proposed covert methods to watermark AI-generated literature to prevent its abuse. AI models would create information that appears normal to humans but would follow a cryptographic formula that would warn other machines that it was AI-made. However, security experts are cautious since manually altering the content interrupts machine and human detection of AI-generated material.

How to Prepare

Cyber security and IT workers can research and use generative AI models to fight spear fishing and extortion. Governments may also launch MDM-defence projects.

In election cycles and global crises, regular people may be the most vulnerable to AI-produced deceit. Until regulation or subsequent technical advances, individuals must recognize exposure to AI-generated fraud, dating scams, other MDM activities.

A three-step verification method of new material in suspicious emails or social media posts can help identify AI content and manipulation. This three-step approach asks about the information's distribution platform (is it reliable? ), author (is the reader familiar with them? ), and plausibility given one's prior knowledge of the topic.

Consider a report by a trusted journalist that makes shocking statements in their typical manner. AI-powered fake news may be released on an unexpected platform, such as a newly created Facebook profile. However, if it links to a known media source, it is more likely to be real.

Though hard and subjective, this verification method may be the only barrier against manipulation for now.

AI language models:

How to Recognize an AI-Generated Article ChatGPT, the popular AI-powered chatbot, can and likely does generate medium.com-style articles.

AI-Generated Text Detectors Fail. Do This. Online tools claim to detect ChatGPT output. Even with superior programming, I tested some of these tools. pub

Why Original Writers Matter Despite AI Language Models Creative writers may never be threatened by AI language models.

Don't know how to start an online business? Here's a guide. By following these recommendations, you can build a lucrative and profitable online business.

What Are Online Businesses Used For?

Most online businesses are websites. A self-created, self-managed website. You may sell things and services online.

To establish an internet business, you must locate a host and set up accounts with numerous companies. Once your accounts are set up, you may start publishing content and selling products or services.

How to Make Money from Your Online Business

Advertising and marketing are the best ways to make money online. You must develop strategies to contact new customers and generate leads. Make sure your website is search engine optimized so people can find you online.

Top 5 Online Business Tips for Startups:

1. Know your target audience's needs.

2. Make your website as appealing as possible.

3. Generate leads and sales with marketing.

4. Track your progress and learn from your mistakes to improve.

5. Be prepared to expand into new markets or regions.

How to Launch a Successful Online Business Without Putting in a Lot of Work

Build with a solid business model to start a profitable online business. By using these tips, you can start your online business without paying much.

First, develop a user-friendly website. You can use an internet marketing platform or create your own website. Once your website is live, optimize it for search engines and add relevant content.

Second, sell online. This can be done through ads or direct sales to website visitors. Finally, use social media to advertise your internet business. By accomplishing these things, you'll draw visitors to your website and make money.

When launching a business, invest long-term. This involves knowing your goals and how you'll pay for them. Volatility can have several effects on your business. If you offer things online, you may need to examine if the market is ready for them.

Invest wisely

Investing all your money in one endeavor can lead to too much risk and little ROI. Diversify your investments to take advantage of all available chances. So, your investments won't encounter unexpected price swings and you'll be immune to economic upheavals.

Financial news updates

When launching or running a thriving online business, financial news is crucial. By knowing current trends and upcoming developments, you can keep your business lucrative.

Keeping up with financial news can also help you avoid potential traps that could harm your bottom line. If you don't know about new legislation that could affect your industry, potential customers may choose another store when they learn about your business's problems.

Volatility ahead

You should expect volatility in the financial sector. Without a plan for coping with volatility, you could run into difficulty. If your organization relies on client input, you may not be able to exploit customer behavior shifts.

Your company could go bankrupt if you don't understand how fickle the stock market can be. By preparing for volatility, you can ensure your organization survives difficult times and market crashes.

Conclusion

Many internet businesses can be profitable. Start quickly with a few straightforward steps. Diversify your investments, follow financial news, and be prepared for volatility to develop a successful business.

Thanks for reading!

Only two of the following three options can be achieved: consistency, availability, and partition tolerance

Someone told me that growing from 30 to 60 is the biggest adjustment for a team or business.

I remember thinking, That's random. Each company is unique. I've seen teams of all types confront the same issues during development periods. With new enterprises starting every year, we should be better at navigating growing difficulties.

As a team grows, its processes and systems break down, requiring reorganization or declining results. Why always? Why isn't there a perfect scaling model? Why hasn't that been found?

The Three Things Productive Organizations Must Have

Any company should be efficient and productive. Three items are needed:

First, it must verify that no two team members have conflicting information about the roadmap, strategy, or any input that could affect execution. Teamwork is required.

Second, it must ensure that everyone can receive the information they need from everyone else quickly, especially as teams become more specialized (an inevitability in a developing organization). It requires everyone's accessibility.

Third, it must ensure that the organization can operate efficiently even if a piece is unavailable. It's partition-tolerant.

From my experience with the many teams I've been on, invested in, or advised, achieving all three is nearly impossible. Why a perfect organization model cannot exist is clear after analysis.

The CAP Theorem: What is it?

Eric Brewer of Berkeley discovered the CAP Theorem, which argues that a distributed data storage should have three benefits. One can only have two at once.

The three benefits are consistency, availability, and partition tolerance, which implies that even if part of the system is offline, the remainder continues to work.

This notion is usually applied to computer science, but I've realized it's also true for human organizations. In a post-COVID world, many organizations are hiring non-co-located staff as they grow. CAP Theorem is more important than ever. Growing teams sometimes think they can develop ways to bypass this law, dooming themselves to a less-than-optimal team dynamic. They should adopt CAP to maximize productivity.

Path 1: Consistency and availability equal no tolerance for partitions

Let's imagine you want your team to always be in sync (i.e., for someone to be the source of truth for the latest information) and to be able to share information with each other. Only division into domains will do.

Numerous developing organizations do this, especially after the early stage (say, 30 people) when everyone may wear many hats and be aware of all the moving elements. After a certain point, it's tougher to keep generalists aligned than to divide them into specialized tasks.

In a specialized, segmented team, leaders optimize consistency and availability (i.e. every function is up-to-speed on the latest strategy, no one is out of sync, and everyone is able to unblock and inform everyone else).

Partition tolerance suffers. If any component of the organization breaks down (someone goes on vacation, quits, underperforms, or Gmail or Slack goes down), productivity stops. There's no way to give the team stability, availability, and smooth operation during a hiccup.

Path 2: Partition Tolerance and Availability = No Consistency

Some businesses avoid relying too heavily on any one person or sub-team by maximizing availability and partition tolerance (the organization continues to function as a whole even if particular components fail). Only redundancy can do that. Instead of specializing each member, the team spreads expertise so people can work in parallel. I switched from Path 1 to Path 2 because I realized too much reliance on one person is risky.

What happens after redundancy? Unreliable. The more people may run independently and in parallel, the less anyone can be the truth. Lack of alignment or updated information can lead to people executing slightly different strategies. So, resources are squandered on the wrong work.

Path 3: Partition and Consistency "Tolerance" equates to "absence"

The third, least-used path stresses partition tolerance and consistency (meaning answers are always correct and up-to-date). In this organizational style, it's most critical to maintain the system operating and keep everyone aligned. No one is allowed to read anything without an assurance that it's up-to-date (i.e. there’s no availability).

Always short-lived. In my experience, a business that prioritizes quality and scalability over speedy information transmission can get bogged down in heavy processes that hinder production. Large-scale, this is unsustainable.

Accepting CAP

When two puzzle pieces fit, the third won't. I've watched developing teams try to tackle these difficulties, only to find, as their ancestors did, that they can never be entirely solved. Idealized solutions fail in reality, causing lost effort, confusion, and lower production.

As teams develop and change, they should embrace CAP, acknowledge there is a limit to productivity in a scaling business, and choose the best two-out-of-three path.