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.

The Traffic Light Approach to Reviewing Personal, Team and Project Development

What needs improvement? If you want to improve, you need to practice your sport, musical instrument, habit, or work project. You need to assess your progress.

Continuous improvement is the foundation of focused practice and a growth mentality. Not just individually. High-performing teams pursue improvement. Right? Why is it hard?

As a leadership coach, senior manager, and high-level athlete, I've found three key questions that may unlock high performance in individuals and teams.

Problems with Reviews

Reviewing and improving performance is crucial, however I hate seeing review sessions in my diary. I rarely respond to questionnaire pop-ups or emails. Why?

Time constrains. Requests to fill out questionnaires often state they will take 10–15 minutes, but I can think of a million other things to do with that time. Next, review overload. Businesses can easily request comments online. No matter what you buy, someone will ask for your opinion. This bombardment might make feedback seem bad, which is bad.

The problem is that we might feel that way about important things like personal growth and work performance. Managers and team leaders face a greater challenge.

When to Conduct a Review

We must be wise about reviewing things that matter to us. Timing and duration matter. Reviewing the experience as quickly as possible preserves information and sentiments. Time must be brief. The review's importance and size will determine its length. We might only take a few seconds to review our morning coffee, but we might require more time for that six-month work project.

These post-event reviews should be supplemented by periodic reflection. Journaling can help with daily reflections, but I also like to undertake personal reviews every six months on vacation or at a retreat.

As an employee or line manager, you don't want to wait a year for a performance assessment. Little and frequently is best, with a more formal and in-depth assessment (typically with a written report) in 6 and 12 months.

The Easiest Method to Conduct a Review Session

I follow Einstein's review process:

“Make things as simple as possible but no simpler.”

Thus, it should be brief but deliver the necessary feedback. Quality critique is hard to receive if the process is overly complicated or long.

I have led or participated in many review processes, from strategic overhauls of big organizations to personal goal coaching. Three key questions guide the process at either end:

• What ought to stop being done?

• What should we do going forward?

• What should we do first?

Following the Rule of 3, I compare it to traffic lights. Red, amber, and green lights:

• Red What ought should we stop?

• Amber What ought to we keep up?

• Green Where should we begin?

This approach is easy to understand and self-explanatory, however below are some examples under each area.

Red What ought should we stop?

As a team or individually, we must stop doing things to improve.

Sometimes they're bad. If we want to lose weight, we should avoid sweets. If a team culture is bad, we may need to stop unpleasant behavior like gossiping instead of having difficult conversations.

Not all things we should stop are wrong. Time matters. Since it is finite, we sometimes have to stop nice things to focus on the most important. Good to Great author Jim Collins famously said:

“Don’t let the good be the enemy of the great.”

Prioritizing requires this idea. Thus, decide what to stop to prioritize.

Amber What ought to we keep up?

Should we continue with the amber light? It helps us decide what to keep doing during review. Many items fall into this category, so focus on those that make the most progress.

Which activities have the most impact? Which behaviors create the best culture? Success-building habits?

Use these questions to find positive momentum. These are the fly-wheel motions, according to Jim Collins. The Compound Effect author Darren Hardy says:

“Consistency is the key to achieving and maintaining momentum.”

What can you do consistently to reach your goal?

Green Where should we begin?

Finally, green lights indicate new beginnings. Red/amber difficulties may be involved. Stopping a red issue may give you more time to do something helpful (in the amber).

This green space inspires creativity. Kolbs learning cycle requires active exploration to progress. Thus, it's crucial to think of new approaches, try them out, and fail if required.

This notion underpins lean start-build, up's measure, learn approach and agile's trying, testing, and reviewing. Try new things until you find what works. Thomas Edison, the lighting legend, exclaimed:

“There is a way to do it better — find it!”

Failure is acceptable, but if you want to fail forward, look back on what you've done.

John Maxwell concurred with Edison:

“Fail early, fail often, but always fail forward”

A good review procedure lets us accomplish that. To avoid failure, we must act, experiment, and reflect.

Use the traffic light system to prioritize queries. Ask:

• Red What needs to stop?

• Amber What should continue to occur?

• Green What might be initiated?

Take a moment to reflect on your day. Check your priorities with these three questions. Even if merely to confirm your direction, it's a terrific exercise!

It's kids' digital crack

TikTok is a destructive social media platform.

• The interface shortens attention spans and dopamine receptors.

• TikTok shares more data than other apps.

• Seeing an endless stream of dancing teens on my glowing box makes me feel like a Blade Runner extra.

TikTok did in one year what MTV, Hollywood, and Warner Music tried to do in 20 years. TikTok has psychotized the two-thirds of society Aldous Huxley said were hypnotizable.

Millions of people, mostly kids, are addicted to learning a new dance, lip-sync, or prank, and those who best dramatize this collective improvisation get likes, comments, and shares.

TikTok is a great app. So what?

The Commercial Magnifying Glass TikTok made me realize my generation's time was up and the teenage Zoomers were the target.

I told my 14-year-old sister, "Enjoy your time under the commercial magnifying glass."

TikTok sells your every move, gesture, and thought. Data is the new oil. If you tell someone, they'll say, "Yeah, they collect data, but who cares? I have nothing to hide."

It's a George Orwell novel's beginning. Look up Big Brother Award winners to see if TikTok won.

TikTok shares your data more than any other social media app, and where it goes is unclear. TikTok uses third-party trackers to monitor your activity after you leave the app.

Consumers can't see what data is shared or how it will be used. — Genius URL

32.5 percent of Tiktok's users are 10 to 19 and 29.5% are 20 to 29.

TikTok is the greatest digital marketing opportunity in history, and they'll use it to sell you things, track you, and control your thoughts. Any of its users will tell you, "I don't care, I just want to be famous."

TikTok manufactures mental illness

TikTok's effect on dopamine and the brain is absurd. Dopamine controls the brain's pleasure and reward centers. It's like a switch that tells your brain "this feels good, repeat."

Dr. Julie Albright, a digital culture and communication sociologist, said TikTok users are "carried away by dopamine." It's hypnotic, you'll keep watching."

TikTok constantly releases dopamine. A guy on TikTok recently said he didn't like books because they were slow and boring.

The US didn't ban Tiktok.

Biden and Trump agree on bad things. Both agree that TikTok threatens national security and children's mental health.

The Chinese Communist Party owns and operates TikTok, but that's not its only problem.

• There’s borderline child porn on TikTok

• It's unsafe for children and violated COPPA.

• It's also Chinese spyware. I'm not a Trump supporter, but I was glad he wanted TikTok regulated and disappointed when he failed.

Full-on internet censorship is rare outside of China, so banning it may be excessive. US should regulate TikTok more.

We must reject a low-quality present for a high-quality future.

TikTok vs YouTube

People got mad when I wrote about YouTube's death.

They didn't like when I said TikTok was YouTube's first real challenger.

Indeed. TikTok is the fastest-growing social network. In three years, the Chinese social media app TikTok has gained over 1 billion active users. In the first quarter of 2020, it had the most downloads of any app in a single quarter.

TikTok is the perfect social media app in many ways. It's brief and direct.

Can you believe they had a YouTube vs TikTok boxing match? We are doomed as a species.

YouTube hosts my favorite videos. That’s why I use it. That’s why you use it. New users expect more. They want something quicker, more addictive.

TikTok's impact on other social media platforms frustrates me. YouTube copied TikTok to compete.

It's all about short, addictive content.

I'll admit I'm probably wrong about TikTok. My friend says his feed is full of videos about food, cute animals, book recommendations, and hot lesbians.

Whatever.

TikTok makes us bad

TikTok is the opposite of what the Ancient Greeks believed about wisdom.

It encourages people to be fake. It's like a never-ending costume party where everyone competes.

It does not mean that Gen Z is doomed.

They could be the saviors of the world for all I know.

TikTok feels like a step towards Mike Judge's "Idiocracy," where the average person is a pleasure-seeking moron.

America should avoid BRICS

Vladimir Putin has watched videos of Muammar Gaddafi's CIA-backed demise.

Gaddafi...

Thief.

Did you know Gaddafi wanted a gold-backed dinar for Africa? Because he considered our global financial system was a Ponzi scheme, he wanted to discontinue trading oil in US dollars.

Or, Gaddafi's Libya enjoyed Africa's highest quality of living before becoming freed. Pictured:

Vladimir Putin is a nasty guy, but he had his reasons for not mentioning NATO assisting Ukraine in resisting US imperialism. Nobody tells you. Sure.

The US dollar's corruption post-2008, debasement by quantitative easing, and lack of value are key factors. BRICS will replace the dollar.

BRICS aren't bricks.

Economy-related.

Brazil, Russia, India, China, and South Africa have cooperated for 14 years to fight U.S. hegemony with a new international currency: BRICS.

BRICS is mostly comical. Now. Saudi Arabia, the second-largest oil hegemon, wants to join.

So what?

The New World Currency is BRICS

Russia was kicked out of G8 for its aggressiveness in Crimea in 2014.

It's now G7.

No biggie, said Putin, he said, and I quote, “Bon appetite.”

He was prepared. China, India, and Brazil lead the New World Order.

Together, they constitute 40% of the world's population and, according to the IMF, 50% of the world's GDP by 2030.

Here’s what the BRICS president Marcos Prado Troyjo had to say earlier this year about no longer needing the US dollar: “We have implemented the mechanism of mutual settlements in rubles and rupees, and there is no need for our countries to use the dollar in mutual settlements. And today a similar mechanism of mutual settlements in rubles and yuan is being developed by China.”

Ick. That's D.C. and NYC warmongers licking their chops for WW3 nasty.

Here's a lovely picture of BRICS to relax you:

If Saudi Arabia joins BRICS, as President Mohammed Bin Salman has expressed interest, a majority of the Middle East will have joined forces to construct a new world order not based on the US currency.

I'm not sure of the new acronym.

SBRICSS? CIRBSS? CRIBSS?

The Reason America Is Harvesting What It Sowed

BRICS began 14 years ago.

14 years ago, what occurred? Concentrate. It involved CDOs, bad subprime mortgages, and Wall Street quants crunching numbers.

2008 recession

When two nations trade, they do so in US dollars, not Euros or gold.

What happened when 2008, an avoidable crisis caused by US banks' cupidity and ignorance, what happened?

Everyone WORLDWIDE felt the pain.

Mostly due to corporate America's avarice.

This should have been a warning that China and Russia had enough of our bs. Like when France sent a battleship to America after Nixon scrapped the gold standard. The US was warned to shape up or be dethroned (or at least try).

Nixon improved in 1971. Kinda. Invented PetroDollar.

Another BS system that unfairly favors America and possibly pushed Russia, China, and Saudi Arabia into BRICS.

The PetroDollar forces oil-exporting nations to trade in US dollars and invest in US Treasury bonds. Brilliant. Genius evil.

Our misdeeds are:

• In conflicts that are not its concern, the USA uses the global reserve currency as a weapon.

• Targeted nations abandon the dollar, and rightfully so, as do nations that depend on them for trade in vital resources.

• The dollar's position as the world's reserve currency is in jeopardy, which could have disastrous economic effects.

• Although we have actually sown our own doom, we appear astonished. According to the Bible, whomever sows to appease his sinful nature will reap destruction from that nature whereas whoever sows to appease the Spirit will reap eternal life from the Spirit.

Americans, even our leaders, lack caution and delayed pleasure. When our unsustainable systems fail, we double down. Bailouts of the banks in 2008 were myopic, puerile, and another nail in America's hegemony.

America has screwed everyone.

We're unpopular.

The BRICS's future

It's happened before.

Saddam Hussein sold oil in Euros in 2000, and the US invaded Iraq a month later. The media has devalued the word conspiracy. The Iraq conspiracy.

There were no WMDs, but NYT journalists like Judy Miller drove Americans into a warmongering frenzy because Saddam would ruin the PetroDollar. Does anyone recall that this war spawned ISIS?

I think America has done good for the world. You can make a convincing case that we're many people's villain.

Learn more in Confessions of an Economic Hitman, The Devil's Chessboard, or Tyranny of the Federal Reserve. Or ignore it. That's easier.

We, America, should extend an olive branch, ask for forgiveness, and learn from our faults, as the Tao Te Ching advises. Unlikely. Our population is apathetic and stupid, and our government is corrupt.

Argentina, Iran, Egypt, and Turkey have also indicated interest in joining BRICS. They're also considering making it gold-backed, making it a new world reserve currency.

You should pay attention.

Thanks for reading!