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 web adds millions or billions of pages per day.

How much Google traffic does this content get?

In 2017, we studied 2 million randomly-published pages to answer this question. Only 5.7% of them ranked in Google's top 10 search results within a year of being published.

94.3 percent of roughly two million pages got no Google traffic.

Two million pages is a small sample compared to the entire web. We did another study.

We analyzed over a billion pages to see how many get organic search traffic and why.

How many pages get search traffic?

90% of pages in our index get no Google traffic, and 5.2% get ten visits or less.

90% of google pages get no organic traffic

How can you join the minority that gets Google organic search traffic?

There are hundreds of SEO problems that can hurt your Google rankings. If we only consider common scenarios, there are only four.

Reason #1: No backlinks

I hate to repeat what most SEO articles say, but it's true:

Backlinks boost Google rankings.

Google's "top 3 ranking factors" include them.

Why don't we divide our studied pages by the number of referring domains?

66.31 percent of pages have no backlinks, and 26.29 percent have three or fewer.

Did you notice the trend already?

Most pages lack search traffic and backlinks.

But are these the same pages?

Let's compare monthly organic search traffic to backlinks from unique websites (referring domains):

More backlinks equals more Google organic traffic.

Referring domains and keyword rankings are correlated.

It's important to note that correlation does not imply causation, and none of these graphs prove backlinks boost Google rankings. Most SEO professionals agree that it's nearly impossible to rank on the first page without backlinks.

You'll need high-quality backlinks to rank in Google and get search traffic. 

Is organic traffic possible without links?

Here are the numbers:

Four million pages get organic search traffic without backlinks. Only one in 20 pages without backlinks has traffic, which is 5% of our sample.

Most get 300 or fewer organic visits per month.

What happens if we exclude high-Domain-Rating pages?

The numbers worsen. Less than 4% of our sample (1.4 million pages) receive organic traffic. Only 320,000 get over 300 monthly organic visits, or 0.1% of our sample.

This suggests high-authority pages without backlinks are more likely to get organic traffic than low-authority pages.

Internal links likely pass PageRank to new pages.

Two other reasons:

1. Our crawler's blocked. Most shady SEOs block backlinks from us. This prevents competitors from seeing (and reporting) PBNs.

2. They choose low-competition subjects. Low-volume queries are less competitive, requiring fewer backlinks to rank.

If the idea of getting search traffic without building backlinks excites you, learn about Keyword Difficulty and how to find keywords/topics with decent traffic potential and low competition.

Reason #2: The page has no long-term traffic potential.

Some pages with many backlinks get no Google traffic.

Why? I filtered Content Explorer for pages with no organic search traffic and divided them into four buckets by linking domains.

Almost 70k pages have backlinks from over 200 domains, but no search traffic.

By manually reviewing these (and other) pages, I noticed two general trends that explain why they get no traffic:

1. They overdid "shady link building" and got penalized by Google;

2. They're not targeting a Google-searched topic.

I won't elaborate on point one because I hope you don't engage in "shady link building"

#2 is self-explanatory:

If nobody searches for what you write, you won't get search traffic.

Consider one of our blog posts' metrics:

No organic traffic despite 337 backlinks from 132 sites.

The page is about "organic traffic research," which nobody searches for.

News articles often have this. They get many links from around the web but little Google traffic.

People can't search for things they don't know about, and most don't care about old events and don't search for them.

Note:

Some news articles rank in the "Top stories" block for relevant, high-volume search queries, generating short-term organic search traffic.

The Guardian's top "Donald Trump" story:

Ahrefs caught on quickly:

"Donald Trump" gets 5.6M monthly searches, so this page got a lot of "Top stories" traffic.

I bet traffic has dropped if you check now.

One of the quickest and most effective SEO wins is:

1. Find your website's pages with the most referring domains;

2. Do keyword research to re-optimize them for relevant topics with good search traffic potential.

Bryan Harris shared this "quick SEO win" during a course interview:

He suggested using Ahrefs' Site Explorer's "Best by links" report to find your site's most-linked pages and analyzing their search traffic. This finds pages with lots of links but little organic search traffic.

We see:

The guide has 67 backlinks but no organic traffic.

We could fix this by re-optimizing the page for "SERP"

A similar guide with 26 backlinks gets 3,400 monthly organic visits, so we should easily increase our traffic.

Don't do this with all low-traffic pages with backlinks. Choose your battles wisely; some pages shouldn't be ranked.

Reason #3: Search intent isn't met

Google returns the most relevant search results.

That's why blog posts with recommendations rank highest for "best yoga mat."

Google knows that most searchers aren't buying.

It's also why this yoga mats page doesn't rank, despite having seven times more backlinks than the top 10 pages:

The page ranks for thousands of other keywords and gets tens of thousands of monthly organic visits. Not being the "best yoga mat" isn't a big deal.

If you have pages with lots of backlinks but no organic traffic, re-optimizing them for search intent can be a quick SEO win.

It was originally a boring landing page describing our product's benefits and offering a 7-day trial.

We realized the problem after analyzing search intent.

People wanted a free tool, not a landing page.

In September 2018, we published a free tool at the same URL. Organic traffic and rankings skyrocketed.

Reason #4: Unindexed page

Google can’t rank pages that aren’t indexed.

If you think this is the case, search Google for site:[url]. You should see at least one result; otherwise, it’s not indexed.

A rogue noindex meta tag is usually to blame. This tells search engines not to index a URL.

Rogue canonicals, redirects, and robots.txt blocks prevent indexing.

Check the "Excluded" tab in Google Search Console's "Coverage" report to see excluded pages.

Google doesn't index broken pages, even with backlinks.

Surprisingly common.

In Ahrefs' Site Explorer, the Best by Links report for a popular content marketing blog shows many broken pages.

One dead page has 131 backlinks:

According to the URL, the page defined content marketing. —a keyword with a monthly search volume of 5,900 in the US.

Luckily, another page ranks for this keyword. Not a huge loss.

At least redirect the dead page's backlinks to a working page on the same topic. This may increase long-tail keyword traffic.

This post is a summary. See the original post here

McMurtry Spéirling outperforms Tesla in speed and efficiency.

EVs were ridiculously slow for decades. However, the 2008 Tesla Roadster revealed that EVs might go extraordinarily fast. The Tesla Model S Plaid and Rimac Nevera are the fastest-accelerating road vehicles, despite combustion-engined road cars dominating the course. A little-known firm beat Tesla and Rimac in the 0-60 race, beat F1 vehicles on a circuit, and boasts a 350-mile driving range. The McMurtry Spéirling is completely insane.

Mat Watson of CarWow, a YouTube megastar, was recently handed a Spéirling and access to Silverstone Circuit (view video above). Mat ran a quarter-mile on Silverstone straight with former F1 driver Max Chilton. The little pocket-rocket automobile touched 100 mph in 2.7 seconds, completed the quarter mile in 7.97 seconds, and hit 0-60 in 1.4 seconds. When looking at autos quickly, 0-60 times can seem near. The Tesla Model S Plaid does 0-60 in 1.99 seconds, which is comparable to the Spéirling. Despite the meager statistics, the Spéirling is nearly 30% faster than Plaid!

My vintage VW Golf 1.4s has an 8.8-second 0-60 time, whereas a BMW Z4 3.0i is 30% faster (with a 0-60 time of 6 seconds). I tried to beat a Z4 off the lights in my Golf, but the Beamer flew away. If they challenge the Spéirling in a Model S Plaid, they'll feel as I did. Fast!

Insane quarter-mile drag time. Its road car record is 7.97 seconds. A Dodge Demon, meant to run extremely fast quarter miles, finishes so in 9.65 seconds, approximately 20% slower. The Rimac Nevera's 8.582-second quarter-mile record was miles behind drag racing. This run hampered the Spéirling. Because it was employing gearing that limited its top speed to 150 mph, it reached there in a little over 5 seconds without accelerating for most of the quarter mile! McMurtry can easily change the gearing, making the Spéirling run quicker.

McMurtry did this how? First, the Spéirling is a tiny single-seater EV with a 60 kWh battery pack, making it one of the lightest EVs ever. The 1,000-hp Spéirling has more than one horsepower per kg. The Nevera has 0.84 horsepower per kg and the Plaid 0.44.

However, you cannot simply construct a car light and power it. Instead of accelerating, it would spin. This makes the Spéirling a fan car. Its huge fans create massive downforce. These fans provide the Spéirling 2 tonnes of downforce while stationary, so you could park it on the ceiling. Its fast 0-60 time comes from its downforce, which lets it deliver all that power without wheel spin.

It also possesses complete downforce at all speeds, allowing it to tackle turns faster than even race vehicles. Spéirlings overcame VW IDRs and F1 cars to set the Goodwood Hill Climb record (read more here). The Spéirling is a dragstrip winner and track dominator, unlike the Plaid and Nevera.

The Spéirling is astonishing for a single-seater. Fan-generated downforce is more efficient than wings and splitters. It also means the vehicle has very minimal drag without the fan. The Spéirling can go 350 miles per charge (WLTP) or 20-30 minutes at full speed on a track despite its 60 kWh battery pack. The G-forces would hurt your neck before the battery died if you drove around a track for longer. The Spéirling can charge at over 200 kW in about 30 minutes. Thus, driving to track days, having fun, and returning is possible. Unlike other high-performance EVs.

Tesla, Rimac, or Lucid will struggle to defeat the Spéirling. They would need to build a fan automobile because adding power to their current vehicle would make it uncontrollable. The EV and automobile industries now have a new, untouchable performance king.

A reading list for successful people

Elon Musk reads and talks. So, one learns. Many brilliant individuals & amazing literature.

This article recommends 3 Elon Musk novels. All of them helped me succeed. Hope they'll help you.

Douglas Adams's The Hitchhiker's Guide to the Galaxy

Page Count: 193
Rating on Goodreads: 4.23

Arthur Dent is pulled off Earth by a buddy seconds before it's razed for a cosmic motorway. The trio hitchhikes through space and gets into problems.

I initially read Hitchhiker's as a child. To evade my mum, I'd read with a flashlight under the covers. She'd scold at me for not sleeping on school nights when she found out. Oops.

The Hitchhiker's Guide to the Galaxy is lighthearted science fiction.

My favorite book quotes are:

• “Space is big. You won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.”

• “Far out in the uncharted backwaters of the unfashionable end of the western spiral arm of the Galaxy lies a small unregarded yellow sun. Orbiting this at a distance of roughly ninety-two million miles is an utterly insignificant little blue-green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea.”

• “On planet Earth, man had always assumed that he was more intelligent than dolphins because he had achieved so much — the wheel, New York, wars, and so on — whilst all the dolphins had ever done was muck about in the water having a good time. But conversely, the dolphins had always believed that they were far more intelligent than man — for precisely the same reasons.”

the Sun Tzu book The Art Of War

Page Count: 273
Rating on Goodreads: 3.97

It's a classic. You may apply The Art of War's ideas to (nearly) every facet of life. Ex:

• Pick your fights.

• Keep in mind that timing is crucial.

• Create a backup plan in case something goes wrong.

• Obstacles provide us a chance to adapt and change.

This book was my first. Since then, I'm a more strategic entrepreneur. Excellent book. And read it ASAP!

My favorite book quotes are:

• “Victorious warriors win first and then go to war, while defeated warriors go to war first and then seek to win.”

• “Engage people with what they expect; it is what they are able to discern and confirms their projections. It settles them into predictable patterns of response, occupying their minds while you wait for the extraordinary moment — that which they cannot anticipate.”

• “If you know the enemy and know yourself, you need not fear the result of a hundred battles. If you know yourself but not the enemy, for every victory gained, you will also suffer a defeat. If you know neither the enemy nor yourself, you will succumb in every battle.”

Peter Thiel's book Zero to One

Page Count: 195
Rating on Goodreads: 4.18

Peter argues the best money-making strategies are typically unproven. Entrepreneurship should never have a defined path to success. Whoever says differently is lying.

Zero to One explores technology and society. Peter is a philosophy major and law school graduate, which informs the work.

Peters' ideas, depth, and intellect stood out in Zero to One. It's a top business book.

My favorite book quotes are:

• “The most valuable businesses of coming decades will be built by entrepreneurs who seek to empower people rather than try to make them obsolete.”

• “The next Bill Gates will not build an operating system. The next Larry Page or Sergey Brin won’t make a search engine. And the next Mark Zuckerberg won’t create a social network. If you are copying these guys, you aren’t learning from them.”

• “If your goal is to never make a mistake in your life, you shouldn’t look for secrets. The prospect of being lonely but right — dedicating your life to something that no one else believes in — is already hard. The prospect of being lonely and wrong can be unbearable.”