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.

Starting a passive income stream with data science and programming

My website is fresh. But how do I monetize it?

Creating a passive-income website is difficult. Advertise first. But what useful are ads without traffic?

Let’s Generate Traffic And Put Our Programming Skills To Use

SEO boosts traffic (Search Engine Optimisation). Traffic generation is complex. Keywords matter more than text, URL, photos, etc.

My Python skills helped here. I wanted to find relevant, Google-trending keywords (tags) for my topic.

First The Code

I wrote the script below here.

import re
from string import punctuation

import nltk
from nltk import TreebankWordTokenizer, sent_tokenize
from nltk.corpus import stopwords


class KeywordsGenerator:
    def __init__(self, pytrends):
        self._pytrends = pytrends

    def generate_tags(self, file_path, top_words=30):
        file_text = self._get_file_contents(file_path)
        clean_text = self._remove_noise(file_text)
        top_words = self._get_top_words(clean_text, top_words)
        suggestions = []
        for top_word in top_words:
            suggestions.extend(self.get_suggestions(top_word))
        suggestions.extend(top_words)
        tags = self._clean_tokens(suggestions)
        return ",".join(list(set(tags)))

    def _remove_noise(self, text):
        #1. Convert Text To Lowercase and remove numbers
        lower_case_text = str.lower(text)
        just_text = re.sub(r'\d+', '', lower_case_text)
        #2. Tokenise Paragraphs To words
        list = sent_tokenize(just_text)
        tokenizer = TreebankWordTokenizer()
        tokens = tokenizer.tokenize(just_text)
        #3. Clean text
        clean = self._clean_tokens(tokens)
        return clean

    def _clean_tokens(self, tokens):
        clean_words = [w for w in tokens if w not in punctuation]
        stopwords_to_remove = stopwords.words('english')
        clean = [w for w in clean_words if w not in stopwords_to_remove and not w.isnumeric()]
        return clean

    def get_suggestions(self, keyword):
        print(f'Searching pytrends for {keyword}')
        result = []
        self._pytrends.build_payload([keyword], cat=0, timeframe='today 12-m')
        data = self._pytrends.related_queries()[keyword]['top']
        if data is None or data.values is None:
            return result
        result.extend([x[0] for x in data.values.tolist()][:2])
        return result

    def _get_file_contents(self, file_path):
        return open(file_path, "r", encoding='utf-8',errors='ignore').read()

    def _get_top_words(self, words, top):
        counts = dict()

        for word in words:
            if word in counts:
                counts[word] += 1
            else:
                counts[word] = 1

        return list({k: v for k, v in sorted(counts.items(), key=lambda item: item[1])}.keys())[:top]


if __name__ == "1__main__":
    from pytrends.request import TrendReq

    nltk.download('punkt')
    nltk.download('stopwords')
    pytrends = TrendReq(hl='en-GB', tz=360)
    tags = KeywordsGenerator(pytrends)\
              .generate_tags('text_file.txt')
    print(tags)

Then The Dependencies

This script requires:

nltk==3.7
pytrends==4.8.0

Analysis of the Script

I copy and paste my article into text file.txt, and the code returns the keywords as a comma-separated string.

To achieve this:

1. A class I made is called KeywordsGenerator.

2. This class has a function: generate_tags

3. The function generate_tags performs the following tasks:

• retrieves text file contents

• uses NLP to clean the text by tokenizing sentences into words, removing punctuation, and other elements.

• identifies the most frequent words that are relevant.

• The pytrends API is then used to retrieve related phrases that are trending for each word from Google.

• finally adds a comma to the end of the word list.

4. I then use the keywords and paste them into the SEO area of my website.

These terms are trending on Google and relevant to my topic. My site's rankings and traffic have improved since I added new keywords. This little script puts our knowledge to work. I shared the script in case anyone faces similar issues.

I hope it helps readers sell their work.

No, it’s not smartphones, overeating, or sugar.

Everyday practices affect brain health. Good brain practices increase memory and cognition.

Bad behaviors increase stress, which destroys brain cells.

Bad behaviors can reverse evolution and diminish the brain. So, avoid these practices for brain health.

1. The silent assassin

Introverts appreciated quarantine.

Before the pandemic, they needed excuses to remain home; thereafter, they had enough.

I am an introvert, and I didn’t hate quarantine. There are billions of people like me who avoid people.

Social relationships are important for brain health. Social anxiety harms your brain.

Antisocial behavior changes brains. It lowers IQ and increases drug abuse risk.

What you can do is as follows:

• Make a daily commitment to engage in conversation with a stranger. Who knows, you might turn out to be your lone mate.

• Get outside for at least 30 minutes each day.

• Shop for food locally rather than online.

• Make a call to a friend you haven't spoken to in a while.

2. Try not to rush things.

People love hustle culture. This economy requires a side gig to save money.

Long hours reduce brain health. A side gig is great until you burn out.

Work ages your wallet and intellect. Overworked brains age faster and lose cognitive function.

Working longer hours can help you make extra money, but it can harm your brain.

Side hustle but don't overwork.

What you can do is as follows:

• Decide what hour you are not permitted to work after.

• Three hours prior to night, turn off your laptop.

• Put down your phone and work.

• Assign due dates to each task.

3. Location is everything!

The environment may cause brain fog. High pollution can cause brain damage.

Air pollution raises Alzheimer's risk. Air pollution causes cognitive and behavioral abnormalities.

Polluted air can trigger early development of incurable brain illnesses, not simply lung harm.

Your city's air quality is uncontrollable. You may take steps to improve air quality.

In Delhi, schools and colleges are closed to protect pupils from polluted air. So I've adapted.

What you can do is as follows:

• To keep your mind healthy and young, make an investment in a high-quality air purifier.

• Enclose your windows during the day.

• Use a N95 mask every day.

4. Don't skip this meal.

Fasting intermittently is trendy. Delaying breakfast to finish fasting is frequent.

Some skip breakfast and have a hefty lunch instead.

Skipping breakfast might affect memory and focus. Skipping breakfast causes low cognition, delayed responsiveness, and irritation.

Breakfast affects mood and productivity.

Intermittent fasting doesn't prevent healthy breakfasts.

What you can do is as follows:

• Try to fast for 14 hours, then break it with a nutritious breakfast.

• So that you can have breakfast in the morning, eat dinner early.

• Make sure your breakfast is heavy in fiber and protein.

5. The quickest way to damage the health of your brain

Brain health requires water. 1% dehydration can reduce cognitive ability by 5%.

Cerebral fog and mental clarity might result from 2% brain dehydration. Dehydration shrinks brain cells.

Dehydration causes midday slumps and unproductivity. Water improves work performance.

Dehydration can harm your brain, so drink water throughout the day.

What you can do is as follows:

• Always keep a water bottle at your desk.

• Enjoy some tasty herbal teas.

• With a big glass of water, begin your day.

• Bring your own water bottle when you travel.

Conclusion

Bad habits can harm brain health. Low cognition reduces focus and productivity.

Unproductive work leads to procrastination, failure, and low self-esteem.

Avoid these harmful habits to optimize brain health and function.