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.

Monkey jpegs and shitcoins have tarnished Web3's reputation. Let’s move on.

Web3 was called "the internet's future."

Well, 'crypto bros' shouted about it loudly.

As quickly as it arrived to be the next internet, it appears to be dead. It's had scandals, turbulence, and crashes galore:

• Web 3.0's cryptocurrencies have crashed. Bitcoin's all-time high was $66,935. This month, Ethereum fell from $2130 to $1117. Six months ago, the cryptocurrency market peaked at $3 trillion. Worst is likely ahead.

• Gas fees make even the simplest Web3 blockchain transactions unsustainable.

• Terra, Luna, and other dollar pegs collapsed, hurting crypto markets. Celsius, a crypto lender backed by VCs and Canada's second-largest pension fund, and Binance, a crypto marketplace, have withheld money and coins. They're near collapse.

• NFT sales are falling rapidly and losing public interest.

Web3 has few real-world uses, like most crypto/blockchain technologies. Web3's image has been tarnished by monkey profile pictures and shitcoins while failing to become decentralized (the whole concept is controlled by VCs).

The damage seems irreparable, leaving Web3 in the gutter.

Step forward our new saviour — Web5

Fear not though, as hero awaits to drag us out of the Web3 hellscape. Jack Dorsey revealed his plan to save the internet quickly.

Dorsey has long criticized Web3, believing that VC capital and silicon valley insiders have created a centralized platform. In a tweet that upset believers and VCs (he was promptly blocked by Marc Andreessen), Dorsey argued, "You don't own "Web3." VCs and LPs do. Their incentives prevent it. It's a centralized organization with a new name.

Dorsey announced Web5 on June 10 in a very Elon-like manner. Block's TBD unit will work on the project (formerly Square).

Web5's pitch is that users will control their own data and identity. Bitcoin-based. Sound familiar? The presentation pack's official definition emphasizes decentralization. Web5 is a decentralized web platform that enables developers to write decentralized web apps using decentralized identifiers, verifiable credentials, and decentralized web nodes, returning ownership and control over identity and data to individuals.

Web5 would be permission-less, open, and token-less. What that means for Earth is anyone's guess. Identity. Ownership. Blockchains. Bitcoin. Different.

Web4 appears to have been skipped, forever destined to wish it could have shown the world what it could have been. (It was probably crap.) As this iteration combines Web2 and Web3, simple math and common sense add up to 5. Or something.

Dorsey and his team have had this idea simmering for a while. Daniel Buchner, a member of Block's Decentralized Identity team, said, "We're finishing up Web5's technical components."

Web5 could be the project that decentralizes the internet. It must be useful to users and convince everyone to drop the countless Web3 projects, products, services, coins, blockchains, and websites being developed as I write this.

Web5 may be too late for Dorsey and the incoming flood of creators.

Web6 is planned!

The next months and years will be hectic and less stable than the transition from Web 1.0 to Web 2.0. 

• Web1 was around 1991-2004.

• Web2 ran from 2004 to 2021. (though the Web3 term was first used in 2014, it only really gained traction years later.)

• Web3 lasted a year.

• Web4 is dead.

Silicon Valley billionaires are turning it into a startup-style race, each disrupting the next iteration until they crack it. Or destroy it completely.

Web5 won't last either.

Elon Musk, Matt Taibbi, the 'Twitter Files,' and Hunter Biden's laptop: what gives?

Explaining Twitter Files

Matt Taibbi released "The Twitter Files," a batch of emails sent by Twitter executives discussing the company's decision to stop an October 2020 New York Post story online.

What's on Twitter? New York Post and Fox News call them "bombshell" documents. Or, as a Post columnist admitted, are they "not the smoking gun"? Onward!

What started this?

The New York Post published an exclusive, potentially explosive story in October 2020: Biden's Secret Emails: Ukrainian executive thanks Hunter Biden for'meeting' veep dad. The story purported to report the contents of a laptop brought to the tabloid by a Delaware computer repair shop owner who said it belonged to President Biden's second son, Hunter Biden. Emails and files on the laptop allegedly showed how Hunter peddled influence with Ukranian businessmen and included a "raunchy 12-minute video" of Hunter smoking crack and having sex.

Twitter banned links to the Post story after it was published, calling it "hacked material." The Post's Twitter account was suspended for multiple days.

Why? Yoel Roth, Twitter's former head of trust and safety, said the company couldn't verify the story, implying they didn't trust the Post.

Twitter's stated purpose rarely includes verifying news stories. This seemed like intentional political interference. This story was hard to verify because the people who claimed to have found the laptop wouldn't give it to other newspapers. (Much of the story, including Hunter's business dealings in Ukraine and China, was later confirmed.)

Roth: "It looked like a hack and leak."

So what are the “Twitter Files?”

Twitter's decision to bury the story became a political scandal, and new CEO Elon Musk promised an explanation. The Twitter Files, named after Facebook leaks.

Musk promised exclusive details of "what really happened" with Hunter Biden late Friday afternoon. The tweet was punctuated with a popcorn emoji.

Explaining Twitter Files

Three hours later, journalist Matt Taibbi tweeted more than three dozen tweets based on internal Twitter documents that revealed "a Frankensteinian tale of a human-built mechanism grown out of its designer's control."

Musk sees this release as a way to shape Twitter's public perception and internal culture in his image. We don't know if the CEO gave Taibbi the documents. Musk hyped the document dump before and during publication, but Taibbi cited "internal sources."

Taibbi shares email screenshots showing Twitter execs discussing the Post story and blocking its distribution. Taibbi says the emails show Twitter's "extraordinary steps" to bury the story.

Twitter communications chief Brandon Borrman has the most damning quote in the Files. Can we say this is policy? The story seemed unbelievable. It seemed like a hack... or not? Could Twitter, which ex-CEO Dick Costolo called "the free speech wing of the free speech party," censor a news story?

Many on the right say the Twitter Files prove the company acted at the behest of Democrats. Both parties had these tools, writes Taibbi. In 2020, both the Trump White House and Biden campaign made requests. He says the system for reporting tweets for deletion is unbalanced because Twitter employees' political donations favor Democrats. Perhaps. These donations may have helped Democrats connect with Twitter staff, but it's also possible they didn't. No emails in Taibbi's cache show these alleged illicit relations or any actions Twitter employees took as a result.

Even Musk's supporters were surprised by the drop. Miranda Devine of the New York Post told Tucker Carlson the documents weren't "the smoking gun we'd hoped for." Sebastian Gorka said on Truth Social, "So far, I'm deeply underwhelmed." DC Democrats collude with Palo Alto Democrats. Whoop!” The Washington Free Beacon's Joe Simonson said the Twitter files are "underwhelming." Twitter was staffed by Democrats who did their bidding. (Why?)

If "The Twitter Files" matter, why?

These emails led Twitter to suppress the Hunter Biden laptop story has real news value. It's rare for a large and valuable company like Twitter to address wrongdoing so thoroughly. Emails resemble FOIA documents. They describe internal drama at a company with government-level power. Katie Notopoulos tweeted, "Any news outlet would've loved this scoop!" It's not a'scandal' as teased."

Twitter's new owner calls it "the de facto public town square," implying public accountability. Like a government agency. Though it's exciting to receive once-hidden documents in response to a FOIA, they may be boring and tell you nothing new. Like Twitter files. We learned how Twitter blocked the Post's story, but not why. Before these documents were released, we knew Twitter had suppressed the story and who was involved.

These people were disciplined and left Twitter. Musk fired Vijaya Gadde, the former CLO who reportedly played a "key role" in the decision. Roth quit over Musk's "dictatorship." Musk arrived after Borrman left. Jack Dorsey, then-CEO, has left. Did those who digitally quarantined the Post's story favor Joe Biden and the Democrats? Republican Party opposition and Trump hatred? New York Post distaste? According to our documents, no. Was there political and press interference? True. We knew.

Taibbi interviewed anonymous ex-Twitter employees about the decision; all expressed shock and outrage. One source said, "Everyone knew this was fucked." Since Taibbi doesn't quote that expletive, we can assume the leaked emails contained few or no sensational quotes. These executives said little to support nefarious claims.

Outlets more invested in the Hunter Biden story than Gizmodo seem vexed by the release and muted headlines. The New York Post, which has never shied away from a blaring headline in its 221-year history, owns the story of Hunter Biden's laptop. Two Friday-night Post alerts about Musk's actions were restrained. Elon Musk will drop Twitter files on NY Post-Hunter Biden laptop censorship today. Elon Musk's Twitter dropped Post censorship details from Biden's laptop. Fox News' Apple News push alert read, "Elon Musk drops Twitter censorship documents."

Bombshell, bombshell, bombshell… what, exactly, is the bombshell? Maybe we've heard this story too much and are missing the big picture. Maybe these documents detail a well-documented decision.

The Post explains why on its website. "Hunter Biden laptop bombshell: Twitter invented reason to censor Post's reporting," its headline says.

Twitter's ad hoc decision to moderate a tabloid's content is not surprising. The social network had done this for years as it battled toxic users—violent white nationalists, virulent transphobes, harassers and bullies of all political stripes, etc. No matter how much Musk crows, the company never had content moderation under control. Buzzfeed's 2016 investigation showed how Twitter has struggled with abusive posters since 2006. Jack Dorsey and his executives improvised, like Musk.

Did the US government interfere with the ex-social VP's media company? That's shocking, a bombshell. Musk said Friday, "Twitter suppressing free speech by itself is not a 1st amendment violation, but acting under government orders with no judicial review is." Indeed! Taibbi believed this. August 2022: "The laptop is secondary." Zeynep Tufecki, a Columbia professor and New York Times columnist, says the FBI is cutting true story distribution. Taibbi retracted the claim Friday night: "I've seen no evidence of government involvement in the laptop story."

What’s the bottom line?

I'm still not sure what's at stake in the Hunter Biden scandal after dozens of New York Post articles, hundreds of hours of Fox News airtime, and thousands of tweets. Briefly: Joe Biden's son left his laptop with a questionable repairman. FBI confiscated it? The repairman made a copy and gave it to Rudy Giuliani's lawyer. The Post got it from Steve Bannon. On that laptop were videos of Hunter Biden smoking crack, cavorting with prostitutes, and emails about introducing his father to a Ukrainian businessman for $50,000 a month. Joe Biden urged Ukraine to fire a prosecutor investigating the company. What? The story seems to be about Biden family business dealings, right?

The discussion has moved past that point anyway. Now, the story is the censorship of it. Adrienne Rich wrote in "Diving Into the Wreck" that she came for "the wreck and not the story of the wreck" No matter how far we go, Hunter Biden's laptop is done. Now, the crash's story matters.

I'm dizzy. Katherine Miller of BuzzFeed wrote, "I know who I believe, and you probably do, too. To believe one is to disbelieve the other, which implicates us in the decision; we're stuck." I'm stuck. Hunter Biden's laptop is a political fabrication. You choose. I've decided.

This could change. Twitter Files drama continues. Taibbi said, "Much more to come." I'm dizzy.

No other calendar is simpler, smaller, and reusable year after year. It works and is used here.

Most of us discard and replace our calendars annually. Each month, we move our calendar ahead another page, thus if we need to know which day of the week corresponds to a given day/month combination, we have to calculate it or flip forward/backward to the corresponding month. Questions like:

• What day does this year's American Thanksgiving fall on?

• Which months contain a Friday the thirteenth?

• When is July 4th? What day of the week?

• Alternatively, what day of the week is Christmas?

They're hard to figure out until you switch to the right month or look up all the months.

However, mathematically, the answers to these questions or any question that requires matching the day of the week with the day/month combination in a year are predictable, basic, and easy to work out. If you use this one-page calendar instead of a 12-month calendar, it lasts the whole year and is easy to alter for future years. Let me explain.

The 2023 one-page calendar is above. The days of the month are on the lower left, which works for all months if you know that:

• There are 31 days in January, March, May, July, August, October, and December.

• All of the months of April, June, September, and November have 30 days.

• And depending on the year, February has either 28 days (in non-leap years) or 29 days (in leap years).

If you know this, this calendar makes it easy to match the day/month of the year to the weekday.

Here are some instances. American Thanksgiving is always on the fourth Thursday of November. You'll always know the month and day of the week, but the date—the day in November—changes each year.

On any other calendar, you'd have to flip to November to see when the fourth Thursday is. This one-page calendar only requires:

• pick the month of November in the top-right corner to begin.

• drag your finger down until Thursday appears,

• then turn left and follow the monthly calendar until you reach the fourth Thursday.

It's obvious: 2023 is the 23rd American Thanksgiving. For every month and day-of-the-week combination, start at the month, drag your finger down to the desired day, and then move to the left to see which dates match.

What if you knew the day of the week and the date of the month, but not the month(s)?

A different method using the same one-page calendar gives the answer. Which months have Friday the 13th this year? Just:

• begin on the 13th of the month, the day you know you desire,

• then swipe right with your finger till Friday appears.

• and then work your way up until you can determine which months the specific Friday the 13th falls under.

One Friday the 13th occurred in January 2023, and another will occur in October.

The most typical reason to consult a calendar is when you know the month/day combination but not the day of the week.

Compared to single-month calendars, the one-page calendar excels here. Take July 4th, for instance. Find the weekday here:

• beginning on the left on the fourth of the month, as you are aware,

• also begin with July, the month of the year you are most familiar with, at the upper right,

• you should move your two fingers in the opposite directions till they meet: on a Tuesday in 2023.

That's how you find your selected day/month combination's weekday.

Another example: Christmas. Christmas Day is always December 25th, however unless your conventional calendar is open to December of your particular year, a question like "what day of the week is Christmas?" difficult to answer.

Unlike the one-page calendar!

Remember the left-hand day of the month. Top-right, you see the month. Put two fingers, one from each hand, on the date (25th) and the month (December). Slide the day hand to the right and the month hand downwards until they touch.

They meet on Monday—December 25, 2023.

For 2023, that's fine, but what happens in 2024? Even worse, what if we want to know the day-of-the-week/day/month combo many years from now?

I think the one-page calendar shines here.

Except for the blue months in the upper-right corner of the one-page calendar, everything is the same year after year. The months also change in a consistent fashion.

Each non-leap year has 365 days—one more than a full 52 weeks (which is 364). Since January 1, 2023 began on a Sunday and 2023 has 365 days, we immediately know that December 31, 2023 will conclude on a Sunday (which you can confirm using the one-page calendar) and that January 1, 2024 will begin on a Monday. Then, reorder the months for 2024, taking in mind that February will have 29 days in a leap year.

Please note the differences between 2023 and 2024 month placement. In 2023:

• October and January began on the same day of the week.

• On the following Monday of the week, May began.

• August started on the next day,

• then the next weekday marked the start of February, March, and November, respectively.

• Unlike June, which starts the following weekday,

• While September and December start on the following day of the week,

• Lastly, April and July start one extra day later.

Since 2024 is a leap year, February has 29 days, disrupting the rhythm. Month placements change to:

• The first day of the week in January, April, and July is the same.

• October will begin the following day.

• Possibly starting the next weekday,

• February and August start on the next weekday,

• beginning on the following day of the week between March and November,

• beginning the following weekday in June,

• and commencing one more day of the week after that, September and December.

Due to the 366-day leap year, 2025 will start two days later than 2024 on January 1st.

Now, looking at the 2025 calendar, you can see that the 2023 pattern of which months start on which days is repeated! The sole variation is a shift of three days-of-the-week ahead because 2023 had one more day (365) than 52 full weeks (364), and 2024 had two more days (366). Again,

• On Wednesday this time, January and October begin on the same day of the week.

• Although May begins on Thursday,

• August begins this Friday.

• March, November, and February all begin on a Saturday.

• Beginning on a Sunday in June

• Beginning on Monday are September and December,

• and on Tuesday, April and July begin.

In 2026 and 2027, the year will commence on a Thursday and a Friday, respectively.

We must return to our leap year monthly arrangement in 2028. Yes, January 1, 2028 begins on a Saturday, but February, which begins on a Tuesday three days before January, will have 29 days. Thus:

• Start dates for January, April, and July are all Saturdays.

• Given that October began on Sunday,

• Although May starts on a Monday,

• beginning on a Tuesday in February and August,

• Beginning on a Wednesday in March and November,

• Beginning on Thursday, June

• and Friday marks the start of September and December.

This is great because there are only 14 calendar configurations: one for each of the seven non-leap years where January 1st begins on each of the seven days of the week, and one for each of the seven leap years where it begins on each day of the week.

The 2023 calendar will function in 2034, 2045, 2051, 2062, 2073, 2079, 2090, 2102, 2113, and 2119. Except when passing over a non-leap year that ends in 00, like 2100, the repeat time always extends to 12 years or shortens to an extra 6 years.

• The pattern is repeated in 2025's calendar in 2031, 2042, 2053, 2059, 2070, 2081, 2087, 2098, 2110, and 2121.

• The extra 6-year repeat at the end of the century on the calendar for 2026 will occur in the years 2037, 2043, 2054, 2065, 2071, 2082, 2093, 2099, 2105, and 2122.

• The 2027s calendar repeats in 2038, 2049, 2055, 2066, 2077, 2083, 2094, 2100, 2106, and 2117, almost exactly matching the 2026s pattern.

For leap years, the recurrence pattern is every 28 years when not passing a non-leap year ending in 00, or 12 or 40 years when we do. 2024's calendar repeats in 2052, 2080, 2120, 2148, 2176, and 2216; 2028's in 2056, 2084, 2124, 2152, 2180, and 2220.

Knowing January 1st and whether it's a leap year lets you construct a one-page calendar for any year. Try it—you might find it easier than any other alternative!

Y Combinators Winter 2022 Demo Day continues the trend of more startups engaging in accelerator Demo Days. Our team evaluated almost 400 projects in Y Combinator's ninth year.

After Winter 2021 Demo Day, we noticed a hurry pushing shorter rounds, inflated valuations, and larger batches.

Despite the batch size, this event's behavior showed a return to normalcy. Our observations show that investors evaluate and fund businesses more carefully. Unlike previous years, more YC businesses gave investors with data rooms and thorough pitch decks in addition to valuation data before Demo Day.

Demo Day pitches were virtual and fast-paced, limiting unplanned meetings. Investors had more time and information to do their due research before meeting founders. Our staff has more time to study diverse areas and engage with interesting entrepreneurs and founders.

This was one of the most regionally diversified YC cohorts to date. This year's Winter Demo Day startups showed some interesting tendencies.

Trends and Industries to Watch Before Demo Day

Demo day events at any accelerator show how investment competition is influencing startups. As startups swiftly become scale-ups and big success stories in fintech, e-commerce, healthcare, and other competitive industries, entrepreneurs and early-stage investors feel pressure to scale quickly and turn a notion into actual innovation.

Too much eagerness can lead founders to focus on market growth and team experience instead of solid concepts, technical expertise, and market validation. Last year, YC Winter Demo Day funding cycles ended too quickly and valuations were unrealistically high.

Scrum Ventures observed a longer funding cycle this year compared to last year's Demo Day. While that seems promising, many factors could be contributing to change, including:

• Market patterns are changing and the economy is becoming worse.

• the industries that investors are thinking about.

• Individual differences between each event batch and the particular businesses and entrepreneurs taking part

The Winter 2022 Batch's Trends

Each year, we also wish to examine trends among early-stage firms and YC event participants. More international startups than ever were anticipated to present at Demo Day.

Less than 50% of demo day startups were from the U.S. For the S21 batch, firms from outside the US were most likely in Latin America or Europe, however this year's batch saw a large surge in startups situated in Asia and Africa.

YC Startup Directory

163 out of 399 startups were B2B software and services companies. Financial, healthcare, and consumer startups were common.

Our team doesn't plan to attend every pitch or speak with every startup's founders or team members. Let's look at cleantech, Web3, and health and wellness startup trends.

Our Opinions Following Conversations with 87 Startups at Demo Day

In the lead-up to Demo Day, we spoke with 87 of the 125 startups going. Compared to B2C enterprises, B2B startups had higher average valuations. A few outliers with high valuations pushed B2B and B2C means above the YC-wide mean and median.

Many of these startups develop business and technology solutions we've previously covered. We've seen API, EdTech, creative platforms, and cybersecurity remain strong and increase each year.

While these persistent tendencies influenced the startups Scrum Ventures looked at and the founders we interacted with on Demo Day, new trends required more research and preparation. Let's examine cleantech, Web3, and health and wellness startups.

Hardware and software that is green

Cleantech enterprises demand varying amounts of funding for hardware and software. Although the same overarching trend is fueling the growth of firms in this category, each subgroup has its own strategy and technique for investigation and identifying successful investments.

Many cleantech startups we spoke to during the YC event are focused on helping industrial operations decrease or recycle carbon emissions.

• Carbon Crusher: Creating carbon negative roads

• Phase Biolabs: Turning carbon emissions into carbon negative products and carbon neutral e-fuels

• Seabound: Capturing carbon dioxide emissions from ships

• Fleetzero: Creating electric cargo ships

• Impossible Mining: Sustainable seabed mining

• Beyond Aero: Creating zero-emission private aircraft

• Verdn: Helping businesses automatically embed environmental pledges for product and service offerings, boost customer engagement

• AeonCharge: Allowing electric vehicle (EV) drivers to more easily locate and pay for EV charging stations

• Phoenix Hydrogen: Offering a hydrogen marketplace and a connected hydrogen hub platform to connect supply and demand for hydrogen fuel and simplify hub planning and partner program expansion

• Aklimate: Allowing businesses to measure and reduce their supply chain’s environmental impact

• Pina Earth: Certifying and tracking the progress of businesses’ forestry projects

• AirMyne: Developing machines that can reverse emissions by removing carbon dioxide from the air

• Unravel Carbon: Software for enterprises to track and reduce their carbon emissions

Web3: NFTs, the metaverse, and cryptocurrency

Web3 technologies handle a wide range of business issues. This category includes companies employing blockchain technology to disrupt entertainment, finance, cybersecurity, and software development.

Many of these startups overlap with YC's FinTech trend. Despite this, B2C and B2B enterprises were evenly represented in Web3. We examined:

• Stablegains: Offering consistent interest on cash balance from the decentralized finance (DeFi) market

• LiquiFi: Simplifying token management with automated vesting contracts, tax reporting, and scheduling. For companies, investors, and finance & accounting

• NFTScoring: An NFT trading platform

• CypherD Wallet: A multichain wallet for crypto and NFTs with a non-custodial crypto debit card that instantly converts coins to USD

• Remi Labs: Allowing businesses to more easily create NFT collections that serve as access to products, memberships, events, and more

• Cashmere: A crypto wallet for Web3 startups to collaboratively manage funds

• Chaingrep: An API that makes blockchain data human-readable and tokens searchable

• Courtyard: A platform for securely storing physical assets and creating 3D representations as NFTs

• Arda: “Banking as a Service for DeFi,” an API that FinTech companies can use to embed DeFi products into their platforms

• earnJARVIS: A premium cryptocurrency management platform, allowing users to create long-term portfolios

• Mysterious: Creating community-specific experiences for Web3 Discords

• Winter: An embeddable widget that allows businesses to sell NFTs to users purchasing with a credit card or bank transaction

• SimpleHash: An API for NFT data that provides compatibility across blockchains, standardized metadata, accurate transaction info, and simple integration

• Lifecast: Tools that address motion sickness issues for 3D VR video

• Gym Class: Virtual reality (VR) multiplayer basketball video game

• WorldQL: An asset API that allows NFT creators to specify multiple in-game interpretations of their assets, increasing their value

• Bonsai Desk: A software development kit (SDK) for 3D analytics

• Campfire: Supporting virtual social experiences for remote teams

• Unai: A virtual headset and Visual World experience

• Vimmerse: Allowing creators to more easily create immersive 3D experiences

Fitness and health

Scrum Ventures encountered fewer health and wellness startup founders than Web3 and Cleantech. The types of challenges these organizations solve are still diverse. Several of these companies are part of a push toward customization in healthcare, an area of biotech set for growth for companies with strong portfolios and experienced leadership.

Here are several startups we considered:

• Syrona Health: Personalized healthcare for women in the workplace

• Anja Health: Personalized umbilical cord blood banking and stem cell preservation

• Alfie: A weight loss program focused on men’s health that coordinates medical care, coaching, and “community-based competition” to help users lose an average of 15% body weight

• Ankr Health: An artificial intelligence (AI)-enabled telehealth platform that provides personalized side effect education for cancer patients and data collection for their care teams

• Koko — A personalized sleep program to improve at-home sleep analysis and training

• Condition-specific telehealth platforms and programs:

• Reviving Mind: Chronic care management covered by insurance and supporting holistic, community-oriented health care

• Equipt Health: At-home delivery of prescription medical equipment to help manage chronic conditions like obstructive sleep apnea

• LunaJoy: Holistic women’s healthcare management for mental health therapy, counseling, and medication

12 Startups from YC's Winter 2022 Demo Day to Watch

Bobidi: 10x faster AI model improvement

Artificial intelligence (AI) models have become a significant tool for firms to improve how well and rapidly they process data. Bobidi helps AI-reliant firms evaluate their models, boosting data insights in less time and reducing data analysis expenditures. The business has created a gamified community that offers a bug bounty for AI, incentivizing community members to test and find weaknesses in clients' AI models.

Magna: DeFi investment management and token vesting

Magna delivers rapid, secure token vesting so consumers may turn DeFi investments into primitives. Carta for Web3 allows enterprises to effortlessly distribute tokens to staff or investors. The Magna team hopes to allow corporations use locked tokens as collateral for loans, facilitate secondary liquidity so investors can sell shares on a public exchange, and power additional DeFi applications.

Perl Street: Funding for infrastructure

This Fintech firm intends to help hardware entrepreneurs get financing by [democratizing] structured finance, unleashing billions for sustainable infrastructure and next-generation hardware solutions. This network has helped hardware entrepreneurs achieve more than $140 million in finance, helping companies working on energy storage devices, EVs, and creating power infrastructure.

CypherD: Multichain cryptocurrency wallet

CypherD seeks to provide a multichain crypto wallet so general customers can explore Web3 products without knowledge hurdles. The startup's beta app lets consumers access crypto from EVM blockchains. The founders have crypto, financial, and startup experience.

Unravel Carbon: Enterprise carbon tracking and offsetting

Unravel Carbon's AI-powered decarbonization technology tracks companies' carbon emissions. Singapore-based startup focuses on Asia. The software can use any company's financial data to trace the supply chain and calculate carbon tracking, which is used to make regulatory disclosures and suggest carbon offsets.

LunaJoy: Precision mental health for women

LunaJoy helped women obtain mental health support throughout life. The platform combines data science to create a tailored experience, allowing women to access psychotherapy, medication management, genetic testing, and health coaching.

Posh: Automated EV battery recycling

Posh attempts to solve one of the EV industry's largest logistical difficulties. Millions of EV batteries will need to be decommissioned in the next decade, and their precious metals and residual capacity will go unused for some time. Posh offers automated, scalable lithium battery disassembly, making EV battery recycling more viable.

Unai: VR headset with 5x higher resolution

Unai stands apart from metaverse companies. Its VR headgear has five times the resolution of existing options and emphasizes human expression and interaction in a remote world. Maxim Perumal's method of latency reduction powers current VR headsets.

Palitronica: Physical infrastructure cybersecurity

Palitronica blends cutting-edge hardware and software to produce networked electronic systems that support crucial physical and supply chain infrastructure. The startup's objective is to build solutions that defend national security and key infrastructure from cybersecurity threats.

Reality Defender: Deepfake detection

Reality Defender alerts firms to bogus users and changed audio, video, and image files. Reality Deference's API and web app score material in real time to prevent fraud, improve content moderation, and detect deception.

Micro Meat: Infrastructure for the manufacture of cell-cultured meat

MicroMeat promotes sustainable meat production. The company has created technologies to scale up bioreactor-grown meat muscle tissue from animal cells. Their goal is to scale up cultured meat manufacturing so cultivated meat products can be brought to market feasibly and swiftly, boosting worldwide meat consumption.

Fleetzero: Electric cargo ships

This startup's battery technology will make cargo ships more sustainable and profitable. Fleetzero's electric cargo ships have five times larger profit margins than fossil fuel ships. Fleetzeros' founder has marine engineering, ship operations, and enterprise sales and business experience.