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Vitalik

Vitalik

4 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

(Edited)

Hackernoon

Hackernoon

4 years ago


👏 Awesome post! When is part 3 coming?

Trent Lapinski

Trent Lapinski

4 years ago

Very complex topic, great explanation

More on Web3 & Crypto

Matt Ward

Matt Ward

3 years ago

Is Web3 nonsense?

Crypto and blockchain have rebranded as web3. They probably thought it sounded better and didn't want the baggage of scam ICOs, STOs, and skirted securities laws.

It was like Facebook becoming Meta. Crypto's biggest players wanted to change public (and regulator) perception away from pump-and-dump schemes.

After the 2018 ICO gold rush, it's understandable. Every project that raised millions (or billions) never shipped a meaningful product.

Like many crazes, charlatans took the money and ran.

Despite its grifter past, web3 is THE hot topic today as more founders, venture firms, and larger institutions look to build the future decentralized internet.

Supposedly.

How often have you heard: This will change the world, fix the internet, and give people power?

Why are most of web3's biggest proponents (and beneficiaries) the same rich, powerful players who built and invested in the modern internet? It's like they want to remake and own the internet.

Something seems off about that.

Why are insiders getting preferential presale terms before the public, allowing early investors and proponents to flip dirt cheap tokens and advisors shares almost immediately after the public sale?

It's a good gig with guaranteed markups, no risk or progress.

If it sounds like insider trading, it is, at least practically. This is clear when people talk about blockchain/web3 launches and tokens.

Fast money, quick flips, and guaranteed markups/returns are common.

Incentives-wise, it's hard to blame them. Who can blame someone for following the rules to win? Is it their fault or regulators' for not leveling the playing field?

It's similar to oil companies polluting for profit, Instagram depressing you into buying a new dress, or pharma pushing an unnecessary pill.

All of that is fair game, at least until we change the playbook, because people (and corporations) change for pain or love. Who doesn't love money?

belief based on money gain

Sinclair:

“It is difficult to get a man to understand something when his salary depends upon his not understanding it.”

Bitcoin, blockchain, and web3 analogies?

Most blockchain and web3 proponents are true believers, not cynical capitalists. They believe blockchain's inherent transparency and permissionless trust allow humanity to evolve beyond our reptilian ways and build a better decentralized and democratic world.

They highlight issues with the modern internet and monopoly players like Google, Facebook, and Apple. Decentralization fixes everything

If we could give power back to the people and get governments/corporations/individuals out of the way, we'd fix everything.

Blockchain solves supply chain and child labor issues in China.

To meet Paris climate goals, reduce emissions. Create a carbon token.

Fixing online hatred and polarization Web3 Twitter and Facebook replacement.

Web3 must just be the answer for everything… your “perfect” silver bullet.

Nothing fits everyone. Blockchain has pros and cons like everything else.

Blockchain's viral, ponzi-like nature has an MLM (mid level marketing) feel. If you bought Taylor Swift's NFT, your investment is tied to her popularity.

Probably makes you promote Swift more. Play music loudly.

Here's another example:

Imagine if Jehovah’s Witnesses (or evangelical preachers…) got paid for every single person they converted to their cause.

It becomes a self-fulfilling prophecy as their faith and wealth grow.

Which breeds extremism? Ultra-Orthodox Jews are an example. maximalists

Bitcoin and blockchain are causes, religions. It's a money-making movement and ideal.

We're good at convincing ourselves of things we want to believe, hence filter bubbles.

I ignore anything that doesn't fit my worldview and seek out like-minded people, which algorithms amplify.

Then what?

Is web3 merely a new scam?

No, never!

Blockchain has many crucial uses.

Sending money home/abroad without bank fees;

Like fleeing a war-torn country and converting savings to Bitcoin;

Like preventing Twitter from silencing dissidents.

Permissionless, trustless databases could benefit society and humanity. There are, however, many limitations.

Lost password?

What if you're cheated?

What if Trump/Putin/your favorite dictator incites a coup d'état?

What-ifs abound. Decentralization's openness brings good and bad.

No gatekeepers or firefighters to rescue you.

ISIS's fundraising is also frictionless.

Community-owned apps with bad interfaces and service.

Trade-offs rule.

So what compromises does web3 make?

What are your trade-offs? Decentralization has many strengths and flaws. Like Bitcoin's wasteful proof-of-work or Ethereum's political/wealth-based proof-of-stake.

To ensure the survival and veracity of the network/blockchain and to safeguard its nodes, extreme measures have been designed/put in place to prevent hostile takeovers aimed at altering the blockchain, i.e., adding money to your own wallet (account), etc.

These protective measures require significant resources and pose challenges. Reduced speed and throughput, high gas fees (cost to submit/write a transaction to the blockchain), and delayed development times, not to mention forked blockchain chains oops, web3 projects.

Protecting dissidents or rogue regimes makes sense. You need safety, privacy, and calm.

First-world life?

What if you assumed EVERYONE you saw was out to rob/attack you? You'd never travel, trust anyone, accomplish much, or live fully. The economy would collapse.

It's like an ant colony where half the ants do nothing but wait to be attacked.

Waste of time and money.

11% of the US budget goes to the military. Imagine what we could do with the $766B+ we spend on what-ifs annually.

Is so much hypothetical security needed?

Blockchain and web3 are similar.

Does your app need permissionless decentralization? Does your scooter-sharing company really need a proof-of-stake system and 1000s of nodes to avoid Russian hackers? Why?

Worst-case scenario? It's not life or death, unless you overstate the what-ifs. Web3 proponents find improbable scenarios to justify decentralization and tokenization.

Do I need a token to prove ownership of my painting? Unless I'm a master thief, I probably bought it.

despite losing the receipt.

I do, however, love Web 3.

Enough Web3 bashing for now. Understand? Decentralization isn't perfect, but it has huge potential when applied to the right problems.

I see many of the right problems as disrupting big tech's ruthless monopolies. I wrote several years ago about how tokenized blockchains could be used to break big tech's stranglehold on platforms, marketplaces, and social media.

Tokenomics schemes can be used for good and are powerful. Here’s how.

Before the ICO boom, I made a series of predictions about blockchain/crypto's future. It's still true.

Here's where I was then and where I see web3 going:

My 11 Big & Bold Predictions for Blockchain

In the near future, people may wear crypto cash rings or bracelets.

  1. While some governments repress cryptocurrency, others will start to embrace it.

  2. Blockchain will fundamentally alter voting and governance, resulting in a more open election process.

  3. Money freedom will lead to a more geographically open world where people will be more able to leave when there is unrest.

  4. Blockchain will make record keeping significantly easier, eliminating the need for a significant portion of government workers whose sole responsibility is paperwork.

  5. Overrated are smart contracts.

6. Tokens will replace company stocks.

7. Blockchain increases real estate's liquidity, value, and volatility.

8. Healthcare may be most affected.

9. Crypto could end privacy and lead to Minority Report.

10. New companies with network effects will displace incumbents.

11. Soon, people will wear rings or bracelets with crypto cash.

Some have already happened, while others are still possible.

Time will tell if they happen.

And finally:

What will web3 be?

Who will be in charge?

Closing remarks

Hope you enjoyed this web3 dive. There's much more to say, but that's for another day.

We're writing history as we go.

Tech regulation, mergers, Bitcoin surge How will history remember us?

What about web3 and blockchain?

Is this a revolution or a tulip craze?

Remember, actions speak louder than words (share them in the comments).

Your turn.

Vitalik

Vitalik

3 years ago

Fairness alternatives to selling below market clearing prices (or community sentiment, or fun)

When a seller has a limited supply of an item in high (or uncertain and possibly high) demand, they frequently set a price far below what "the market will bear." As a result, the item sells out quickly, with lucky buyers being those who tried to buy first. This has happened in the Ethereum ecosystem, particularly with NFT sales and token sales/ICOs. But this phenomenon is much older; concerts and restaurants frequently make similar choices, resulting in fast sell-outs or long lines.

Why do sellers do this? Economists have long wondered. A seller should sell at the market-clearing price if the amount buyers are willing to buy exactly equals the amount the seller has to sell. If the seller is unsure of the market-clearing price, they should sell at auction and let the market decide. So, if you want to sell something below market value, don't do it. It will hurt your sales and it will hurt your customers. The competitions created by non-price-based allocation mechanisms can sometimes have negative externalities that harm third parties, as we will see.

However, the prevalence of below-market-clearing pricing suggests that sellers do it for good reason. And indeed, as decades of research into this topic has shown, there often are. So, is it possible to achieve the same goals with less unfairness, inefficiency, and harm?

Selling at below market-clearing prices has large inefficiencies and negative externalities

An item that is sold at market value or at an auction allows someone who really wants it to pay the high price or bid high in the auction. So, if a seller sells an item below market value, some people will get it and others won't. But the mechanism deciding who gets the item isn't random, and it's not always well correlated with participant desire. It's not always about being the fastest at clicking buttons. Sometimes it means waking up at 2 a.m. (but 11 p.m. or even 2 p.m. elsewhere). Sometimes it's just a "auction by other means" that's more chaotic, less efficient, and has far more negative externalities.

There are many examples of this in the Ethereum ecosystem. Let's start with the 2017 ICO craze. For example, an ICO project would set the price of the token and a hard maximum for how many tokens they are willing to sell, and the sale would start automatically at some point in time. The sale ends when the cap is reached.

So what? In practice, these sales often ended in 30 seconds or less. Everyone would start sending transactions in as soon as (or just before) the sale started, offering higher and higher fees to encourage miners to include their transaction first. Instead of the token seller receiving revenue, miners receive it, and the sale prices out all other applications on-chain.

The most expensive transaction in the BAT sale set a fee of 580,000 gwei, paying a fee of $6,600 to get included in the sale.

Many ICOs after that tried various strategies to avoid these gas price auctions; one ICO notably had a smart contract that checked the transaction's gasprice and rejected it if it exceeded 50 gwei. But that didn't solve the issue. Buyers hoping to game the system sent many transactions hoping one would get through. An auction by another name, clogging the chain even more.

ICOs have recently lost popularity, but NFTs and NFT sales have risen in popularity. But the NFT space didn't learn from 2017; they do fixed-quantity sales just like ICOs (eg. see the mint function on lines 97-108 of this contract here). So what?

That's not the worst; some NFT sales have caused gas price spikes of up to 2000 gwei.

High gas prices from users fighting to get in first by sending higher and higher transaction fees. An auction renamed, pricing out all other applications on-chain for 15 minutes.

So why do sellers sometimes sell below market price?

Selling below market value is nothing new, and many articles, papers, and podcasts have written (and sometimes bitterly complained) about the unwillingness to use auctions or set prices to market-clearing levels.

Many of the arguments are the same for both blockchain (NFTs and ICOs) and non-blockchain examples (popular restaurants and concerts). Fairness and the desire not to exclude the poor, lose fans or create tension by being perceived as greedy are major concerns. The 1986 paper by Kahneman, Knetsch, and Thaler explains how fairness and greed can influence these decisions. I recall that the desire to avoid perceptions of greed was also a major factor in discouraging the use of auction-like mechanisms in 2017.

Aside from fairness concerns, there is the argument that selling out and long lines create a sense of popularity and prestige, making the product more appealing to others. Long lines should have the same effect as high prices in a rational actor model, but this is not the case in reality. This applies to ICOs and NFTs as well as restaurants. Aside from increasing marketing value, some people find the game of grabbing a limited set of opportunities first before everyone else is quite entertaining.

But there are some blockchain-specific factors. One argument for selling ICO tokens below market value (and one that persuaded the OmiseGo team to adopt their capped sale strategy) is community dynamics. The first rule of community sentiment management is to encourage price increases. People are happy if they are "in the green." If the price drops below what the community members paid, they are unhappy and start calling you a scammer, possibly causing a social media cascade where everyone calls you a scammer.

This effect can only be avoided by pricing low enough that post-launch market prices will almost certainly be higher. But how do you do this without creating a rush for the gates that leads to an auction?

Interesting solutions

It's 2021. We have a blockchain. The blockchain is home to a powerful decentralized finance ecosystem, as well as a rapidly expanding set of non-financial tools. The blockchain also allows us to reset social norms. Where decades of economists yelling about "efficiency" failed, blockchains may be able to legitimize new uses of mechanism design. If we could use our more advanced tools to create an approach that more directly solves the problems, with fewer side effects, wouldn't that be better than fiddling with a coarse-grained one-dimensional strategy space of selling at market price versus below market price?

Begin with the goals. We'll try to cover ICOs, NFTs, and conference tickets (really a type of NFT) all at the same time.

1. Fairness: don't completely exclude low-income people from participation; give them a chance. The goal of token sales is to avoid high initial wealth concentration and have a larger and more diverse initial token holder community.

2. Don’t create races: Avoid situations where many people rush to do the same thing and only a few get in (this is the type of situation that leads to the horrible auctions-by-another-name that we saw above).

3. Don't require precise market knowledge: the mechanism should work even if the seller has no idea how much demand exists.

4. Fun: The process of participating in the sale should be fun and game-like, but not frustrating.

5. Give buyers positive expected returns: in the case of a token (or an NFT), buyers should expect price increases rather than decreases. This requires selling below market value.
Let's start with (1). From Ethereum's perspective, there is a simple solution. Use a tool designed for the job: proof of personhood protocols! Here's one quick idea:

Mechanism 1 Each participant (verified by ID) can buy up to ‘’X’’ tokens at price P, with the option to buy more at an auction.

With the per-person mechanism, buyers can get positive expected returns for the portion sold through the per-person mechanism, and the auction part does not require sellers to understand demand levels. Is it race-free? The number of participants buying through the per-person pool appears to be high. But what if the per-person pool isn't big enough to accommodate everyone?

Make the per-person allocation amount dynamic.

Mechanism 2 Each participant can deposit up to X tokens into a smart contract to declare interest. Last but not least, each buyer receives min(X, N / buyers) tokens, where N is the total sold through the per-person pool (some other amount can also be sold by auction). The buyer gets their deposit back if it exceeds the amount needed to buy their allocation.
No longer is there a race condition based on the number of buyers per person. No matter how high the demand, it's always better to join sooner rather than later.

Here's another idea if you like clever game mechanics with fancy quadratic formulas.

Mechanism 3 Each participant can buy X units at a price P X 2 up to a maximum of C tokens per buyer. C starts low and gradually increases until enough units are sold.

The quantity allocated to each buyer is theoretically optimal, though post-sale transfers will degrade this optimality over time. Mechanisms 2 and 3 appear to meet all of the above objectives. They're not perfect, but they're good starting points.

One more issue. For fixed and limited supply NFTs, the equilibrium purchased quantity per participant may be fractional (in mechanism 2, number of buyers > N, and in mechanism 3, setting C = 1 may already lead to over-subscription). With fractional sales, you can offer lottery tickets: if there are N items available, you have a chance of N/number of buyers of getting the item, otherwise you get a refund. For a conference, groups could bundle their lottery tickets to guarantee a win or a loss. The certainty of getting the item can be auctioned.

The bottom tier of "sponsorships" can be used to sell conference tickets at market rate. You may end up with a sponsor board full of people's faces, but is that okay? After all, John Lilic was on EthCC's sponsor board!

Simply put, if you want to be reliably fair to people, you need an input that explicitly measures people. Authentication protocols do this (and if desired can be combined with zero knowledge proofs to ensure privacy). So we should combine the efficiency of market and auction-based pricing with the equality of proof of personhood mechanics.

Answers to possible questions

Q: Won't people who don't care about your project buy the item and immediately resell it?

A: Not at first. Meta-games take time to appear in practice. If they do, making them untradeable for a while may help mitigate the damage. Using your face to claim that your previous account was hacked and that your identity, including everything in it, should be moved to another account works because proof-of-personhood identities are untradeable.

Q: What if I want to make my item available to a specific community?

A: Instead of ID, use proof of participation tokens linked to community events. Another option, also serving egalitarian and gamification purposes, is to encrypt items within publicly available puzzle solutions.

Q: How do we know they'll accept? Strange new mechanisms have previously been resisted.

A: Having economists write screeds about how they "should" accept a new mechanism that they find strange is difficult (or even "equity"). However, abrupt changes in context effectively reset people's expectations. So the blockchain space is the best place to try this. You could wait for the "metaverse", but it's possible that the best version will run on Ethereum anyway, so start now.

Farhan Ali Khan

Farhan Ali Khan

2 years ago

Introduction to Zero-Knowledge Proofs: The Art of Proving Without Revealing

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.

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Scott Duke Kominers

3 years ago

NFT Creators Go Creative Commons Zero (cc0)


On January 1, "Public Domain Day," thousands of creative works immediately join the public domain. The original creator or copyright holder loses exclusive rights to reproduce, adapt, or publish the work, and anybody can use it. It happens with movies, poems, music, artworks, books (where creative rights endure 70 years beyond the author's death), and sometimes source code.

Public domain creative works open the door to new uses. 400,000 sound recordings from before 1923, including Winnie-the-Pooh, were released this year.  With most of A.A. Milne's 1926 Winnie-the-Pooh characters now available, we're seeing innovative interpretations Milne likely never planned. The ancient hyphenated version of the honey-loving bear is being adapted for a horror movie: "Winnie-the-Pooh: Blood and Honey"... with Pooh and Piglet as the baddies.

Counterintuitively, experimenting and recombination can occasionally increase IP value. Open source movements allow the public to build on (or fork and duplicate) existing technologies. Permissionless innovation helps Android, Linux, and other open source software projects compete. Crypto's success at attracting public development is also due to its support of open source and "remix culture," notably in NFT forums.

Production memes

NFT projects use several IP strategies to establish brands, communities, and content. Some preserve regular IP protections; others offer NFT owners the opportunity to innovate on connected IP; yet others have removed copyright and other IP safeguards.

By using the "Creative Commons Zero" (cc0) license, artists can intentionally select for "no rights reserved." This option permits anyone to benefit from derivative works without legal repercussions. There's still a lot of confusion between copyrights and NFTs, so nothing here should be considered legal, financial, tax, or investment advice. Check out this post for an overview of copyright vulnerabilities with NFTs and how authors can protect owners' rights. This article focuses on cc0.

Nouns, a 2021 project, popularized cc0 for NFTs. Others followed, including: A Common Place, Anonymice, Blitmap, Chain Runners, Cryptoadz, CryptoTeddies, Goblintown, Gradis, Loot, mfers, Mirakai, Shields, and Terrarium Club are cc0 projects.

Popular crypto artist XCOPY licensed their 1-of-1 NFT artwork "Right-click and Save As Guy" under cc0 in January, exactly one month after selling it. cc0 has spawned many derivatives.

"Right-click Save As Guy" by XCOPY (1)/derivative works (2)

"Right-click Save As Guy" by XCOPY (1)/derivative works (2)

XCOPY said Monday he would apply cc0 to "all his existing art." "We haven't seen a cc0 summer yet, but I think it's approaching," said the artist. - predicting a "DeFi summer" in 2020, when decentralized finance gained popularity.

Why do so many NFT authors choose "no rights"?

Promoting expansions of the original project to create a more lively and active community is one rationale. This makes sense in crypto, where many value open sharing and establishing community.

Creativity depends on cultural significance. NFTs may allow verifiable ownership of any digital asset, regardless of license, but cc0 jumpstarts "meme-ability" by actively, not passively, inviting derivative works. As new derivatives are made and shared, attention might flow back to the original, boosting its reputation. This may inspire new interpretations, leading in a flywheel effect where each derivative adds to the original's worth - similar to platform network effects, where platforms become more valuable as more users join them.

cc0 licence allows creators "seize production memes."

"SEASON 1 MEME CARD 2"

Physical items are also using cc0 NFT assets, thus it's not just a digital phenomenon. The Nouns Vision initiative turned the square-framed spectacles shown on each new NounsDAO NFT ("one per day, forever") into luxury sunglasses. Blitmap's pixel-art has been used on shoes, apparel, and caps. In traditional IP regimes, a single owner controls creation, licensing, and production.

The physical "blitcap" (3rd level) is a descendant of the trait in the cc0 Chain Runners collection (2nd), which uses the "logo" from cc0 Blitmap (1st)! The Logo is Blitmap token #84 and has been used as a trait in various collections. The "Dom Rose" is another popular token. These homages reference Blitmap's influence as a cc0 leader, as one of the earliest NFT projects to proclaim public domain intents. A new collection, Citizens of Tajigen, emerged last week with a Blitcap characteristic.

These derivatives can be a win-win for everyone, not just the original inventors, especially when using NFT assets to establish unique brands. As people learn about the derivative, they may become interested in the original. If you see someone wearing Nouns glasses on the street (or in a Super Bowl ad), you may desire a pair, but you may also be interested in buying an original NounsDAO NFT or related derivative.

Blitmap Logo Hat (1), Chain Runners #780 ft. Hat (2), and Blitmap Original "Logo #87" (3)

Blitmap Logo Hat (1), Chain Runners #780 ft. Hat (2), and Blitmap Original "Logo #87" (3)

Co-creating open source

NFTs' power comes from smart contract technology's intrinsic composability. Many smart contracts can be integrated or stacked to generate richer applications.

"Money Legos" describes how decentralized finance ("DeFi") smart contracts interconnect to generate new financial use cases. Yearn communicates with MakerDAO's stablecoin $DAI and exchange liquidity provider Curve by calling public smart contract methods. NFTs and their underlying smart contracts can operate as the base-layer framework for recombining and interconnecting culture and creativity.

cc0 gives an NFT's enthusiast community authority to develop new value layers whenever, wherever, and however they wish.

Multiple cc0 projects are playable characters in HyperLoot, a Loot Project knockoff.

Open source and Linux's rise are parallels. When the internet was young, Microsoft dominated the OS market with Windows. Linux (and its developer Linus Torvalds) championed a community-first mentality, freely available the source code without restrictions. This led to developers worldwide producing new software for Linux, from web servers to databases. As people (and organizations) created world-class open source software, Linux's value proposition grew, leading to explosive development and industry innovation. According to Truelist, Linux powers 96.3% of the top 1 million web servers and 85% of smartphones.

With cc0 licensing empowering NFT community builders, one might hope for long-term innovation. Combining cc0 with NFTs "turns an antagonistic game into a co-operative one," says NounsDAO cofounder punk4156. It's important on several levels. First, decentralized systems from open source to crypto are about trust and coordination, therefore facilitating cooperation is crucial. Second, the dynamics of this cooperation work well in the context of NFTs because giving people ownership over their digital assets allows them to internalize the results of co-creation through the value that accrues to their assets and contributions, which incentivizes them to participate in co-creation in the first place.

Licensed to create

If cc0 projects are open source "applications" or "platforms," then NFT artwork, metadata, and smart contracts provide the "user interface" and the underlying blockchain (e.g., Ethereum) is the "operating system." For these apps to attain Linux-like potential, more infrastructure services must be established and made available so people may take advantage of cc0's remixing capabilities.

These services are developing. Zora protocol and OpenSea's open source Seaport protocol enable open, permissionless NFT marketplaces. A pixel-art-rendering engine was just published on-chain to the Ethereum blockchain and integrated into OKPC and ICE64. Each application improves blockchain's "out-of-the-box" capabilities, leading to new apps created from the improved building blocks.

Web3 developer growth is at an all-time high, yet it's still a small fraction of active software developers globally. As additional developers enter the field, prospective NFT projects may find more creative and infrastructure Legos for cc0 and beyond.

Electric Capital Developer Report (2021), p. 122

Electric Capital Developer Report (2021), p. 122

Growth requires composability. Users can easily integrate digital assets developed on public standards and compatible infrastructure into other platforms. The Loot Project is one of the first to illustrate decentralized co-creation, worldbuilding, and more in NFTs. This example was low-fi or "incomplete" aesthetically, providing room for imagination and community co-creation.

Loot began with a series of Loot bag NFTs, each listing eight "adventure things" in white writing on a black backdrop (such as Loot Bag #5726's "Katana, Divine Robe, Great Helm, Wool Sash, Divine Slippers, Chain Gloves, Amulet, Gold Ring"). Dom Hofmann's free Loot bags served as a foundation for the community.

Several projects have begun metaphorical (lore) and practical (game development) world-building in a short time, with artists contributing many variations to the collective "Lootverse." They've produced games (Realms & The Crypt), characters (Genesis Project, Hyperloot, Loot Explorers), storytelling initiatives (Banners, OpenQuill), and even infrastructure (The Rift).

Why cc0 and composability? Because consumers own and control Loot bags, they may use them wherever they choose by connecting their crypto wallets. This allows users to participate in multiple derivative projects, such as  Genesis Adventurers, whose characters appear in many others — creating a decentralized franchise not owned by any one corporation.

Genesis Project's Genesis Adventurer (1) with HyperLoot (2) and Loot Explorer (3) versions

Genesis Project's Genesis Adventurer (1) with HyperLoot (2) and Loot Explorer (3) versions

When to go cc0

There are several IP development strategies NFT projects can use. When it comes to cc0, it’s important to be realistic. The public domain won't make a project a runaway success just by implementing the license. cc0 works well for NFT initiatives that can develop a rich, enlarged ecosystem.

Many of the most successful cc0 projects have introduced flexible intellectual property. The Nouns brand is as obvious for a beer ad as for real glasses; Loot bags are simple primitives that make sense in all adventure settings; and the Goblintown visual style looks good on dwarfs, zombies, and cranky owls as it does on Val Kilmer.

The ideal cc0 NFT project gives builders the opportunity to add value:

  • vertically, by stacking new content and features directly on top of the original cc0 assets (for instance, as with games built on the Loot ecosystem, among others), and

  • horizontally, by introducing distinct but related intellectual property that helps propagate the original cc0 project’s brand (as with various Goblintown derivatives, among others).

These actions can assist cc0 NFT business models. Because cc0 NFT projects receive royalties from secondary sales, third-party extensions and derivatives can boost demand for the original assets.

Using cc0 license lowers friction that could hinder brand-reinforcing extensions or lead to them bypassing the original. Robbie Broome recently argued (in the context of his cc0 project A Common Place) that giving away his IP to cc0 avoids bad rehashes down the line. If UrbanOutfitters wanted to put my design on a tee, they could use the actual work instead of hiring a designer. CC0 can turn competition into cooperation.

Community agreement about core assets' value and contribution can help cc0 projects. Cohesion and engagement are key. Using the above examples: Developers can design adventure games around whatever themes and item concepts they desire, but many choose Loot bags because of the Lootverse's community togetherness. Flipmap shared half of its money with the original Blitmap artists in acknowledgment of that project's core role in the community. This can build a healthy culture within a cc0 project ecosystem. Commentator NiftyPins said it was smart to acknowledge the people that constructed their universe. Many OG Blitmap artists have popped into the Flipmap discord to share information.

cc0 isn't a one-size-fits-all answer; NFTs formed around well-established brands may prefer more restrictive licenses to preserve their intellectual property and reinforce exclusivity. cc0 has some superficial similarities to permitting NFT owners to market the IP connected with their NFTs (à la Bored Ape Yacht Club), but there is a significant difference: cc0 holders can't exclude others from utilizing the same IP. This can make it tougher for holders to develop commercial brands on cc0 assets or offer specific rights to partners. Holders can still introduce enlarged intellectual property (such as backstories or derivatives) that they control.


Blockchain technologies and the crypto ethos are decentralized and open-source. This makes it logical for crypto initiatives to build around cc0 content models, which build on the work of the Creative Commons foundation and numerous open source pioneers.

NFT creators that choose cc0 must select how involved they want to be in building the ecosystem. Some cc0 project leaders, like Chain Runners' developers, have kept building on top of the initial cc0 assets, creating an environment derivative projects can plug into. Dom Hofmann stood back from Loot, letting the community lead. (Dom is also working on additional cc0 NFT projects for the company he formed to build Blitmap.) Other authors have chosen out totally, like sartoshi, who announced his exit from the cc0 project he founded, mfers, and from the NFT area by publishing a final edition suitably named "end of sartoshi" and then deactivating his Twitter account. A multi-signature wallet of seven mfers controls the project's smart contract. 

cc0 licensing allows a robust community to co-create in ways that benefit all members, regardless of original creators' continuous commitment. We foresee more organized infrastructure and design patterns as NFT matures. Like open source software, value capture frameworks may see innovation. (We could imagine a variant of the "Sleepycat license," which requires commercial software to pay licensing fees when embedding open source components.) As creators progress the space, we expect them to build unique rights and licensing strategies. cc0 allows NFT producers to bootstrap ideas that may take off.

Edward Williams

Edward Williams

3 years ago

I currently manage 4 profitable online companies. I find all the generic advice and garbage courses very frustrating. The only advice you need is this.

A man playing chess.

This is for young entrepreneurs, especially in tech.

People give useless success advice on TikTok and Reddit. Early risers, bookworms, etc. Entrepreneurship courses. Work hard and hustle.

False. These aren't successful traits.

I mean, organization is good. As someone who founded several businesses and now works at a VC firm, I find these tips to be clichés.

Based on founding four successful businesses and working with other successful firms, here's my best actionable advice:

1. Choose a sector or a niche and become an expert in it.

This is more generic than my next tip, but it's a must-do that's often overlooked. Become an expert in the industry or niche you want to enter. Discover everything.

Buy (future) competitors' products. Understand consumers' pain points. Market-test. Target keyword combos. Learn technical details.

The most successful businesses I've worked with were all formed by 9-5 employees. They knew the industry's pain points. They started a business targeting these pain points.

2. Choose a niche or industry crossroads to target.

How do you choose an industry or niche? What if your industry is too competitive?

List your skills and hobbies. Randomness is fine. Find an intersection between two interests or skills.

Say you build websites well. You like cars.

Web design is a *very* competitive industry. Cars and web design?

Instead of web design, target car dealers and mechanics. Build a few fake demo auto mechanic websites, then cold call shops with poor websites. Verticalize.

I've noticed a pattern:

  • Person works in a particular industry for a corporation.

  • Person gains expertise in the relevant industry.

  • Person quits their job and launches a small business to address a problem that their former employer was unwilling to address.

I originally posted this on Reddit and it seemed to have taken off so I decided to share it with you all.

Focus on the product. When someone buys from you, you convince them the product's value exceeds the price. It's not fair and favors the buyer.

Creating a superior product or service will win. Narrowing this helps you outcompete others.

You may be their only (lucky) option.

Web3Lunch

Web3Lunch

3 years ago

An employee of OpenSea might get a 40-year prison sentence for insider trading using NFTs.

GM Friens

The space had better days. Those greenish spikes...oh wow, haven't felt that in ages. Cryptocurrencies and NFTs have lost popularity. Google agrees. Both are declining.

As seen below, crypto interest spiked in May because of the Luna fall. NFT interest is similar to early October last year.

Google Trends

This makes me think NFTs are mostly hype and FOMO. No art or community. I've seen enough initiatives to know that communities stick around if they're profitable. Once it starts falling, they move on to the next project. The space has no long-term investments. Flip everything.

OpenSea trading volume has stayed steady for months. May's volume is 1.8 million ETH ($3.3 billion).

Source: Dune

Despite this, I think NFTs and crypto will stick around. In bad markets, builders gain most.

Only 4k developers are active on Ethereum blockchain. It's low. A great chance for the space enthusiasts.

An employee of OpenSea might get a 40-year prison sentence for insider trading using NFTs.

Nathaniel Chastian, an OpenSea employee, traded on insider knowledge. He'll serve 40 years for that.

Here's what happened if you're unfamiliar.

OpenSea is a secondary NFT marketplace. Their homepage featured remarkable drops. Whatever gets featured there, NFT prices will rise 5x.

Chastian was at OpenSea. He chose forthcoming NFTs for OpenSeas' webpage.

Using anonymous digital currency wallets and OpenSea accounts, he would buy NFTs before promoting them on the homepage, showcase them, and then sell them for at least 25 times the price he paid.

From June through September 2021, this happened. Later caught, fired. He's charged with wire fraud and money laundering, each carrying a 20-year maximum penalty.

Although web3 space is all about decentralization, a step like this is welcomed since it restores faith in the area. We hope to see more similar examples soon.

Here's the press release.

Source from Justice.gov

Understanding smart contracts

@cantino.eth has a Twitter thread on smart contracts. Must-read. Also, he appears educated about the space, so follow him.