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.

Creating 10,000 auto-generated artworks, deploying a smart contract to the Ethereum / Polygon blockchain, setting up some tools, etc.

There is so much to do from launching to running an NFT project. Creating parts for artworks, generating 10,000 unique artworks and metadata, creating a smart contract and deploying it to a blockchain network, creating a website, creating a Twitter account, setting up a Discord server, setting up an OpenSea collection. In addition, you need to have MetaMask installed in your browser and have some ETH / MATIC. Did you get tired of doing all this? Don’t worry, once you know what you need to do, all you have to do is do it one by one.

To be honest, it’s best to run an NFT project in a team of three or more, including artists, developers, and marketers. However, depending on your motivation, you can do it by yourself. Some people might come later to offer help with your project. The most important thing is to take a step as soon as possible.

Creating Parts for Artworks

There are lots of free/paid software for drawing, but after all, I think Adobe Illustrator or Photoshop is the best. The images of Skulls In Love are a composite of 48x48 pixel parts created using Photoshop.

How Many Parts Should You Create?

Are you wondering how many parts you should create to avoid duplication as much as possible when generating your artworks? My friend Stephane, a developer, has created a great tool to help with that.

Generating 10,000 Unique Artworks and Metadata

I highly recommend using the HashLips Art Engine to generate your artworks and metadata. Perhaps there is no better artworks generation tool at the moment.

GitHub: https://github.com/HashLips/hashlips_art_engine
YouTube:

Storing Artworks and Metadata

Ideally, the generated artworks and metadata should be stored on-chain, but if you want to store them off-chain, you should use IPFS. Do not store in centralized storage. This is because data will be lost if the server goes down or if the company goes down. On the other hand, IPFS is a more secure way to find data because it utilizes a distributed, decentralized system.

Storing to IPFS is easy with Pinata, NFT.Storage, and so on. The Skulls In Love uses Pinata. It’s very easy to use, just upload the folder containing your artworks.

Creating and Deploying a Smart Contract

You don’t have to create a smart contract from scratch. There are many great NFT projects, many of which publish their contract source code on Etherscan / PolygonScan. You can choose the contract you like and reuse it. Of course, that requires some knowledge of Solidity, but it depends on your efforts. If you don’t know which contract to choose, use the HashLips smart contract. It’s very simple, but it has almost all the functions you need.

GitHub: https://github.com/HashLips/hashlips_nft_contract

Creating a Website

The website here is not just a static site to showcase your project, it’s a so-called dApp that allows you to access your smart contract and mint NFTs. In fact, this level of dApp is not too difficult for anyone who has ever created a website. Because the ethers.js / web3.js libraries make it easy to interact with your smart contract. There’s also no problem connecting wallets, as MetaMask has great documentation.

The Skulls In Love uses a simple, fast, and modern dApp that I built from scratch using Next.js. It is published on GitHub, so feel free to use it.

Why do people mint NFTs on a website?

Ethereum’s gas fees are high, so if you mint all your NFTs, there will be a huge initial cost. So it makes sense to get the buyers to help with the gas fees for minting.
What about Polygon? Polygon’s gas fees are super cheap, so even if you mint 10,000 NFTs, it’s not a big deal. But we don’t do that. Since NFT projects are a kind of game, it involves the fun of not knowing what will come out after minting.

Creating a Twitter Account

I highly recommend creating a Twitter account. Twitter is an indispensable tool for announcing giveaways and reaching more people. It’s better to announce your project and your artworks little by little, 1–2 weeks before launching your project.

Creating and Setting Up a Discord Server

I highly recommend creating a Discord server as well as a Twitter account. The Discord server is a community and its home. Fans of your NFT project will want to join your community and interact with many other members. So, carefully create each channel on your Discord server to make it a cozy place for your community members.

If you are unfamiliar with Discord, you may be particularly confused by the following:
What bots should I use?
How should I set roles and permissions?
But don’t worry. There are lots of great YouTube videos and blog posts about these.
It’s also a good idea to join the Discord servers of some NFT projects and see how they’re made. Our Discord server is so simple that even beginners will find it easy to understand. Please join us and see it!

UPDATED: As your Discord server grows, you cannot manage it on your own. In this case, you will be hiring several moderators, but choose carefully before hiring. And don’t give them important role permissions right after hiring. Initially, the same permissions as other members are sufficient. After a while, you can add permissions as needed, such as kicking/banning, using the “@every” tag, and adding roles. Again, don’t immediately give significant permissions to your Mod role. Your server can be messed up by fake moderators.

Setting Up Your OpenSea Collection

Before you start selling your NFTs, you need to reserve some for airdrops, giveaways, staff, and more. It’s up to you whether it’s 100, 500, or how many.

After minting some of your NFTs, your account and collection should have been created in OpenSea. Go to OpenSea, connect to your wallet, and set up your collection. Just set your logo, banner image, description, links, royalties, and more. It’s not that difficult.

Promoting Your Project

After all, promotion is the most important thing. In fact, almost every successful NFT project spends a lot of time and effort on it.

In addition to Twitter and Discord, it’s even better to use Instagram, Reddit, and Medium. Also, register your project in NFTCalendar and DISBOARD

About Promoters

You’ll probably get lots of contacts from promoters on your Discord, Twitter, Instagram, and more. But most of them are scams, so don’t pay right away. If you have a promoter that looks attractive to you, be sure to check the promoter’s social media accounts or website to see who he/she is. They basically charge in dollars. The amount they charge isn’t cheap, but promoters with lots of followers may have some temporary effect on your project. Some promoters accept 50% prepaid and 50% postpaid. If you can afford it, it might be worth a try. I never ask them, though.

When Should the Promotion Activities Start?

You may be worried that if you promote your project before it starts, someone will copy your project (artworks). It is true that some projects have actually suffered such damage. I don’t have a clear answer to this question right now, but:

• Do not publish all the information about your project too early
• The information should be released little by little
• Creating artworks that no one can easily copy
  I think these are important.
  If anyone has a good idea, please share it!

About Giveaways

When hosting giveaways, you’ll probably use multiple social media platforms. You may want to grow your Discord server faster. But if joining the Discord server is included in the giveaway requirements, some people hate it. I recommend holding giveaways for each platform. On Twitter and Reddit, you should just add the words “Discord members-only giveaway is being held now! Please join us if you like!”.

If you want to easily pick a giveaway winner in your browser, I recommend Twitter Picker.

Precautions for Distributing Free NFTs

If you want to increase your Twitter followers and Discord members, you can actually get a lot of people by holding events such as giveaways and invite contests. However, distributing many free NFTs at once can be dangerous. Some people who want free NFTs, as soon as they get a free one, sell it at a very low price on marketplaces such as OpenSea. They don’t care about your project and are only thinking about replacing their own “free” NFTs with Ethereum. The lower the floor price of your NFTs, the lower the value of your NFTs (project). Try to think of ways to get people to “buy” your NFTs as much as possible.

Ethereum vs. Polygon

Even though Ethereum has high gas fees, NFT projects on the Ethereum network are still mainstream and popular. On the other hand, Polygon has very low gas fees and fast transaction processing, but NFT projects on the Polygon network are not very popular.

Why? There are several reasons, but the biggest one is that it’s a lot of work to get MATIC (on Polygon blockchain, use MATIC instead of ETH) ready to use. Simply put, you need to bridge your tokens to the Polygon chain. So people need to do this first before minting your NFTs on your website. It may not be a big deal for those who are familiar with crypto and blockchain, but it may be complicated for those who are not. I hope that the tedious work will be simplified in the near future.

If you are confident that your NFTs will be purchased even if they are expensive, or if the total supply of your NFTs is low, you may choose Ethereum. If you just want to save money, you should choose Polygon. Keep in mind that gas fees are incurred not only when minting, but also when performing some of your smart contract functions and when transferring your NFTs.
If I were to launch a new NFT project, I would probably choose Ethereum or Solana.

Conclusion

Some people may want to start an NFT project to make money, but don’t forget to enjoy your own project. Several months ago, I was playing with creating generative art by imitating the CryptoPunks. I found out that auto-generated artworks would be more interesting than I had imagined, and since then I’ve been completely absorbed in generative art.

This is one of the Skulls In Love artworks:

This character wears a cowboy hat, black slim sunglasses, and a kimono. If anyone looks like this, I can’t help laughing!

The Skulls In Love NFTs can be minted for a small amount of MATIC on the official website. Please give it a try to see what kind of unique characters will appear 💀💖

Thank you for reading to the end. I hope this article will be helpful to those who want to launch an NFT project in the future ✨

What Is the Howey Test?

To determine whether a transaction qualifies as a "investment contract" and thus qualifies as a security, the Howey Test refers to the U.S. Supreme Court cass: the Securities Act of 1933 and the Securities Exchange Act of 1934. According to the Howey Test, an investment contract exists when "money is invested in a common enterprise with a reasonable expectation of profits from others' efforts." 

The test applies to any contract, scheme, or transaction. The Howey Test helps investors and project backers understand blockchain and digital currency projects. ICOs and certain cryptocurrencies may be found to be "investment contracts" under the test.

Understanding the Howey Test

The Howey Test comes from the 1946 Supreme Court case SEC v. W.J. Howey Co. The Howey Company sold citrus groves to Florida buyers who leased them back to Howey. The company would maintain the groves and sell the fruit for the owners. Both parties benefited. Most buyers had no farming experience and were not required to farm the land. 

The SEC intervened because Howey failed to register the transactions. The court ruled that the leaseback agreements were investment contracts.

This established four criteria for determining an investment contract. Investing contract:

1. An investment of money
2. n a common enterprise
3. With the expectation of profit
4. To be derived from the efforts of others

In the case of Howey, the buyers saw the transactions as valuable because others provided the labor and expertise. An income stream was obtained by only investing capital. As a result of the Howey Test, the transaction had to be registered with the SEC.

Howey Test and Cryptocurrencies

Bitcoin is notoriously difficult to categorize. Decentralized, they evade regulation in many ways. Regardless, the SEC is looking into digital assets and determining when their sale qualifies as an investment contract.

The SEC claims that selling digital assets meets the "investment of money" test because fiat money or other digital assets are being exchanged. Like the "common enterprise" test. 

Whether a digital asset qualifies as an investment contract depends on whether there is a "expectation of profit from others' efforts."

For example, buyers of digital assets may be relying on others' efforts if they expect the project's backers to build and maintain the digital network, rather than a dispersed community of unaffiliated users. Also, if the project's backers create scarcity by burning tokens, the test is met. Another way the "efforts of others" test is met is if the project's backers continue to act in a managerial role.

These are just a few examples given by the SEC. If a project's success is dependent on ongoing support from backers, the buyer of the digital asset is likely relying on "others' efforts."

Special Considerations

If the SEC determines a cryptocurrency token is a security, many issues arise. It means the SEC can decide whether a token can be sold to US investors and forces the project to register. 

In 2017, the SEC ruled that selling DAO tokens for Ether violated federal securities laws. Instead of enforcing securities laws, the SEC issued a warning to the cryptocurrency industry. 

Due to the Howey Test, most ICOs today are likely inaccessible to US investors. After a year of ICOs, then-SEC Chair Jay Clayton declared them all securities. 

Howey Test FAQs

How Do You Determine If Something Is a Security?

The Howey Test determines whether certain transactions are "investment contracts." Securities are transactions that qualify as "investment contracts" under the Securities Act of 1933 and the Securities Exchange Act of 1934.

The Howey Test looks for a "investment of money in a common enterprise with a reasonable expectation of profits from others' efforts." If so, the Securities Act of 1933 and the Securities Exchange Act of 1934 require disclosure and registration.

Why Is Bitcoin Not a Security?

Former SEC Chair Jay Clayton clarified in June 2018 that bitcoin is not a security: "Cryptocurrencies: Replace the dollar, euro, and yen with bitcoin. That type of currency is not a security," said Clayton.

Bitcoin, which has never sought public funding to develop its technology, fails the SEC's Howey Test. However, according to Clayton, ICO tokens are securities. 

A Security Defined by the SEC

In the public and private markets, securities are fungible and tradeable financial instruments. The SEC regulates public securities sales.

The Supreme Court defined a security offering in SEC v. W.J. Howey Co. In its judgment, the court defines a security using four criteria:

• An investment contract's existence
• The formation of a common enterprise
• The issuer's profit promise
• Third-party promotion of the offering

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Many believe Putin is simply sabre rattling and intimidating us. They see no threat of nuclear war. We can send NATO troops into Ukraine without risking a nuclear war.

I keep reading that Putin is just using nuclear blackmail and that a strong leader will call the bluff. That, in my opinion, misunderstands the danger of sending NATO into Ukraine.
It assumes that once NATO moves in, Putin can either push the red nuclear button or not.
Sure, Putin won't go nuclear if NATO invades Ukraine. So we're safe? Can't we just move NATO?

No, because history has taught us that wars often escalate far beyond our initial expectations. One domino falls, knocking down another. That's why having clear boundaries is vital. Crossing a seemingly harmless line can set off a chain of events that are unstoppable once started.
One example is WWI. The assassin of Archduke Franz Ferdinand could not have known that his actions would kill millions. They couldn't have known that invading Serbia to punish them for not handing over the accomplices would start a world war. Every action triggered a counter-action, plunging Europe into a brutal and bloody war. Each leader saw their actions as limited, not realizing how they kept the dominos falling.

Nobody can predict the future, but it's easy to imagine how NATO intervention could trigger a chain of events leading to a total war. Let me suggest some outcomes.
NATO creates a no-fly-zone. In retaliation, Russia bombs NATO airfields. Russia may see this as a limited counter-move that shouldn't cause further NATO escalation. They think it's a reasonable response to force NATO out of Ukraine. Nobody has yet thought to use the nuke.
Will NATO act? Polish airfields bombed, will they be stuck? Is this an article 5 event? If so, what should be done?

It could happen. Maybe NATO sends troops into Ukraine to punish Russia. Maybe NATO will bomb Russian airfields.

Putin's response Is bombing Russian airfields an invasion or an attack? Remember that Russia has always used nuclear weapons for defense, not offense. But let's not panic, let's assume Russia doesn't go nuclear.

Maybe Russia retaliates by attacking NATO military bases with planes. Maybe they use ships to attack military targets. How does NATO respond? Will they fight Russia in Ukraine or escalate? Will they invade Russia or attack more military installations there?
Seen the pattern? As each nation responds, smaller limited military operations can grow in scope.

So far, the Russian military has shown that they begin with less brutal methods. As losses and failures increase, brutal means are used. Syria had the same. Assad used chemical weapons and attacked hospitals, schools, residential areas, etc.
A NATO invasion of Ukraine would cost Russia dearly. “Oh, this isn't looking so good, better pull out and finish this war,” do you think? No way. Desperate, they will resort to more brutal tactics. If desperate, Russia has a huge arsenal of ugly weapons. They have nerve agents, chemical weapons, and other nasty stuff.

What happens if Russia uses chemical weapons? What if Russian nerve agents kill NATO soldiers horribly? West calls for retaliation will grow. Will we invade Russia? Will we bomb them?

We are angry and determined to punish war criminal Putin, so NATO tanks may be heading to Moscow. We want vengeance for his chemical attacks and bombing of our cities.
Do you think the distance between that red nuclear button and Putin's finger will be that far once NATO tanks are on their way to Moscow?

We might avoid a nuclear apocalypse. A NATO invasion force or even Western cities may be used by Putin. Not as destructive as ICBMs. Putin may think we won't respond to tactical nukes with a full nuclear counterattack. Why would we risk a nuclear Holocaust by launching ICBMs on Russia?

Maybe. My point is that at every stage of the escalation, one party may underestimate the other's response. This war is spiraling out of control and the chances of a nuclear exchange are increasing. Nobody really wants it.

Fear, anger, and resentment cause it. If Putin and his inner circle decide their time is up, they may no longer care about the rest of the world. We saw it with Hitler. Hitler, seeing the end of his empire, ordered the destruction of Germany. Nobody should win if he couldn't. He wanted to destroy everything, including Paris.

In other words, the danger isn't what happens after NATO intervenes The danger is the potential chain reaction. Gambling has a psychological equivalent. It's best to exit when you've lost less. We humans are willing to take small risks for big rewards. To avoid losses, we are willing to take high risks. Daniel Kahneman describes this behavior in his book Thinking, Fast and Slow.

And so bettors who have lost a lot begin taking bigger risks to make up for it. We get a snowball effect. NATO involvement in the Ukraine conflict is akin to entering a casino and placing a bet. We'll start taking bigger risks as we start losing to Russian retaliation. That's the game's psychology.

It's impossible to stop. So will politicians and citizens from both Russia and the West, until we risk the end of human civilization.

You can avoid spiraling into ever larger bets in the Casino by drawing a hard line and declaring “I will not enter that Casino.” We're doing it now. We supply Ukraine. We send money and intelligence but don't cross that crucial line.

It's difficult to watch what happened in Bucha without demanding NATO involvement. What should we do? Of course, I'm not in charge. I'm a writer. My hope is that people will think about the consequences of the actions we demand. My hope is that you think ahead not just one step but multiple dominos.

More and more, we are driven by our emotions. We cannot act solely on emotion in matters of life and death. If we make the wrong choice, more people will die.

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