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.

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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Everyone is talking about the $UST attack right now, including Janet Yellen. But no one is talking about how much money the attacker made (or how brilliant it was). Lets dig in.

Our story starts in late March, when the Luna Foundation Guard (or LFG) starts buying BTC to help back $UST. LFG started accumulating BTC on 3/22, and by March 26th had a $1bn+ BTC position. This is leg #1 that made this trade (or attack) brilliant.

The second leg comes in the form of the 4pool Frax announcement for $UST on April 1st. This added the second leg needed to help execute the strategy in a capital efficient way (liquidity will be lower and then the attack is on).

We don't know when the attacker borrowed 100k BTC to start the position, other than that it was sold into Kwon's buying (still speculation). LFG bought 15k BTC between March 27th and April 11th, so lets just take the average price between these dates ($42k).

So you have a ~$4.2bn short position built. Over the same time, the attacker builds a $1bn OTC position in $UST. The stage is now set to create a run on the bank and get paid on your BTC short. In anticipation of the 4pool, LFG initially removes $150mm from 3pool liquidity.

The liquidity was pulled on 5/8 and then the attacker uses $350mm of UST to drain curve liquidity (and LFG pulls another $100mm of liquidity).

But this only starts the de-pegging (down to 0.972 at the lows). LFG begins selling $BTC to defend the peg, causing downward pressure on BTC while the run on $UST was just getting started.

With the Curve liquidity drained, the attacker used the remainder of their $1b OTC $UST position ($650mm or so) to start offloading on Binance. As withdrawals from Anchor turned from concern into panic, this caused a real de-peg as people fled for the exits

So LFG is selling $BTC to restore the peg while the attacker is selling $UST on Binance. Eventually the chain gets congested and the CEXs suspend withdrawals of $UST, fueling the bank run panic. $UST de-pegs to 60c at the bottom, while $BTC bleeds out.

The crypto community panics as they wonder how much $BTC will be sold to keep the peg. There are liquidations across the board and LUNA pukes because of its redemption mechanism (the attacker very well could have shorted LUNA as well). BTC fell 25% from $42k on 4/11 to $31.3k

So how much did our attacker make? There aren't details on where they covered obviously, but if they are able to cover (or buy back) the entire position at ~$32k, that means they made $952mm on the short.

On the $350mm of $UST curve dumps I don't think they took much of a loss, lets assume 3% or just $11m. And lets assume that all the Binance dumps were done at 80c, thats another $125mm cost of doing business. For a grand total profit of $815mm (bf borrow cost).

BTC was the perfect playground for the trade, as the liquidity was there to pull it off. While having LFG involved in BTC, and foreseeing they would sell to keep the peg (and prevent LUNA from dying) was the kicker.

Lastly, the liquidity being low on 3pool in advance of 4pool allowed the attacker to drain it with only $350mm, causing the broader panic in both BTC and $UST. Any shorts on LUNA would've added a lot of P&L here as well, with it falling -65% since 5/7.

And for the reply guys, yes I know a lot of this involves some speculation & assumptions. But a lot of money was made here either way, and I thought it would be cool to dive into how they did it.

I've sold over 40,000 non-fiction books on Amazon and made over $160,000 in six years while writing on the side.

I have a full-time job and three young sons; I can't spend 40 hours a week writing. This article describes my journey.

I write mainly tech books:

Thanks to my readers, many wrote positive evaluations. Several are bestsellers.

A few have been adopted by universities as textbooks:

My books' passive income allows me more time with my family.

Knowing I could quit my job and write full time gave me more confidence. And I find purpose in my work (i am in christian ministry).

I'm always eager to write. When work is a dread or something bad happens, writing gives me energy. Writing isn't scary. In fact, I can’t stop myself from writing!

Writing has also established my tech authority. Universities use my books, as I've said. Traditional publishers have asked me to write books.

These mindsets helped me become a successful nonfiction author:

1. You don’t have to be an Authority

Yes, I have computer science experience. But I'm no expert on my topics. Before authoring "Beginning Node.js, Express & MongoDB," my most profitable book, I had no experience with those topics. Node was a new server-side technology for me. Would that stop me from writing a book? It can. I liked learning a new technology. So I read the top three Node books, took the top online courses, and put them into my own book (which makes me know more than 90 percent of people already).

I didn't have to worry about using too much jargon because I was learning as I wrote. An expert forgets a beginner's hardship.

2. Solve a micro-problem (Niching down)

I didn't set out to write a definitive handbook. I found a market with several challenges and wrote one book. Ex:

• Instead of web development, what about web development using Angular?
• Instead of Blockchain, what about Blockchain using Solidity and React?
• Instead of cooking recipes, how about a recipe for a specific kind of diet?
• Instead of Learning math, what about Learning Singapore Math?

3. Piggy Backing Trends

The above topics may still be a competitive market. E.g.  Angular, React.   To stand out, include the latest technologies or trends in your book. Learn iOS 15 instead of iOS programming. Instead of personal finance, what about personal finance with NFTs.

Even though you're a newbie author, your topic is well-known.

4. Publish short books

My books are known for being direct. Many people like this:

Your reader will appreciate you cutting out the fluff and getting to the good stuff. A reader can finish and review your book.

Second, short books are easier to write. Instead of creating a 500-page book for $50 (which few will buy), write a 100-page book that answers a subset of the problem and sell it for less. (You make less, but that's another subject). At least it got published instead of languishing. Less time spent creating a book means less time wasted if it fails. Write a small-bets book portfolio like Daniel Vassallo!

Third, it's $2.99-$9.99 on Amazon (gets 70 percent royalties for ebooks). Anything less receives 35% royalties. $9.99 books have 20,000–30,000 words. If you write more and charge more over $9.99, you get 35% royalties. Why not make it a $9.99 book?

(This is the ebook version.) Paperbacks cost more. Higher royalties allow for higher prices.

5. Validate book idea

Amazon will tell you if your book concept, title, and related phrases are popular. See? Check its best-sellers list.

150,000 is preferable. It sells 2–3 copies daily. Consider your rivals. Profitable niches have high demand and low competition.

Don't be afraid of competitive niches. First, it shows high demand. Secondly, what are the ways you can undercut the completion? Better book? Or cheaper option? There was lots of competition in my NodeJS book's area. None received 4.5 stars or more. I wrote a NodeJS book. Today, it's a best-selling Node book.

What’s Next

So long. Part II follows. Meanwhile, I will continue to write more books!

Follow my journey on Twitter.

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