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.

The international law firm Holland & Knight used an NFT built and airdropped by its asset recovery team to serve a defendant in a hacking case.

The law firms Holland & Knight and Bluestone used a nonfungible token to serve a defendant in a hacking case with a temporary restraining order, marking the first documented legal process assisted by an NFT.

The so-called "service token" or "service NFT" was served to an unknown defendant in a hacking case involving LCX, a cryptocurrency exchange based in Liechtenstein that was hacked for over $8 million in January. The attack compromised the platform's hot wallets, resulting in the loss of Ether (ETH), USD Coin (USDC), and other cryptocurrencies, according to Cointelegraph at the time.

On June 7, LCX claimed that around 60% of the stolen cash had been frozen, with investigations ongoing in Liechtenstein, Ireland, Spain, and the United States. Based on a court judgment from the New York Supreme Court, Centre Consortium, a company created by USDC issuer Circle and crypto exchange Coinbase, has frozen around $1.3 million in USDC.

The monies were laundered through Tornado Cash, according to LCX, but were later tracked using "algorithmic forensic analysis." The organization was also able to identify wallets linked to the hacker as a result of the investigation.

In light of these findings, the law firms representing LCX, Holland & Knight and Bluestone, served the unnamed defendant with a temporary restraining order issued on-chain using an NFT. According to LCX, this system "was allowed by the New York Supreme Court and is an example of how innovation can bring legitimacy and transparency to a market that some say is ungovernable."

I quit my 8-year Amazon job last week. I wasn't motivated to do another year despite promotions, pay, recognition, and praise.

In AWS, I built developer tools. I could have worked in that field forever.

I became an Amazon developer. Within 3.5 years, I was promoted twice to senior engineer and would have been promoted to principal engineer if I stayed. The company said I had great potential.

Over time, I became a reputed expert and leader within the company. I was respected.

First year I made $75K, last year $511K. If I stayed another two years, I could have made $1M.

Despite Amazon's reputation, my work–life balance was good. I no longer needed to prove myself and could do everything in 40 hours a week. My team worked from home once a week, and I rarely opened my laptop nights or weekends.

My coworkers were great. I had three generous, empathetic managers. I’m very grateful to everyone I worked with.

Everything was going well and getting better. My motivation to go to work each morning was declining despite my career and income growth.

Another promotion, pay raise, or big project wouldn't have boosted my motivation. Motivation was also waning. It was my freedom.

Demotivation

My motivation was high in the beginning. I worked with someone on an internal tool with little scrutiny. I had more freedom to choose how and what to work on than in recent years. Me and another person improved it, talked to users, released updates, and tested it. Whatever we wanted, we did. We did our best and were mostly self-directed.

In recent years, things have changed. My department's most important project had many stakeholders and complex goals. What I could do depended on my ability to convince others it was the best way to achieve our goals.

Amazon was always someone else's terms. The terms started out simple (keep fixing it), but became more complex over time (maximize all goals; satisfy all stakeholders). Working in a large organization imposed restrictions on how to do the work, what to do, what goals to set, and what business to pursue. This situation forced me to do things I didn't want to do.

Finding New Motivation

What would I do forever? Not something I did until I reached a milestone (an exit), but something I'd do until I'm 80. What could I do for the next 45 years that would make me excited to wake up and pay my bills? Is that too unambitious? Nope. Because I'm motivated by two things.

One is an external carrot or stick. I'm not forced to file my taxes every April, but I do because I don't want to go to jail. Or I may not like something but do it anyway because I need to pay the bills or want a nice car. Extrinsic motivation

One is internal. When there's no carrot or stick, this motivates me. This fuels hobbies. I wanted a job that was intrinsically motivated.

Is this too low-key? Extrinsic motivation isn't sustainable. Getting promoted felt good for a week, then it was over. When I hit $100K, I admired my W2 for a few days, but then it wore off. Same thing happened at $200K, $300K, $400K, and $500K. Earning $1M or $10M wouldn't change anything. I feel the same about every material reward or possession. Getting them feels good at first, but quickly fades.

Things I've done since I was a kid, when no one forced me to, don't wear off. Coding, selling my creations, charting my own path, and being honest. Why not always use my strengths and motivation? I'm lucky to live in a time when I can work independently in my field without large investments. So that’s what I’m doing.

What’s Next?

I'm going all-in on independence and will make a living from scratch. I won't do only what I like, but on my terms. My goal is to cover my family's expenses before my savings run out while doing something I enjoy. What more could I want from my work?

You can now follow me on Twitter as I continue to document my journey.

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