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.

Everywhere we turn, there's a new metaverse or Web3 debut. Microsoft recently announced a $68.7 BILLION cash purchase of Activision.

Like AI in 2013 and blockchain in 2014, NFT growth in 2021 feels like this year's metaverse and Web3 growth. We are all bombarded with information, conflicting signals, and a sensation of FOMO.

How can we evaluate the metaverse and Web3 in a noisy, new world? My framework for evaluating upcoming technologies and themes is shown below. I hope you will also find them helpful.

Understand the “pipes” in a new space. 

Whatever people say, Metaverse and Web3 will have to coexist with the current Internet. Companies who host, move, and store data over the Internet have a lot of intriguing use cases in Metaverse and Web3, whether in infrastructure, data analytics, or compliance. Hence the following point.

## Understand the apps layer and their infrastructure.

Gaming, crypto exchanges, and NFT marketplaces would not exist today if not for technology that enables rapid app creation. Yes, according to Chainalysis and other research, 30–40% of Ethereum is self-hosted, with the rest hosted by large cloud providers. For Microsoft to acquire Activision makes strategic sense. It's not only about the games, but also the infrastructure that supports them.

Follow the money

Understanding how money and wealth flow in a complex and dynamic environment helps build clarity. Unless you are exceedingly wealthy, you have limited ability to significantly engage in the Web3 economy today. Few can just buy 10 ETH and spend it in one day. You must comprehend who benefits from the process, and how that 10 ETH circulates now and possibly tomorrow. Major holders and players control supply and liquidity in any market. Today, most Web3 apps are designed to increase capital inflow so existing significant holders can utilize it to create a nascent Web3 economy. When you see a new Metaverse or Web3 application, remember how money flows.

What is the use case? 

What does the app do? If there is no clear use case with clear makers and consumers solving a real problem, then the euphoria soon fades, and the only stakeholders who remain enthused are those who have too much to lose.

Time is a major competition that is often overlooked.

We're only busier, but each day is still 24 hours. Using new apps may mean that time is lost doing other things. The user must be eager to learn. Metaverse and Web3 vs. our time?  I don't think we know the answer yet (at least for working adults whose cost of time is higher).
I don't think we know the answer yet (at least for working adults whose cost of time is higher).

People and organizations need security and transparency.

For new technologies or apps to be widely used, they must be safe, transparent, and trustworthy. What does secure Metaverse and Web3 mean? This is an intriguing subject for both the business and public sectors. Cloud adoption grew in part due to improved security and data protection regulations.

I write on VC, startups, and leadership.

More on https://www.linkedin.com/in/joycejshen/ and https://joyceshen.substack.com/

This writing is my own opinion and does not represent investment advice.

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Grow Your Donations Tax-Free.

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Avoid a Windfall Tax Year.

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Make a contribution to reduce or eliminate capital gains.

One of the most common ways to fund a DAF is by gifting publicly traded securities. Securities held for more than a year can be donated at fair market value and are not subject to capital gains tax. If a donor liquidates assets and then donates the proceeds to their DAF, capital gains tax reduces the amount available for philanthropy. Gifts of appreciated securities, mutual funds, real estate, and other assets are immediately tax deductible up to 30% of Adjusted gross income (AGI), with a five-year carry-forward for gifts that exceed AGI limits.

Using Appreciated Stock as a Gift

Donating appreciated stock directly to a DAF rather than liquidating it and donating the proceeds reduces philanthropists' tax liability by eliminating capital gains tax and lowering marginal income tax.

In the example below, a donor has $100,000 in long-term appreciated stock with a cost basis of $10,000:

Using a DAF would allow this donor to give more to charity while paying less taxes. This strategy often allows donors to give more than 20% more to their favorite causes.

The information provided here is general and educational in nature. It is not intended to be, nor should it be construed as, legal or tax advice. NPT does not provide legal or tax advice. Furthermore, the content provided here is related to taxation at the federal level only. NPT strongly encourages you to consult with your tax advisor or attorney before making charitable contributions.