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.

Financial regulators have for years attempted to apply existing laws to the multitude of issues created by digital assets. In 2021, leading federal regulators and members of Congress have begun to call for legislation to address these issues. As a result, 2022 may be the year when federal legislation finally addresses digital asset issues that have been growing since the mining of the first Bitcoin block in 2009.

Digital Asset Regulation in the Absence of Legislation

So far, Congress has left the task of addressing issues created by digital assets to regulatory agencies. Although a Congressional Blockchain Caucus formed in 2016, House and Senate members introduced few bills addressing digital assets until 2018. As of October 2021, Congress has not amended federal laws on financial regulation, which were last significantly revised by the Dodd-Frank Act in 2010, to address digital asset issues.

In the absence of legislation, issues that do not fit well into existing statutes have created problems. An example is the legal status of digital assets, which can be considered to be either securities or commodities, and can even shift from one to the other over time. Years after the SEC’s 2017 report applying the definition of a security to digital tokens, the SEC and the CFTC have yet to clarify the distinction between securities and commodities for the thousands of digital assets in existence.

SEC Chair Gary Gensler has called for Congress to act, stating in August, “We need additional Congressional authorities to prevent transactions, products, and platforms from falling between regulatory cracks.” Gensler has reached out to Sen. Elizabeth Warren (D-Ma.), who has expressed her own concerns about the need for legislation.

Legislation on Digital Assets in 2021

While regulators and members of Congress talked about the need for legislation, and the debate over cryptocurrency tax reporting in the 2021 infrastructure bill generated headlines, House and Senate bills proposing specific solutions to various issues quietly started to emerge.

Digital Token Sales

Several House bills attempt to address securities law barriers to digital token sales—some of them by building on ideas proposed by regulators in past years.

Exclusion from the definition of a security. Congressional Blockchain Caucus members have been introducing bills to exclude digital tokens from the definition of a security since 2018, and they have revived those bills in 2021. They include the Token Taxonomy Act of 2021 (H.R. 1628), successor to identically named bills in 2018 and 2019, and the Securities Clarity Act (H.R. 4451), successor to a 2020 namesake.

Safe harbor. SEC Commissioner Hester Peirce proposed a regulatory safe harbor for token sales in 2020, and two 2021 bills have proposed statutory safe harbors. Rep. Patrick McHenry (R-N.C.), Republican leader of the House Financial Services Committee, introduced a Clarity for Digital Tokens Act of 2021 (H.R. 5496) that would amend the Securities Act to create a safe harbor providing a grace period of exemption from Securities Act registration requirements. The Digital Asset Market Structure and Investor Protection Act (H.R. 4741) from Rep. Don Beyer (D-Va.) would amend the Securities Exchange Act to define a new type of security—a “digital asset security”—and add issuers of digital asset securities to an existing provision for delayed registration of securities.

Stablecoins

Stablecoins—digital currencies linked to the value of the U.S. dollar or other fiat currencies—have not yet been the subject of regulatory action, although Treasury Secretary Janet Yellen and Federal Reserve Chair Jerome Powell have each underscored the need to create a regulatory framework for them. The Beyer bill proposes to create a regulatory regime for stablecoins by amending Title 31 of the U.S. Code. Treasury Department approval would be required for any “digital asset fiat-based stablecoin” to be issued or used, under an application process to be established by Treasury in consultation with the Federal Reserve, the SEC, and the CFTC.

Serious consideration for any of these proposals in the current session of Congress may be unlikely. A spate of autumn bills on crypto ransom payments (S. 2666, S. 2923, S. 2926, H.R. 5501) shows that Congress is more inclined to pay attention first to issues that are more spectacular and less arcane. Moreover, the arcaneness of digital asset regulatory issues is likely only to increase further, now that major industry players such as Coinbase and Andreessen Horowitz are starting to roll out their own regulatory proposals.

Digital Dollar vs. Digital Yuan

Impetus to pass legislation on another type of digital asset, a central bank digital currency (CBDC), may come from a different source: rivalry with China.
China established itself as a world leader in developing a CBDC with a pilot project launched in 2020, and in 2021, the People’s Bank of China announced that its CBDC will be used at the Beijing Winter Olympics in February 2022. Republican Senators responded by calling for the U.S. Olympic Committee to forbid use of China’s CBDC by U.S. athletes in Beijing and introducing a bill (S. 2543) to require a study of its national security implications.

The Beijing Olympics could motivate a legislative mandate to accelerate implementation of a U.S. digital dollar, which the Federal Reserve has been in the process of considering in 2021. Antecedents to such legislation already exist. A House bill sponsored by 46 Republicans (H.R. 4792) has a provision that would require the Treasury Department to assess China’s CBDC project and report on the status of Federal Reserve work on a CBDC, and the Beyer bill includes a provision amending the Federal Reserve Act to authorize issuing a digital dollar.

Both parties are likely to support creating a digital dollar. The Covid-19 pandemic made a digital dollar for delivery of relief payments a popular idea in 2020, and House Democrats introduced bills with provisions for creating one in 2020 and 2021. Bipartisan support for a bill on a digital dollar, based on concerns both foreign and domestic in nature, could result.

International rivalry and bipartisan support may make the digital dollar a gateway issue for digital asset legislation in 2022. Legislative work on a digital dollar may open the door for considering further digital asset issues—including the regulatory issues that have been emerging for years—in 2022 and beyond.

Let's take a look back at the internet's history and see where we're going — and why.

Tim Berners Lee had a problem. He was at CERN, the world's largest particle physics factory, at the time. The institute's stated goal was to study the simplest particles with the most sophisticated scientific instruments. The institute completed the LEP Tunnel in 1988, a 27 kilometer ring. This was Europe's largest civil engineering project (to study smaller particles — electrons).

The problem Tim Berners Lee found was information loss, not particle physics. CERN employed a thousand people in 1989. Due to team size and complexity, people often struggled to recall past project information. While these obstacles could be overcome, high turnover was nearly impossible. Berners Lee addressed the issue in a proposal titled ‘Information Management'.

He had an idea. Create an information management system that allowed users to access data in a decentralized manner using a new technology called ‘hypertext'.
To quote Berners Lee, his proposal was “vague but exciting...”. The paper eventually evolved into the internet we know today. Here are three popular W3C standards used by billions of people today:

(credit: CERN)

HTML (Hypertext Markup)

A web formatting language.

URI (Unique Resource Identifier)

Each web resource has its own “address”. Known as ‘a URL'.

HTTP (Hypertext Transfer Protocol)

Retrieves linked resources from across the web.

These technologies underpin all computer work. They were the seeds of our quest to reorganize information, a task as fruitful as particle physics.

Tim Berners-Lee would probably think the three decades from 1989 to 2018 were eventful. He'd be amazed by the billions, the inspiring, the novel. Unlocking innovation at CERN through ‘Information Management'.
The fictional character would probably need a drink, walk, and a few deep breaths to fully grasp the internet's impact. He'd be surprised to see a few big names in the mix.

Then he'd say, "Something's wrong here."

We should review the web's history before going there. Was it a success after Berners Lee made it public? Web1 and Web2: What is it about what we are doing now that so many believe we need a new one, web3?

Per Outlier Ventures' Jamie Burke:

Let's explore.

Web1: The Read-Only Web

Web1 was the digital age. We put our books, research, and lives ‘online'. The web made information retrieval easier than any filing cabinet ever. Massive amounts of data were stored online. Encyclopedias, medical records, and entire libraries were put away into floppy disks and hard drives.

In 2015, the web had around 305,500,000,000 pages of content (280 million copies of Atlas Shrugged).

Initially, one didn't expect to contribute much to this database. Web1 was an online version of the real world, but not yet a new way of using the invention.

That doesn't mean developers weren't building. The web was being advanced by great minds. Web2 was born as technology advanced.

Web2: Read-Write Web

Remember when you clicked something on a website and the whole page refreshed? Is it too early to call the mid-2000s ‘the good old days'?
Browsers improved gradually, then suddenly. AJAX calls augmented CGI scripts, and applications began sending data back and forth without disrupting the entire web page. One button to ‘digg' a post (see below). Web experiences blossomed.

In 2006, Digg was the most active ‘Web 2.0' site. (Photo: Ethereum Foundation Taylor Gerring)

Interaction was the focus of new applications. Posting, upvoting, hearting, pinning, tweeting, liking, commenting, and clapping became a lexicon of their own. It exploded in 2004. Easy ways to ‘write' on the internet grew, and continue to grow.

Facebook became a Web2 icon, where users created trillions of rows of data. Google and Amazon moved from Web1 to Web2 by better understanding users and building products and services that met their needs.

Business models based on Software-as-a-Service and then managing consumer data within them for a fee have exploded.

Web2 Emerging Issues

Unbelievably, an intriguing dilemma arose. When creating this read-write web, a non-trivial question skirted underneath the covers. Who owns it all?

A good plot line emerges. Many amazing, world-changing software products quietly lost users' data control.
For example: Facebook owns much of your social graph data. Even if you hate Facebook, you can't leave without giving up that data. There is no ‘export' or ‘exit'. The platform owns ownership.

While many companies can pull data on you, you cannot do so.

On the surface, this isn't an issue. These companies use my data better than I do! A complex group of stakeholders, each with their own goals. One is maximizing shareholder value for public companies. Tim Berners-Lee (and others) dislike the incentives created.

It's easy to see what the read-write web has allowed in retrospect. We've been given the keys to create content instead of just consume it. On Facebook and Twitter, anyone with a laptop and internet can participate. But the engagement isn't ours. Platforms own themselves.

Web3: The ‘Unmediated’ Read-Write Web

Tim Berners Lee proposed a decade ago that ‘linked data' could solve the internet's data problem.

The Web of Data also allows for new domain-specific applications. Unlike Web 2.0 mashups, Linked Data applications work with an unbound global data space. As new data sources appear on the Web, they can provide more complete answers.

At around the same time as linked data research began, Satoshi Nakamoto created Bitcoin. After ten years, it appears that Berners Lee's ideas ‘link' spiritually with cryptocurrencies.

What should Web 3 do?

Here are some quick predictions for the web's future.

Users' data:
Users own information and provide it to corporations, businesses, or services that will benefit them.

Defying censorship:

No government, company, or institution should control your access to information (1, 2, 3)

Connect users and platforms:

Create symbiotic rather than competitive relationships between users and platform creators.

Open networks:

“First, the cryptonetwork-participant contract is enforced in open source code. Their voices and exits are used to keep them in check.” Dixon, Chris (4)

Global interactivity:

Transacting value, information, or assets with anyone with internet access, anywhere, at low cost

Self-determination:

Giving you the ability to own, see, and understand your entire digital identity.

Not pull, push:

‘Push' your data to trusted sources instead of ‘pulling' it from others.

Where Does This Leave Us?

People believe web3 can help build a better, fairer system. This is not the same as equal pay or outcomes, but more equal opportunity.

It should be noted that some of these advantages have been discussed previously. Will the changes work? Will they make a difference? These unanswered questions are technical, economic, political, and philosophical. Unintended consequences are likely.

We hope Web3 is a more democratic web. And we think incentives help the user. If there’s one thing that’s on our side, it’s that open has always beaten closed, given a long enough timescale.

We are at the start.