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.

"Web 3.0" is a trendy buzzword with a vague definition. Everyone agrees it has to do with a blockchain-based internet evolution, but what is it?

Yet, the meaning and prospects for Web3 have become hot topics in crypto communities. Big corporations use the term to gain a foothold in the space while avoiding the negative connotations of “crypto.”

But it can't be evaluated without a definition.

Among those criticizing Web3's vagueness is Cobie:

“Even the crypto community is split on whether Bitcoin is Web3,” he adds.

The phrase was coined by an early crypto thinker, and the community has had years to figure out what it means. Many ideologies and commercial realities have driven reverse engineering.

Web3 is becoming clearer as a concept. It contains ideas. It was probably coined by Ethereum co-founder Gavin Wood in 2014. His definition of Web3 included “trustless transactions” as part of its tech stack. Wood founded the Web3 Foundation and the Polkadot network, a Web3 alternative future.

The 2013 Ethereum white paper had previously allowed devotees to imagine a DAO, for example.

Web3 now has concepts like decentralized autonomous organizations, sovereign digital identity, censorship-free data storage, and data divided by multiple servers. They intertwine discussions about the “Web3” movement and its viability.

These ideas are linked by Cobie's initial Web3 definition. A key component of Web3 should be “ownership of value” for one's own content and data.

Noting that “late-stage capitalism greedcorps that make you buy a fractionalized micropayment NFT on Cardano to operate your electric toothbrush” may build the new web, he notes that “crypto founders are too rich to care anymore.”

Very Important

Many critics of Web3 claim it isn't practical or achievable. Web3 critics like Moxie Marlinspike (creator of sslstrip and Signal/TextSecure) can never see people running their own servers. Early in January, he argued that protocols are more difficult to create than platforms.

While this is true, some projects, like the file storage protocol IPFS, allow users to choose which jurisdictions their data is shared between.

But full decentralization is a difficult problem. Suhaza, replying to Moxie, said:

So, here are the questions: Web3: Is it a go? Is it truly decentralized?

Web3 history is shaped by Web2 failure.

This is the story of how the Internet was turned upside down...

Then came the vision. Everyone can create content for free. Decentralized open-source believers like Tim Berners-Lee popularized it.

Real-world data trade-offs for content creation and pricing.

A giant Wikipedia page married to a giant Craig's List. No ads, no logins, and a private web carve-up. For free usage, you give up your privacy and data to the algorithmic targeted advertising of Web 2.

Our data is centralized and savaged by giant corporations. Data localization rules and geopolitical walls like China's Great Firewall further fragment the internet.

The decentralized Web3 reflects Berners-original Lee's vision: "No permission is required from a central authority to post anything... there is no central controlling node and thus no single point of failure." Now he runs Solid, a Web3 data storage startup.

So Web3 starts with decentralized servers and data privacy.

Web3 begins with decentralized storage.

Data decentralization is a key feature of the Web3 tech stack. Web2 has closed databases. Large corporations like Facebook, Google, and others go to great lengths to collect, control, and monetize data. We want to change it.

Amazon, Google, Microsoft, Alibaba, and Huawei, according to Gartner, currently control 80% of the global cloud infrastructure market. Web3 wants to change that.

Decentralization enlarges power structures by giving participants a stake in the network. Users own data on open encrypted networks in Web3. This area has many projects.

Apps like Filecoin and IPFS have led the way. Data is replicated across multiple nodes in Web3 storage providers like Filecoin.

But the new tech stack and ideology raise many questions.

Giving users control over their data

According to Ryan Kris, COO of Verida, his “Web3 vision” is “empowering people to control their own data.”

Verida targets SDKs that address issues in the Web3 stack: identity, messaging, personal storage, and data interoperability.

A big app suite? “Yes, but it's a frontier technology,” he says. They are currently building a credentialing system for decentralized health in Bermuda.

By empowering individuals, how will Web3 create a fairer internet? Kris, who has worked in telecoms, finance, cyber security, and blockchain consulting for decades, admits it is difficult:

“The viability of Web3 raises some good business questions,” he adds. “How can users regain control over centralized personal data? How are startups motivated to build products and tools that support this transition? How are existing Web2 companies encouraged to pivot to a Web3 business model to compete with market leaders?

Kris adds that new technologies have regulatory and practical issues:

Each project has varying degrees of decentralization. The diehards say DApps that use centralized storage are no longer “Web3” companies. But fully decentralized technology is hard to build.

Web2.5?

Some argue that we're actually building Web2.5 businesses, which are crypto-native but not fully decentralized. This is vital. For example, the NFT may be on a blockchain, but it is linked to centralized data repositories like OpenSea. A server failure could result in data loss.

However, according to Apollo Capital crypto analyst David Angliss, OpenSea is “not exactly community-led”. Also in 2021, much to the chagrin of crypto enthusiasts, OpenSea tried and failed to list on the Nasdaq.

This is where Web2.5 is defined.

“Web3 isn't a crypto segment. “Anything that uses a blockchain for censorship resistance is Web3,” Angliss tells us.

“Web3 gives users control over their data and identity. This is not possible in Web2.”

“Web2 is like feudalism, with walled-off ecosystems ruled by a few. For example, an honest user owned the Instagram account “Meta,” which Facebook rebranded and then had to make up a reason to suspend. Not anymore with Web3. If I buy ‘Ethereum.ens,' Ethereum cannot take it away from me.”

Angliss uses OpenSea as a Web2.5 business example. Too decentralized, i.e. censorship resistant, can be unprofitable for a large company like OpenSea. For example, OpenSea “enables NFT trading”. But it also stopped the sale of stolen Bored Apes.”

Web3 (or Web2.5, depending on the context) has been described as a new way to privatize internet.

“Being in the crypto ecosystem doesn't make it Web3,” Angliss says. The biggest risk is centralized closed ecosystems rather than a growing Web3.

LooksRare and OpenDAO are two community-led platforms that are more decentralized than OpenSea. LooksRare has even been “vampire attacking” OpenSea, indicating a Web3 competitor to the Web2.5 NFT king could find favor.

The addition of a token gives these new NFT platforms more options for building customer loyalty. For example, OpenSea charges a fee that goes nowhere. Stakeholders of LOOKS tokens earn 100% of the trading fees charged by LooksRare on every basic sale.

Maybe Web3's time has come.

So whose data is it?

Continuing criticisms of Web3 platforms' decentralization may indicate we're too early. Users want to own and store their in-game assets and NFTs on decentralized platforms like the Metaverse and play-to-earn games. Start-ups like Arweave, Sia, and Aleph.im  propose an alternative.

To be truly decentralized, Web3 requires new off-chain models that sidestep cloud computing and Web2.5.

“Arweave and Sia emerged as formidable competitors this year,” says the Messari Report. They seek to reduce the risk of an NFT being lost due to a data breach on a centralized server.

Aleph.im, another Web3 cloud competitor, seeks to replace cloud computing with a service network. It is a decentralized computing network that supports multiple blockchains by retrieving and encrypting data.

“The Aleph.im network provides a truly decentralized alternative where it is most needed: storage and computing,” says Johnathan Schemoul, founder of Aleph.im. For reasons of consensus and security, blockchains are not designed for large storage or high-performance computing.

As a result, large data sets are frequently stored off-chain, increasing the risk for centralized databases like OpenSea

Aleph.im enables users to own digital assets using both blockchains and off-chain decentralized cloud technologies.

Aleph.im raised $10 million in mid-January 2022, and Ubisoft uses its network for NFT storage. This is the first time a big-budget gaming studio has given users this much control.

It also suggests Web3 could work as a B2B model, even if consumers aren't concerned about “decentralization.” Starting with gaming is common.

Can Tokenomics help Web3 adoption?

Web3 consumer adoption is another story. The average user may not be interested in all this decentralization talk. Still, how much do people value privacy over convenience? Can tokenomics solve the privacy vs. convenience dilemma?

Holon Global Investments' Jonathan Hooker tells us that human internet behavior will change. “Do you own Bitcoin?” he asks in his Web3 explanation. How does it feel to own and control your own sovereign wealth? Then:

“The business model must find what that person values,” he says. Putting their own health records on centralized systems they don't control?

“How vital are those medical records to that person at a critical time anywhere in the world? Filecoin and IPFS can help.”

Web3 adoption depends on NFT storage competition. A free off-chain storage of NFT metadata and assets was launched by Filecoin in April 2021.

Denationalization and blockchain technology have significant implications for data ownership and compensation for lending, staking, and using data. 

Tokenomics can change human behavior, but many people simply sign into Web2 apps using a Facebook API without hesitation. Our data is already owned by Google, Baidu, Tencent, and Facebook (and its parent company Meta). Is it too late to recover?

Maybe. “Data is like fruit, it starts out fresh but ages,” he says. "Big Tech's data on us will expire."

Web3 founder Kris agrees with Hooker that “value for data is the issue, not privacy.” People accept losing their data privacy, so tokenize it. People readily give up data, so why not pay for it?

"Personalized data offering is valuable in personalization. “I will sell my social media data but not my health data.”

Purists and mass consumer adoption struggle with key management.

Others question data tokenomics' optimism. While acknowledging its potential, Box founder Aaron Levie questioned the viability of Web3 models in a Tweet thread:

“Why? Because data almost always works in an app. A product and APIs that moved quickly to build value and trust over time.”

Levie contends that tokenomics may complicate matters. In addition to community governance and tokenomics, Web3 ideals likely add a new negotiation vector.

“These are hard problems about human coordination, not software or blockchains,”. Using a Facebook API is simple. The business model and user interface are crucial.

For example, the crypto faithful have a common misconception about logging into Web3. It goes like this: Web 1 had usernames and passwords. Web 2 uses Google, Facebook, or Twitter APIs, while Web 3 uses your wallet. Pay with Ethereum on MetaMask, for example.

But Levie is correct. Blockchain key management is stressed in this meme. Even seasoned crypto enthusiasts have heart attacks, let alone newbies.

Web3 requires a better user experience, according to Kris, the company's founder. “How does a user recover keys?”

And at this point, no solution is likely to be completely decentralized. So Web3 key management can be improved. ”The moment someone loses control of their keys, Web3 ceases to exist.”

That leaves a major issue for Web3 purists. Put this one in the too-hard basket.

Is 2022 the Year of Web3?

Web3 must first solve a number of issues before it can be mainstreamed. It must be better and cheaper than Web2.5, or have other significant advantages.

Web3 aims for scalability without sacrificing decentralization protocols. But decentralization is difficult and centralized services are more convenient.

Ethereum co-founder Vitalik Buterin himself stated recently"

This is why (centralized) Binance to Binance transactions trump Ethereum payments in some places because they don't have to be verified 12 times."

“I do think a lot of people care about decentralization, but they're not going to take decentralization if decentralization costs $8 per transaction,” he continued.

“Blockchains need to be affordable for people to use them in mainstream applications... Not for 2014 whales, but for today's users."

For now, scalability, tokenomics, mainstream adoption, and decentralization believers seem to be holding Web3 hostage.

Much like crypto's past.

But stay tuned.