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.

Azuki Banner (www.azuki.com)

Disclaimer: This is my personal viewpoint. I'm not on the Azuki team. Please keep in mind that I am merely a fan, community member, and holder. Please do your own research and pardon my grammar. Thanks!

Azuki has changed my view of NFTs.

When I first entered the NFT world, I had no idea what to expect. I liked the idea. So I invested in some projects, fought for whitelists, and discovered some cool NFTs projects (shout-out to CATC). I lost more money than I earned at one point, but I hadn't invested excessively (only put in what you can afford to lose). Despite my losses, I kept looking. I almost waited for the “ah-ha” moment. A NFT project that changed my perspective on NFTs. What makes an NFT project more than a work of art?

Answer: Azuki.

The Art

The Azuki art drew me in as an anime fan. It looked like something out of an anime, and I'd never seen it before in NFT.
The project was still new. The first two animated teasers were released with little fanfare, but I was impressed with their quality. You can find them on Instagram or in their earlier Tweets.

The teasers hinted that this project could be big and that the team could deliver. It was amazing to see Shao cut the Azuki posters with her katana. Especially at the end when she sheaths her sword and the music cues. Then the live action video of the young boy arranging the Azuki posters seemed movie-like. I felt like I was entering the Azuki story, brand, and dope theme.

The team did not disappoint with the Azuki NFTs. The level of detail in the art is stunning. There were Azukis of all genders, skin and hair types, and more. These 10,000 Azukis have so much representation that almost anyone can find something that resonates. Rather than me rambling on, I suggest you visit the Azuki gallery

The Team

If the art is meant to draw you in and be the project's face, the team makes it more. The NFT would be a JPEG without a good team leader. Not that community isn't important, but no community would rally around a bad team.

Because I've been rugged before, I'm very focused on the team when considering a project. While many project teams are anonymous, I try to find ones that are doxxed (public) or at least appear to be established. Unlike Azuki, where most of the Azuki team is anonymous, Steamboy is public. He is (or was) Overwatch's character art director and co-creator of Azuki. I felt reassured and could trust the project after seeing someone from a major game series on the team.

Then I tried to learn as much as I could about the team. Following everyone on Twitter, reading their tweets, and listening to recorded AMAs. I was impressed by the team's professionalism and dedication to their vision for Azuki, led by ZZZAGABOND.
I believe the phrase “actions speak louder than words” applies to Azuki. I can think of a few examples of what the Azuki team has done, but my favorite is ERC721A.

I was ecstatic when the dev team announced it. This fascinates me as a self-taught developer. Azuki released a product that saves people money, improves the NFT space, and is open source. It showed their love for Azuki and the NFT community.

The Community

Community, community, community. It's almost a chant in the NFT space now. A community, like a team, can make or break a project. We are the project's consumers, shareholders, core, and lifeblood. The team builds the house, and we fill it. We stay for the community.
When I first entered the Azuki Discord, I was surprised by the calm atmosphere. There was no news about the project. No release date, no whitelisting requirements. No grinding or spamming either. People just wanted to hangout, get to know each other, and talk. It was nice. So the team could pick genuine people for their mintlist (aka whitelist).
But nothing fundamental has changed since the release. It has remained an authentic, fun, and helpful community. I'm constantly logging into Discord to chat with others or follow conversations. I see the community's openness to newcomers. Everyone respects each other (barring a few bad apples) and the variety of people passing through is fascinating. This human connection and interaction is what I enjoy about this place. Being a part of a group that supports a cause.
Finally, I want to thank the amazing Azuki mod team and the kissaten channel for their contributions.

The Brand

So, what sets Azuki apart from other projects? They are shaping a brand or identity. The Azuki website, I believe, best captures their vision. (This is me gushing over the site.)

If you go to the website, turn on the dope playlist in the bottom left. The playlist features a mix of Asian and non-Asian hip-hop and rap artists, with some lo-fi thrown in. The songs on the playlist change, but I think you get the vibe Azuki embodies just by turning on the music.
The Garden is our next stop where we are introduced to Azuki.

A brand.

The text begins with Azuki's intention in the space. It's a community-made metaverse brand. Then it goes into more detail about Azuki's plans. Initiation of a story or journey. "Would you like to take the red bean and jump down the rabbit hole with us?" I love the Matrix red pill or blue pill play they used. (Azuki in Japanese means red bean.)

It's intriguing and draws you in. Taking the red bean causes what? Where am I going? I think they did well in piqueing a newcomer's interest.
Not convinced by the Garden? Read the Manifesto. It reinforces Azuki's role.

Not to mention the Mindmap, it sets Azuki apart from other projects and overused Roadmaps. I like how the team recognizes that the NFT space is not linear. So many of us are still trying to figure it out. It is Azuki's vision to adapt to changing environments while maintaining their values. I admire their commitment to long-term growth.

Conclusion

To be honest, I have no idea what the future holds. Azuki is still new and could fail. But I'm a long-term Azuki fan. I don't care about quick gains. The future looks bright for Azuki. I believe in the team's output. I love being an Azuki.
Thank you! IKUZO!

Full post here

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.