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.

NFTs look ready to rule 2022, and the recent pivot toward NFT utility in P2E gaming could make blockchain gaming this year’s sector darling.

After the popularity of decentralized finance (DeFi) came the rise of nonfungible tokens (NFTs), and to the surprise of many, NFTs took the spotlight and now remain front and center with the highest volume in sales occurring at the start of January 2022.
While 2021 became the year of NFTs, GameFi applications did surpass DeFi in terms of user popularity. According to data from DappRadar, Bloomberg gathered:

Blockchain play-to-earn (P2E) game Axie infinity skyrocketed and kicked off a gaming craze that is expected to continue all throughout 2022. Crypto pundits and gaming advocates have high expectations for P2E blockchain-based games and there’s bound to be a few sleeping giants that will dominate the sector.

Let’s take a look at five blockchain games that could make waves in 2022.

DeFi Kingdoms

The inspiration for DeFi Kingdoms came from simple beginnings — a passion for investing that lured the developers to blockchain technology. DeFi Kingdoms was born as a visualization of liquidity pool investing where in-game ‘gardens’ represent literal and figurative token pairings and liquidity pool mining.

As shown in the game, investors have a portion of their LP share within a plot filled with blooming plants. By attaching the concept of growth to DeFi protocols within a play-and-earn model, DeFi Kingdoms puts a twist on “playing” a game.

Built on the Harmony Network, DeFi Kingdoms became the first project on the network to ever top the DappRadar charts. This could be attributed to an influx of individuals interested in both DeFi and blockchain games or it could be attributed to its recent in-game utility token JEWEL surging.

JEWEL is a utility token that allows users to purchase NFTs in-game buffs to increase a base-level stat. It is also used for liquidity mining to grant users the opportunity to make more JEWEL through staking.

JEWEL is also a governance token that gives holders a vote in the growth and evolution of the project. In the past four months, the token price surged from $1.23 to an all-time high of $22.52. At the time of writing, JEWEL is down by nearly 16%, trading at $19.51.

Surging approximately 1,487% from its humble start of $1.23 four months ago in September, JEWEL token price has increased roughly 165% this last month alone, according to data from CoinGecko.

Guild of Guardians

Guild of Guardians is one of the more anticipated blockchain games in 2022 and it is built on ImmutableX, the first layer-two solution built on Ethereum that focuses on NFTs. Aiming to provide more access, it will operate as a free-to-play mobile role-playing game, modeling the P2E mechanics.

Similar to blockchain games like Axie Infinity, Guild of Guardians in-game assets can be exchanged. The project seems to be of interest to many gamers and investors with its NFT founder sale and token launch generating nearly $10 million in volume.

Launching its in-game token in October of 2021, the Guild of Guardians (GOG) tokens are ERC-20 tokens known as ‘gems’ inside the game. Gems are what power key features in the game such as minting in-game NFTs and interacting with the marketplace, and are available to earn while playing.

For the last month, the Guild of Guardians token has performed rather steadily after spiking to its all-time high of $2.81 after its launch. Despite the token being down over 50% from its all-time high, at the time of writing, some members of the community are looking forward to the possibility of staking and liquidity pools, which are features that tend to help stabilize token prices.

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