More on NFTs & Art
Jayden Levitt
3 years ago
Starbucks' NFT Project recently defeated its rivals.
The same way Amazon killed bookstores. You just can’t see it yet.
Shultz globalized coffee. Before Starbucks, coffee sucked.
All accounts say 1970s coffee was awful.
Starbucks had three stores selling ground Indonesian coffee in the 1980s.
What a show!
A year after joining the company at 29, Shultz traveled to Italy for R&D.
He noticed the coffee shops' sense of theater and community and realized Starbucks was in the wrong business.
Integrating coffee and destination created a sense of community in the store.
Brilliant!
He told Starbucks' founders about his experience.
They disapproved.
For two years.
Shultz left and opened an Italian coffee shop chain like any good entrepreneur.
Starbucks ran into financial trouble, so the founders offered to sell to Shultz.
Shultz bought Starbucks in 1987 for $3.8 million, including six stores and a payment plan.
Starbucks is worth $100.79Billion, per Google Finance.
26,500 times Shultz's initial investment
Starbucks is releasing its own NFT Platform under Shultz and his early Vision.
This year, Starbucks Odyssey launches. The new digital experience combines a Loyalty Rewards program with NFT.
The side chain Polygon-based platform doesn't require a Crypto Wallet. Customers can earn and buy digital assets to unlock incentives and experiences.
They've removed all friction, making it more immersive and convenient than a coffee shop.
Brilliant!
NFTs are the access coupon to their digital community, but they don't highlight the technology.
They prioritize consumer experience by adding non-technical users to Web3. Their collectables are called journey stamps, not NFTs.
No mention of bundled gas fees.
Brady Brewer, Starbucks' CMO, said;
“It happens to be built on blockchain and web3 technologies, but the customer — to be honest — may very well not even know that what they’re doing is interacting with blockchain technology. It’s just the enabler,”
Rewards members will log into a web app using their loyalty program credentials to access Starbucks Odyssey. They won't know about blockchain transactions.
Starbucks has just dealt its rivals a devastating blow.
It generates more than ten times the revenue of its closest competitor Costa Coffee.
The coffee giant is booming.
Starbucks is ahead of its competitors. No wonder.
They have an innovative, adaptable leadership team.
Starbucks' DNA challenges the narrative, especially when others reject their ideas.
I’m off for a cappuccino.
Jennifer Tieu
3 years ago
Why I Love Azuki
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.
With ERC721A, Azuki has created a new algorithm that allows minting multiple NFTs for essentially the same cost as minting one NFT.
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.
We're creating a new brand together.
A metaverse brand. By the people.
A collection of 10,000 avatars that grant Garden membership. It starts with exclusive streetwear collabs, NFT drops, live events, and more. Azuki allows for a new media genre that the world has yet to discover. Let's build together an Azuki, your metaverse identity.
The Garden is a magical internet corner where art, community, and culture collide. The boundaries between the physical and digital worlds are blurring.
Try a Red Bean.
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.)
Morpheus, the rebel leader, offers Neo the choice of a red or blue pill in The Matrix. “You take the blue pill... After the story, you go back to bed and believe whatever you want. Your red pill... Let me show you how deep the rabbit hole goes.” Aware that the red pill will free him from the enslaving control of the machine-generated dream world and allow him to escape into the real world, he takes it. However, living the “truth of reality” is harsher and more difficult.
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.
Here comes a new wave…
And surfing here is different.
Breaking down barriers.
Building open communities.
Creating magic internet money with our friends.
To those who don’t get it, we tell them: gm.
They’ll come around eventually.
Here’s to the ones with the courage to jump down a peculiar rabbit hole.
One that pulls you away from a world that’s created by many and owned by few…
To a world that’s created by more and owned by all.
From The Garden come the human beans that sprout into your family.
We rise together.
We build together.
We grow together.
Ready to take the red bean?
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
Scott Duke Kominers
3 years ago
NFT Creators Go Creative Commons Zero (cc0)
On January 1, "Public Domain Day," thousands of creative works immediately join the public domain. The original creator or copyright holder loses exclusive rights to reproduce, adapt, or publish the work, and anybody can use it. It happens with movies, poems, music, artworks, books (where creative rights endure 70 years beyond the author's death), and sometimes source code.
Public domain creative works open the door to new uses. 400,000 sound recordings from before 1923, including Winnie-the-Pooh, were released this year. With most of A.A. Milne's 1926 Winnie-the-Pooh characters now available, we're seeing innovative interpretations Milne likely never planned. The ancient hyphenated version of the honey-loving bear is being adapted for a horror movie: "Winnie-the-Pooh: Blood and Honey"... with Pooh and Piglet as the baddies.
Counterintuitively, experimenting and recombination can occasionally increase IP value. Open source movements allow the public to build on (or fork and duplicate) existing technologies. Permissionless innovation helps Android, Linux, and other open source software projects compete. Crypto's success at attracting public development is also due to its support of open source and "remix culture," notably in NFT forums.
Production memes
NFT projects use several IP strategies to establish brands, communities, and content. Some preserve regular IP protections; others offer NFT owners the opportunity to innovate on connected IP; yet others have removed copyright and other IP safeguards.
By using the "Creative Commons Zero" (cc0) license, artists can intentionally select for "no rights reserved." This option permits anyone to benefit from derivative works without legal repercussions. There's still a lot of confusion between copyrights and NFTs, so nothing here should be considered legal, financial, tax, or investment advice. Check out this post for an overview of copyright vulnerabilities with NFTs and how authors can protect owners' rights. This article focuses on cc0.
Nouns, a 2021 project, popularized cc0 for NFTs. Others followed, including: A Common Place, Anonymice, Blitmap, Chain Runners, Cryptoadz, CryptoTeddies, Goblintown, Gradis, Loot, mfers, Mirakai, Shields, and Terrarium Club are cc0 projects.
Popular crypto artist XCOPY licensed their 1-of-1 NFT artwork "Right-click and Save As Guy" under cc0 in January, exactly one month after selling it. cc0 has spawned many derivatives.
"Right-click Save As Guy" by XCOPY (1)/derivative works (2)
XCOPY said Monday he would apply cc0 to "all his existing art." "We haven't seen a cc0 summer yet, but I think it's approaching," said the artist. - predicting a "DeFi summer" in 2020, when decentralized finance gained popularity.
Why do so many NFT authors choose "no rights"?
Promoting expansions of the original project to create a more lively and active community is one rationale. This makes sense in crypto, where many value open sharing and establishing community.
Creativity depends on cultural significance. NFTs may allow verifiable ownership of any digital asset, regardless of license, but cc0 jumpstarts "meme-ability" by actively, not passively, inviting derivative works. As new derivatives are made and shared, attention might flow back to the original, boosting its reputation. This may inspire new interpretations, leading in a flywheel effect where each derivative adds to the original's worth - similar to platform network effects, where platforms become more valuable as more users join them.
cc0 licence allows creators "seize production memes."
Physical items are also using cc0 NFT assets, thus it's not just a digital phenomenon. The Nouns Vision initiative turned the square-framed spectacles shown on each new NounsDAO NFT ("one per day, forever") into luxury sunglasses. Blitmap's pixel-art has been used on shoes, apparel, and caps. In traditional IP regimes, a single owner controls creation, licensing, and production.
The physical "blitcap" (3rd level) is a descendant of the trait in the cc0 Chain Runners collection (2nd), which uses the "logo" from cc0 Blitmap (1st)! The Logo is Blitmap token #84 and has been used as a trait in various collections. The "Dom Rose" is another popular token. These homages reference Blitmap's influence as a cc0 leader, as one of the earliest NFT projects to proclaim public domain intents. A new collection, Citizens of Tajigen, emerged last week with a Blitcap characteristic.
These derivatives can be a win-win for everyone, not just the original inventors, especially when using NFT assets to establish unique brands. As people learn about the derivative, they may become interested in the original. If you see someone wearing Nouns glasses on the street (or in a Super Bowl ad), you may desire a pair, but you may also be interested in buying an original NounsDAO NFT or related derivative.
Blitmap Logo Hat (1), Chain Runners #780 ft. Hat (2), and Blitmap Original "Logo #87" (3)
Co-creating open source
NFTs' power comes from smart contract technology's intrinsic composability. Many smart contracts can be integrated or stacked to generate richer applications.
"Money Legos" describes how decentralized finance ("DeFi") smart contracts interconnect to generate new financial use cases. Yearn communicates with MakerDAO's stablecoin $DAI and exchange liquidity provider Curve by calling public smart contract methods. NFTs and their underlying smart contracts can operate as the base-layer framework for recombining and interconnecting culture and creativity.
cc0 gives an NFT's enthusiast community authority to develop new value layers whenever, wherever, and however they wish.
Multiple cc0 projects are playable characters in HyperLoot, a Loot Project knockoff.
Open source and Linux's rise are parallels. When the internet was young, Microsoft dominated the OS market with Windows. Linux (and its developer Linus Torvalds) championed a community-first mentality, freely available the source code without restrictions. This led to developers worldwide producing new software for Linux, from web servers to databases. As people (and organizations) created world-class open source software, Linux's value proposition grew, leading to explosive development and industry innovation. According to Truelist, Linux powers 96.3% of the top 1 million web servers and 85% of smartphones.
With cc0 licensing empowering NFT community builders, one might hope for long-term innovation. Combining cc0 with NFTs "turns an antagonistic game into a co-operative one," says NounsDAO cofounder punk4156. It's important on several levels. First, decentralized systems from open source to crypto are about trust and coordination, therefore facilitating cooperation is crucial. Second, the dynamics of this cooperation work well in the context of NFTs because giving people ownership over their digital assets allows them to internalize the results of co-creation through the value that accrues to their assets and contributions, which incentivizes them to participate in co-creation in the first place.
Licensed to create
If cc0 projects are open source "applications" or "platforms," then NFT artwork, metadata, and smart contracts provide the "user interface" and the underlying blockchain (e.g., Ethereum) is the "operating system." For these apps to attain Linux-like potential, more infrastructure services must be established and made available so people may take advantage of cc0's remixing capabilities.
These services are developing. Zora protocol and OpenSea's open source Seaport protocol enable open, permissionless NFT marketplaces. A pixel-art-rendering engine was just published on-chain to the Ethereum blockchain and integrated into OKPC and ICE64. Each application improves blockchain's "out-of-the-box" capabilities, leading to new apps created from the improved building blocks.
Web3 developer growth is at an all-time high, yet it's still a small fraction of active software developers globally. As additional developers enter the field, prospective NFT projects may find more creative and infrastructure Legos for cc0 and beyond.
Electric Capital Developer Report (2021), p. 122
Growth requires composability. Users can easily integrate digital assets developed on public standards and compatible infrastructure into other platforms. The Loot Project is one of the first to illustrate decentralized co-creation, worldbuilding, and more in NFTs. This example was low-fi or "incomplete" aesthetically, providing room for imagination and community co-creation.
Loot began with a series of Loot bag NFTs, each listing eight "adventure things" in white writing on a black backdrop (such as Loot Bag #5726's "Katana, Divine Robe, Great Helm, Wool Sash, Divine Slippers, Chain Gloves, Amulet, Gold Ring"). Dom Hofmann's free Loot bags served as a foundation for the community.
Several projects have begun metaphorical (lore) and practical (game development) world-building in a short time, with artists contributing many variations to the collective "Lootverse." They've produced games (Realms & The Crypt), characters (Genesis Project, Hyperloot, Loot Explorers), storytelling initiatives (Banners, OpenQuill), and even infrastructure (The Rift).
Why cc0 and composability? Because consumers own and control Loot bags, they may use them wherever they choose by connecting their crypto wallets. This allows users to participate in multiple derivative projects, such as Genesis Adventurers, whose characters appear in many others — creating a decentralized franchise not owned by any one corporation.
Genesis Project's Genesis Adventurer (1) with HyperLoot (2) and Loot Explorer (3) versions
When to go cc0
There are several IP development strategies NFT projects can use. When it comes to cc0, it’s important to be realistic. The public domain won't make a project a runaway success just by implementing the license. cc0 works well for NFT initiatives that can develop a rich, enlarged ecosystem.
Many of the most successful cc0 projects have introduced flexible intellectual property. The Nouns brand is as obvious for a beer ad as for real glasses; Loot bags are simple primitives that make sense in all adventure settings; and the Goblintown visual style looks good on dwarfs, zombies, and cranky owls as it does on Val Kilmer.
The ideal cc0 NFT project gives builders the opportunity to add value:
vertically, by stacking new content and features directly on top of the original cc0 assets (for instance, as with games built on the Loot ecosystem, among others), and
horizontally, by introducing distinct but related intellectual property that helps propagate the original cc0 project’s brand (as with various Goblintown derivatives, among others).
These actions can assist cc0 NFT business models. Because cc0 NFT projects receive royalties from secondary sales, third-party extensions and derivatives can boost demand for the original assets.
Using cc0 license lowers friction that could hinder brand-reinforcing extensions or lead to them bypassing the original. Robbie Broome recently argued (in the context of his cc0 project A Common Place) that giving away his IP to cc0 avoids bad rehashes down the line. If UrbanOutfitters wanted to put my design on a tee, they could use the actual work instead of hiring a designer. CC0 can turn competition into cooperation.
Community agreement about core assets' value and contribution can help cc0 projects. Cohesion and engagement are key. Using the above examples: Developers can design adventure games around whatever themes and item concepts they desire, but many choose Loot bags because of the Lootverse's community togetherness. Flipmap shared half of its money with the original Blitmap artists in acknowledgment of that project's core role in the community. This can build a healthy culture within a cc0 project ecosystem. Commentator NiftyPins said it was smart to acknowledge the people that constructed their universe. Many OG Blitmap artists have popped into the Flipmap discord to share information.
cc0 isn't a one-size-fits-all answer; NFTs formed around well-established brands may prefer more restrictive licenses to preserve their intellectual property and reinforce exclusivity. cc0 has some superficial similarities to permitting NFT owners to market the IP connected with their NFTs (à la Bored Ape Yacht Club), but there is a significant difference: cc0 holders can't exclude others from utilizing the same IP. This can make it tougher for holders to develop commercial brands on cc0 assets or offer specific rights to partners. Holders can still introduce enlarged intellectual property (such as backstories or derivatives) that they control.
Blockchain technologies and the crypto ethos are decentralized and open-source. This makes it logical for crypto initiatives to build around cc0 content models, which build on the work of the Creative Commons foundation and numerous open source pioneers.
NFT creators that choose cc0 must select how involved they want to be in building the ecosystem. Some cc0 project leaders, like Chain Runners' developers, have kept building on top of the initial cc0 assets, creating an environment derivative projects can plug into. Dom Hofmann stood back from Loot, letting the community lead. (Dom is also working on additional cc0 NFT projects for the company he formed to build Blitmap.) Other authors have chosen out totally, like sartoshi, who announced his exit from the cc0 project he founded, mfers, and from the NFT area by publishing a final edition suitably named "end of sartoshi" and then deactivating his Twitter account. A multi-signature wallet of seven mfers controls the project's smart contract.
cc0 licensing allows a robust community to co-create in ways that benefit all members, regardless of original creators' continuous commitment. We foresee more organized infrastructure and design patterns as NFT matures. Like open source software, value capture frameworks may see innovation. (We could imagine a variant of the "Sleepycat license," which requires commercial software to pay licensing fees when embedding open source components.) As creators progress the space, we expect them to build unique rights and licensing strategies. cc0 allows NFT producers to bootstrap ideas that may take off.
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INTΞGRITY team
3 years ago
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Categories of Recipients: Analytics Providers, Communication Providers, Custom Service Providers, Fraud Prevention and Security Providers, Infrastructure Providers, Marketing Providers, Payment Processors
Category of Personal Information: Commercial Information
Categories of Recipients: Analytics Providers, Infrastructure Providers, Payment Processors
Category of Personal Information: Internet or Other Electronic Network Activity Information
Categories of Recipients: Analytics Providers, Infrastructure Providers
Category of Personal Information: Inferences
Categories of Recipients: Analytics Providers, Infrastructure Providers
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You have the right, subject to certain limitations: (1) to request more information about the categories and specific pieces of personal information we collect, use, and disclose about you; (2) to request the deletion of your personal information; (3) to opt out of any future sales of your personal information; and (4) to not be discriminated against for exercising these rights. You may submit these requests by email to hello@int3grity.com. We shall not treat you differently if you exercise your rights under the CCPA.
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This section applies to you if you are based in the European Economic Area ("EEA"), the United Kingdom, or Switzerland and have specific rights and safeguards regarding the processing of your personal data under relevant law.
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We will process your personal information based on the following legal grounds:
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We retain the personal information associated with your account so long as your account is active. If you close your account, your account information will be deleted within 14 days. We retain other personal data for as long as is required to fulfill the objectives for which it was obtained and for other legitimate business purposes, such as to meet our legal, regulatory, or other compliance responsibilities.
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You have the right to request access to the personal data we hold on you and to get your data in a portable format, to request that your personal data be rectified or erased, and to object to or request that we restrict particular processing, subject to certain limitations. To assert your legal rights:
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If we are unable to settle your concern over our processing of personal data, you have the right to file a complaint with the Data Protection Authority in your country. The links below provide access to the contact information for your Data Protection Authority.
For people in the EEA, please visit https://edpb.europa.eu/about-edpb/board/members en.
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For people in Switzerland: https://www.edoeb.admin.ch/edoeb/en/home/the-fdpic/contact.html
CONTACT US
Please contact us at hello@int3grity.com if you have any queries regarding this Privacy Statement.
Sam Hickmann
3 years ago
What is headline inflation?
Headline inflation is the raw Consumer price index (CPI) reported monthly by the Bureau of labour statistics (BLS). CPI measures inflation by calculating the cost of a fixed basket of goods. The CPI uses a base year to index the current year's prices.
Explaining Inflation
As it includes all aspects of an economy that experience inflation, headline inflation is not adjusted to remove volatile figures. Headline inflation is often linked to cost-of-living changes, which is useful for consumers.
The headline figure doesn't account for seasonality or volatile food and energy prices, which are removed from the core CPI. Headline inflation is usually annualized, so a monthly headline figure of 4% inflation would equal 4% inflation for the year if repeated for 12 months. Top-line inflation is compared year-over-year.
Inflation's downsides
Inflation erodes future dollar values, can stifle economic growth, and can raise interest rates. Core inflation is often considered a better metric than headline inflation. Investors and economists use headline and core results to set growth forecasts and monetary policy.
Core Inflation
Core inflation removes volatile CPI components that can distort the headline number. Food and energy costs are commonly removed. Environmental shifts that affect crop growth can affect food prices outside of the economy. Political dissent can affect energy costs, such as oil production.
From 1957 to 2018, the U.S. averaged 3.64 percent core inflation. In June 1980, the rate reached 13.60%. May 1957 had 0% inflation. The Fed's core inflation target for 2022 is 3%.
Central bank:
A central bank has privileged control over a nation's or group's money and credit. Modern central banks are responsible for monetary policy and bank regulation. Central banks are anti-competitive and non-market-based. Many central banks are not government agencies and are therefore considered politically independent. Even if a central bank isn't government-owned, its privileges are protected by law. A central bank's legal monopoly status gives it the right to issue banknotes and cash. Private commercial banks can only issue demand deposits.
What are living costs?
The cost of living is the amount needed to cover housing, food, taxes, and healthcare in a certain place and time. Cost of living is used to compare the cost of living between cities and is tied to wages. If expenses are higher in a city like New York, salaries must be higher so people can live there.
What's U.S. bureau of labor statistics?
BLS collects and distributes economic and labor market data about the U.S. Its reports include the CPI and PPI, both important inflation measures.
Vitalik
3 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
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.
see part 3