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.

Maybe you'll work for a DAO next! Over $1 Billion in NFTs in the Flamingo DAO Another DAO tried to buy the NFL team Denver Broncos. The UkraineDAO raised over $7 Million for Ukraine. The PleasrDAO paid $4m for a Wu-Tang Clan album that belonged to the “pharma bro.”
DAOs move billions and employ thousands. So learn what a DAO is, how it works, and how to create one!

DAO? So, what? Why is it better?

A Decentralized Autonomous Organization (DAO). Some people like to also refer to it as Digital Autonomous Organization, but I prefer the former.
They are virtual organizations. In the real world, you have organizations or companies right? These firms have shareholders and a board. Usually, anyone with authority makes decisions. It could be the CEO, the Board, or the HIPPO. If you own stock in that company, you may also be able to influence decisions. It's now possible to do something similar but much better and more equitable in the cryptocurrency world.

This article informs you:

DAOs- What are the most common DAOs, their advantages and disadvantages over traditional companies? What are they if any?
Is a DAO legally recognized?
How secure is a DAO?
I’m ready whenever you are!

A DAO is a type of company that is operated by smart contracts on the blockchain. Smart contracts are computer code that self-executes our commands. Those contracts can be any. Most second-generation blockchains support smart contracts. Examples are Ethereum, Solana, Polygon, Binance Smart Chain, EOS, etc. I think I've gone off topic. Back on track.   Now let's go!
Unlike traditional corporations, DAOs are governed by smart contracts. Unlike traditional company governance, DAO governance is fully transparent and auditable. That's one of the things that sets it apart. The clarity!
A DAO, like a traditional company, has one major difference. In other words, it is decentralized. DAOs are more ‘democratic' than traditional companies because anyone can vote on decisions. Anyone! In a DAO, we (you and I) make the decisions, not the top-shots. We are the CEO and investors. A DAO gives its community members power. We get to decide.
As long as you are a stakeholder, i.e. own a portion of the DAO tokens, you can participate in the DAO. Tokens are open to all. It's just a matter of exchanging it. Ownership of DAO tokens entitles you to exclusive benefits such as governance, voting, and so on. You can vote for a move, a plan, or the DAO's next investment. You can even pitch for funding. Any ‘big' decision in a DAO requires a vote from all stakeholders. In this case, ‘token-holders'! In other words, they function like stock.

What are the 5 DAO types?

Different DAOs exist. We will categorize decentralized autonomous organizations based on their mode of operation, structure, and even technology. Here are a few. You've probably heard of them:

1. DeFi DAO

These DAOs offer DeFi (decentralized financial) services via smart contract protocols. They use tokens to vote protocol and financial changes. Uniswap, Aave, Maker DAO, and Olympus DAO are some examples. Most DAOs manage billions.

Maker DAO was one of the first protocols ever created. It is a decentralized organization on the Ethereum blockchain that allows cryptocurrency lending and borrowing without a middleman.
Maker DAO issues DAI, a stable coin. DAI is a top-rated USD-pegged stable coin.
Maker DAO has an MKR token. These token holders are in charge of adjusting the Dai stable coin policy. Simply put, MKR tokens represent DAO “shares”.

2. Investment DAO

Investors pool their funds and make investment decisions. Investing in new businesses or art is one example. Investment DAOs help DeFi operations pool capital. The Meta Cartel DAO is a community of people who want to invest in new projects built on the Ethereum blockchain. Instead of investing one by one, they want to pool their resources and share ideas on how to make better financial decisions.

Other investment DAOs include the LAO and Friends with Benefits.

3. DAO Grant/Launchpad

In a grant DAO, community members contribute funds to a grant pool and vote on how to allocate and distribute them. These DAOs fund new DeFi projects. Those in need only need to apply. The Moloch DAO is a great Grant DAO. The tokens are used to allocate capital. Also see Gitcoin and Seedify.

4. DAO Collector

I debated whether to put it under ‘Investment DAO' or leave it alone. It's a subset of investment DAOs. This group buys non-fungible tokens, artwork, and collectibles. The market for NFTs has recently exploded, and it's time to investigate. The Pleasr DAO is a collector DAO. One copy of Wu-Tang Clan's "Once Upon a Time in Shaolin" cost the Pleasr DAO $4 million. Pleasr DAO is known for buying Doge meme NFT. Collector DAOs include the Flamingo, Mutant Cats DAO, and Constitution DAOs. Don't underestimate their websites' "childish" style. They have millions.

5. Social DAO

These are social networking and interaction platforms. For example, Decentraland DAO and Friends With Benefits DAO.

What are the DAO Benefits?

Here are some of the benefits of a decentralized autonomous organization:

• They are trustless. You don’t need to trust a CEO or management team
• It can’t be shut down unless a majority of the token holders agree. The government can't shut - It down because it isn't centralized.
• It's fully democratic
• It is open-source and fully transparent.

What about DAO drawbacks?

We've been saying DAOs are the bomb? But are they really the shit? What could go wrong with DAO?
DAOs may contain bugs. If they are hacked, the results can be catastrophic.
No trade secrets exist. Because the smart contract is transparent and coded on the blockchain, it can be copied. It may be used by another organization without credit. Maybe DAOs should use Secret, Oasis, or Horizen blockchain networks.

Are DAOs legally recognized??

In most counties, DAO regulation is inexistent. It's unclear. Most DAOs don’t have a legal personality. The Howey Test and the Securities Act of 1933 determine whether DAO tokens are securities. Although most countries follow the US, this is only considered for the US. Wyoming became the first state to recognize DAOs as legal entities in July 2021 after passing a DAO bill. DAOs registered in Wyoming are thus legally recognized as business entities in the US and thus receive the same legal protections as a Limited Liability Company.

In terms of cyber-security, how secure is a DAO?

Blockchains are secure. However, smart contracts may have security flaws or bugs. This can be avoided by third-party smart contract reviews, testing, and auditing

Finally, Decentralized Autonomous Organizations are timeless. Let us examine the current situation: Ukraine's invasion. A DAO was formed to help Ukrainian troops fighting the Russians. It was named Ukraine DAO. Pleasr DAO, NFT studio Trippy Labs, and Russian art collective Pussy Riot organized this fundraiser. Coindesk reports that over $3 million has been raised in Ethereum-based tokens. AidForUkraine, a DAO aimed at supporting Ukraine's defense efforts, has launched. Accepting Solana token donations. They are fully transparent, uncensorable, and can’t be shut down or sanctioned.
DAOs are undeniably the future of blockchain. Everyone is paying attention. Personally, I believe traditional companies will soon have to choose between adapting or being left behind.

WWDC 2022 began early Tuesday morning. WWDC brought a ton of new features (which went for just shy of two hours).

With so many announcements, we thought we'd compile them. And now...

WWDC?

WWDC is Apple's developer conference. This includes iOS, macOS, watchOS, and iPadOS (all of its iPads). It's where Apple announces new features for developers to use. It's also where Apple previews new software.

Virtual WWDC runs June 6-10.  You can rewatch the stream on Apple's website.

WWDC 2022 news:

Completely everything. Really. iOS 16 first.

iOS 16.

iOS 16 is a major iPhone update. iOS 16 adds the ability to customize the Lock Screen's color/theme. And widgets. It also organizes notifications and pairs Lock Screen with Focus themes. Edit or recall recently sent messages, recover recently deleted messages, and mark conversations as unread. Apple gives us yet another reason to stay in its walled garden with iMessage.

New iOS includes family sharing. Parents can set up a child's account with parental controls to restrict apps, movies, books, and music. iOS 16 lets large families and friend pods share iCloud photos. Up to six people can contribute photos to a separate iCloud library.

Live Text is getting creepier. Users can interact with text in any video frame. Touch and hold an image's subject to remove it from its background and place it in apps like messages. Dictation offers a new on-device voice-and-touch experience. Siri can run app shortcuts without setup in iOS 16. Apple also unveiled a new iOS 16 feature to help people break up with abusive partners who track their locations or read their messages. Safety Check.

Apple Pay Later allows iPhone users to buy products and pay for them later. iOS 16 pushes Mail. Users can schedule emails and cancel delivery before it reaches a recipient's inbox (be quick!). Mail now detects if you forgot an attachment, as Gmail has for years. iOS 16's Maps app gets "Multi-Stop Routing," .

Apple News also gets an iOS 16 update. Apple News adds My Sports. With iOS 16, the Apple Watch's Fitness app is also coming to iOS and the iPhone, using motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person). 

iOS 16 includes accessibility updates like Door Detection.

watchOS9

Many of Apple's software updates are designed to take advantage of the larger screens in recent models, but they also improve health and fitness tracking.

The most obvious reason to upgrade watchOS every year is to get new watch faces from Apple. WatchOS 9 will add four new faces.

Runners' workout metrics improve.
Apple quickly realized that fitness tracking would be the Apple Watch's main feature, even though it's been the killer app for wearables since their debut. For watchOS 9, the Apple Watch will use its accelerometer and gyroscope to track a runner's form, stride length, and ground contact time. It also introduces the ability to specify heart rate zones, distance, and time intervals, with vibrating haptic feedback and voice alerts.

The Apple Watch's Fitness app is coming to iOS and the iPhone, using the smartphone's motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person).

We'll get sleep tracking, medication reminders, and drug interaction alerts. Your watch can create calendar events. A new Week view shows what meetings or responsibilities stand between you and the weekend.

iPadOS16

WWDC 2022 introduced iPad updates. iPadOS 16 is similar to iOS for the iPhone, but has features for larger screens and tablet accessories. The software update gives it many iPhone-like features.

iPadOS 16's Home app, like iOS 16, will have a new design language. iPad users who want to blame it on the rain finally have a Weather app. iPadOS 16 will have iCloud's Shared Photo Library, Live Text and Visual Look Up upgrades, and FaceTime Handoff, so you can switch between devices during a call.

Apple highlighted iPadOS 16's multitasking at WWDC 2022. iPad's Stage Manager sounds like a community theater app. It's a powerful multitasking tool for tablets and brings them closer to emulating laptops. Apple's iPadOS 16 supports multi-user collaboration. You can share content from Files, Keynote, Numbers, Pages, Notes, Reminders, Safari, and other third-party apps in Apple Messages.

M2-chip

WWDC 2022 revealed Apple's M2 chip. Apple has started the next generation of Apple Silicon for the Mac with M2. Apple says this device improves M1's performance.

M2's second-generation 5nm chip has 25% more transistors than M1's. 100GB/s memory bandwidth (50 per cent more than M1). M2 has 24GB of unified memory, up from 16GB but less than some ultraportable PCs' 32GB. The M2 chip has 10% better multi-core CPU performance than the M2, and it's nearly twice as fast as the latest 10-core PC laptop chip at the same power level (CPU performance is 18 per cent greater than M1).

New MacBooks

Apple introduced the M2-powered MacBook Air. Apple's entry-level laptop has a larger display, a new processor, new colors, and a notch.

M2 also powers the 13-inch MacBook Pro. The 13-inch MacBook Pro has 24GB of unified memory and 50% more memory bandwidth. New MacBook Pro batteries last 20 hours. As I type on the 2021 MacBook Pro, I can only imagine how much power the M2 will add.

macOS 13.0 (or, macOS Ventura)

macOS Ventura will take full advantage of M2 with new features like Stage Manager and Continuity Camera and Handoff for FaceTime. Safari, Mail, Messages, Spotlight, and more get updates in macOS Ventura.

Apple hasn't run out of California landmarks to name its OS after yet. macOS 13 will be called Ventura when it's released in a few months, but it's more than a name change and new wallpapers. 

Stage Manager organizes windows

Stage Manager is a new macOS tool that organizes open windows and applications so they're still visible while focusing on a specific task. The main app sits in the middle of the desktop, while other apps and documents are organized and piled up to the side.

Improved Searching

Spotlight is one of macOS's least appreciated features, but with Ventura, it's becoming even more useful. Live Text lets you extract text from Spotlight results without leaving the window, including images from the photo library and the web.

Mail lets you schedule or unsend emails.

We've all sent an email we regret, whether it contained regrettable words or was sent at the wrong time. In macOS Ventura, Mail users can cancel or reschedule a message after sending it. Mail will now intelligently determine if a person was forgotten from a CC list or if a promised attachment wasn't included. Procrastinators can set a reminder to read a message later.

Safari adds tab sharing and password passkeys

Apple is updating Safari to make it more user-friendly... mostly. Users can share a group of tabs with friends or family, a useful feature when researching a topic with too many tabs. Passkeys will replace passwords in Safari's next version. Instead of entering random gibberish when creating a new account, macOS users can use TouchID to create an on-device passkey. Using an iPhone's camera and a QR system, Passkey syncs and works across all Apple devices and Windows computers.

Continuity adds Facetime device switching and iPhone webcam.

With macOS Ventura, iPhone users can transfer a FaceTime call from their phone to their desktop or laptop using Handoff, or vice versa if they started a call at their desk and need to continue it elsewhere. Apple finally admits its laptop and monitor webcams aren't the best. Continuity makes the iPhone a webcam. Apple demonstrated a feature where the wide-angle lens could provide a live stream of the desk below, while the standard zoom lens could focus on the speaker's face. New iPhone laptop mounts are coming.

System Preferences

System Preferences is Now System Settings and Looks Like iOS
Ventura's System Preferences has been renamed System Settings and is much more similar in appearance to iOS and iPadOS. As the iPhone and iPad are gateway devices into Apple's hardware ecosystem, new Mac users should find it easier to adjust.

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