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.

Musk’s purchase of Twitter has sealed his bond with the American right—whether the platform’s left-leaning employees and users like it or not.

Elon Musk's pursuit of Twitter Inc. began earlier this month as a joke. It started slowly, then spiraled out of control, culminating on April 25 with the world's richest man agreeing to spend $44 billion on one of the most politically significant technology companies ever. There have been bigger financial acquisitions, but Twitter's significance has always outpaced its balance sheet. This is a unique Silicon Valley deal.

To recap: Musk announced in early April that he had bought a stake in Twitter, citing the company's alleged suppression of free speech. His complaints were vague, relying heavily on the dog whistles of the ultra-right. A week later, he announced he'd buy the company for $54.20 per share, four days after initially pledging to join Twitter's board. Twitter's directors noticed the 420 reference as well, and responded with a “shareholder rights” plan (i.e., a poison pill) that included a 420 joke.

Musk - Patrick Pleul/Getty Images

No one knew if the bid was genuine. Musk's Twitter plans seemed implausible or insincere. In a tweet, he referred to automated accounts that use his name to promote cryptocurrency. He enraged his prospective employees by suggesting that Twitter's San Francisco headquarters be turned into a homeless shelter, renaming the company Titter, and expressing solidarity with his growing conservative fan base. “The woke mind virus is making Netflix unwatchable,” he tweeted on April 19.

But Musk got funding, and after a frantic weekend of negotiations, Twitter said yes. Unlike most buyouts, Musk will personally fund the deal, putting up up to $21 billion in cash and borrowing another $12.5 billion against his Tesla stock.

Free Speech and Partisanship

Percentage of respondents who agree with the following

The deal is expected to replatform accounts that were banned by Twitter for harassing others, spreading misinformation, or inciting violence, such as former President Donald Trump's account. As a result, Musk is at odds with his own left-leaning employees, users, and advertisers, who would prefer more content moderation rather than less.

Dorsey - Photographer: Joe Raedle/Getty Images

Previously, the company's leadership had similar issues. Founder Jack Dorsey stepped down last year amid concerns about slowing growth and product development, as well as his dual role as CEO of payments processor Block Inc. Compared to Musk, a father of seven who already runs four companies (besides Tesla and SpaceX), Dorsey is laser-focused.

Musk's motivation to buy Twitter may be political. Affirming the American far right with $44 billion spent on “free speech” Right-wing activists have promoted a series of competing upstart Twitter competitors—Parler, Gettr, and Trump's own effort, Truth Social—since Trump was banned from major social media platforms for encouraging rioters at the US Capitol on Jan. 6, 2021. But Musk can give them a social network with lax content moderation and a real user base. Trump said he wouldn't return to Twitter after the deal was announced, but he wouldn't be the first to do so.

Trump - Eli Hiller/Bloomberg

Conservative activists and lawmakers are already ecstatic. “A great day for free speech in America,” said Missouri Republican Josh Hawley. The day the deal was announced, Tucker Carlson opened his nightly Fox show with a 10-minute laudatory monologue. “The single biggest political development since Donald Trump's election in 2016,” he gushed over Musk.

But Musk's supporters and detractors misunderstand how much his business interests influence his political ideology. He marketed Tesla's cars as carbon-saving machines that were faster and cooler than gas-powered luxury cars during George W. Bush's presidency. Musk gained a huge following among wealthy environmentalists who reserved hundreds of thousands of Tesla sedans years before they were made during Barack Obama's presidency. Musk in the Trump era advocated for a carbon tax, but he also fought local officials (and his own workers) over Covid rules that slowed the reopening of his Bay Area factory.

Teslas at the Las Vegas Convention Center Loop Central Station in April 2021. The Las Vegas Convention Center Loop was Musk's first commercial project. Ethan Miller/Getty Images

Musk's rightward shift matched the rise of the nationalist-populist right and the desire to serve a growing EV market. In 2019, he unveiled the Cybertruck, a Tesla pickup, and in 2018, he announced plans to manufacture it at a new plant outside Austin. In 2021, he decided to move Tesla's headquarters there, citing California's "land of over-regulation." After Ford and General Motors beat him to the electric truck market, Musk reframed Tesla as a company for pickup-driving dudes.

Similarly, his purchase of Twitter will be entwined with his other business interests. Tesla has a factory in China and is friendly with Beijing. This could be seen as a conflict of interest when Musk's Twitter decides how to treat Chinese-backed disinformation, as Amazon.com Inc. founder Jeff Bezos noted.

Musk has focused on Twitter's product and social impact, but the company's biggest challenges are financial: Either increase cash flow or cut costs to comfortably service his new debt. Even if Musk can't do that, he can still benefit from the deal. He has recently used the increased attention to promote other business interests: Boring has hyperloops and Neuralink brain implants on the way, Musk tweeted. Remember Tesla's long-promised robotaxis!

Musk may be comfortable saying he has no expectation of profit because it benefits his other businesses. At the TED conference on April 14, Musk insisted that his interest in Twitter was solely charitable. “I don't care about money.”

The rockets and weed jokes make it easy to see Musk as unique—and his crazy buyout will undoubtedly add to that narrative. However, he is a megabillionaire who is risking a small amount of money (approximately 13% of his net worth) to gain potentially enormous influence. Musk makes everything seem new, but this is a rehash of an old media story.

The new Bitcoin Academy will teach Jay-Marcy Z's Houses neighbors "What is Cryptocurrency."
Jay-Z grew up in Brooklyn's Marcy Houses. The rapper and Block CEO Jack Dorsey are giving back to his hometown by creating the Bitcoin Academy.

The Bitcoin Academy will offer online and in-person classes, including "What is Money?" and "What is Blockchain?"
The program will provide participants with a mobile hotspot and a small amount of Bitcoin for hands-on learning.

Students will receive dinner and two evenings of instruction until early September. The Shawn Carter Foundation will help with on-the-ground instruction.

Jay-Z and Dorsey announced the program Thursday morning. It will begin at Marcy Houses but may be expanded.

Crypto Blockchain Plug and Black Bitcoin Billionaire, which has received a grant from Block, will teach the classes.

Jay-Z, Dorsey reunite

Jay-Z and Dorsey have previously worked together to promote a Bitcoin and crypto-based future.

In 2021, Dorsey's Block (then Square) acquired the rapper's streaming music service Tidal, which they propose using for NFT distribution.

Dorsey and Jay-Z launched an endowment in 2021 to fund Bitcoin development in Africa and India.

Dorsey is funding the new Bitcoin Academy out of his own pocket (as is Jay-Z), but he's also pushed crypto-related charitable endeavors at Block, including a $5 million fund backed by corporate Bitcoin interest.

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