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.

With cryptocurrencies like Bitcoin, Ether, and Dogecoin gyrating in value over the past few months, many people are looking at so-called stablecoins like Terra to invest in because of their more predictable prices.

Terraform Labs, which oversees the Terra cryptocurrency project, has benefited from its rising popularity. The company said recently that investors like Arrington Capital, Lightspeed Venture Partners, and Pantera Capital have pledged $150 million to help it incubate various crypto projects that are connected to Terra.

Terraform Labs and its partners have built apps that operate on the company’s blockchain technology that helps keep a permanent and shared record of the firm’s crypto-related financial transactions.

Here’s what you need to know about Terra and the company behind it.

What is Terra?

Terra is a blockchain project developed by Terraform Labs that powers the startup’s cryptocurrencies and financial apps. These cryptocurrencies include the Terra U.S. Dollar, or UST, that is pegged to the U.S. dollar through an algorithm.

Terra is a stablecoin that is intended to reduce the volatility endemic to cryptocurrencies like Bitcoin. Some stablecoins, like Tether, are pegged to more conventional currencies, like the U.S. dollar, through cash and cash equivalents as opposed to an algorithm and associated reserve token.

To mint new UST tokens, a percentage of another digital token and reserve asset, Luna, is “burned.” If the demand for UST rises with more people using the currency, more Luna will be automatically burned and diverted to a community pool. That balancing act is supposed to help stabilize the price, to a degree.

“Luna directly benefits from the economic growth of the Terra economy, and it suffers from contractions of the Terra coin,” Terraform Labs CEO Do Kwon said.

Each time someone buys something—like an ice cream—using UST, that transaction generates a fee, similar to a credit card transaction. That fee is then distributed to people who own Luna tokens, similar to a stock dividend.

Who leads Terra?

The South Korean firm Terraform Labs was founded in 2018 by Daniel Shin and Kwon, who is now the company’s CEO. Kwon is a 29-year-old former Microsoft employee; Shin now heads the Chai online payment service, a Terra partner. Kwon said many Koreans have used the Chai service to buy goods like movie tickets using Terra cryptocurrency.

Terraform Labs does not make money from transactions using its crypto and instead relies on outside funding to operate, Kwon said. It has raised $57 million in funding from investors like HashKey Digital Asset Group, Divergence Digital Currency Fund, and Huobi Capital, according to deal-tracking service PitchBook. The amount raised is in addition to the latest $150 million funding commitment announced on July 16.

What are Terra’s plans?

Terraform Labs plans to use Terra’s blockchain and its associated cryptocurrencies—including one pegged to the Korean won—to create a digital financial system independent of major banks and fintech-app makers. So far, its main source of growth has been in Korea, where people have bought goods at stores, like coffee, using the Chai payment app that’s built on Terra’s blockchain. Kwon said the company’s associated Mirror trading app is experiencing growth in China and Thailand.

Meanwhile, Kwon said Terraform Labs would use its latest $150 million in funding to invest in groups that build financial apps on Terra’s blockchain. He likened the scouting and investing in other groups as akin to a “Y Combinator demo day type of situation,” a reference to the popular startup pitch event organized by early-stage investor Y Combinator.

The combination of all these Terra-specific financial apps shows that Terraform Labs is “almost creating a kind of bank,” said Ryan Watkins, a senior research analyst at cryptocurrency consultancy Messari.

In addition to cryptocurrencies, Terraform Labs has a number of other projects including the Anchor app, a high-yield savings account for holders of the group’s digital coins. Meanwhile, people can use the firm’s associated Mirror app to create synthetic financial assets that mimic more conventional ones, like “tokenized” representations of corporate stocks. These synthetic assets are supposed to be helpful to people like “a small retail trader in Thailand” who can more easily buy shares and “get some exposure to the upside” of stocks that they otherwise wouldn’t have been able to obtain, Kwon said. But some critics have said the U.S. Securities and Exchange Commission may eventually crack down on synthetic stocks, which are currently unregulated.

What do critics say?

Terra still has a long way to go to catch up to bigger cryptocurrency projects like Ethereum.

Most financial transactions involving Terra-related cryptocurrencies have originated in Korea, where its founders are based. Although Terra is becoming more popular in Korea thanks to rising interest in its partner Chai, it’s too early to say whether Terra-related currencies will gain traction in other countries.

Terra’s blockchain runs on a “limited number of nodes,” said Messari’s Watkins, referring to the computers that help keep the system running. That helps reduce latency that may otherwise slow processing of financial transactions, he said.

But the tradeoff is that Terra is less “decentralized” than other blockchain platforms like Ethereum, which is powered by thousands of interconnected computing nodes worldwide. That could make Terra less appealing to some blockchain purists.

Madoff who?

Bernie Madoff ran the largest Ponzi scheme in history, defrauding thousands of investors over at least 17 years, and possibly longer. He pioneered electronic trading and chaired Nasdaq in the 1990s. On April 14, 2021, he died while serving a 150-year sentence for money laundering, securities fraud, and other crimes.

Understanding Madoff

Madoff claimed to generate large, steady returns through a trading strategy called split-strike conversion, but he simply deposited client funds into a single bank account and paid out existing clients. He funded redemptions by attracting new investors and their capital, but the market crashed in late 2008. He confessed to his sons, who worked at his firm, on Dec. 10, 2008. Next day, they turned him in. The fund reported $64.8 billion in client assets.

Madoff pleaded guilty to 11 federal felony counts, including securities fraud, wire fraud, mail fraud, perjury, and money laundering. Ponzi scheme became a symbol of Wall Street's greed and dishonesty before the financial crisis. Madoff was sentenced to 150 years in prison and ordered to forfeit $170 billion, but no other Wall Street figures faced legal ramifications.

Bernie Madoff's Brief Biography

Bernie Madoff was born in Queens, New York, on April 29, 1938. He began dating Ruth (née Alpern) when they were teenagers. Madoff told a journalist by phone from prison that his father's sporting goods store went bankrupt during the Korean War: "You watch your father, who you idolize, build a big business and then lose everything." Madoff was determined to achieve "lasting success" like his father "whatever it took," but his career had ups and downs.

Early Madoff investments

At 22, he started Bernard L. Madoff Investment Securities LLC. First, he traded penny stocks with $5,000 he earned installing sprinklers and as a lifeguard. Family and friends soon invested with him. Madoff's bets soured after the "Kennedy Slide" in 1962, and his father-in-law had to bail him out.

Madoff felt he wasn't part of the Wall Street in-crowd. "We weren't NYSE members," he told Fishman. "It's obvious." According to Madoff, he was a scrappy market maker. "I was happy to take the crumbs," he told Fishman, citing a client who wanted to sell eight bonds; a bigger firm would turn it down.

Recognition

Success came when he and his brother Peter built electronic trading capabilities, or "artificial intelligence," that attracted massive order flow and provided market insights. "I had all these major banks coming down, entertaining me," Madoff told Fishman. "It was mind-bending."

By the late 1980s, he and four other Wall Street mainstays processed half of the NYSE's order flow. Controversially, he paid for much of it, and by the late 1980s, Madoff was making in the vicinity of $100 million a year.  He was Nasdaq chairman from 1990 to 1993.

Madoff's Ponzi scheme

It is not certain exactly when Madoff's Ponzi scheme began. He testified in court that it began in 1991, but his account manager, Frank DiPascali, had been at the firm since 1975.

Why Madoff did the scheme is unclear. "I had enough money to support my family's lifestyle. "I don't know why," he told Fishman." Madoff could have won Wall Street's respect as a market maker and electronic trading pioneer.

Madoff told Fishman he wasn't solely responsible for the fraud. "I let myself be talked into something, and that's my fault," he said, without saying who convinced him. "I thought I could escape eventually. I thought it'd be quick, but I couldn't."

Carl Shapiro, Jeffry Picower, Stanley Chais, and Norm Levy have been linked to Bernard L. Madoff Investment Securities LLC for years. Madoff's scheme made these men hundreds of millions of dollars in the 1960s and 1970s.

Madoff told Fishman, "Everyone was greedy, everyone wanted to go on." He says the Big Four and others who pumped client funds to him, outsourcing their asset management, must have suspected his returns or should have. "How can you make 15%-18% when everyone else is making less?" said Madoff.

How Madoff Got Away with It for So Long

Madoff's high returns made clients look the other way. He deposited their money in a Chase Manhattan Bank account, which merged to become JPMorgan Chase & Co. in 2000. The bank may have made $483 million from those deposits, so it didn't investigate.

When clients redeemed their investments, Madoff funded the payouts with new capital he attracted by promising unbelievable returns and earning his victims' trust. Madoff created an image of exclusivity by turning away clients. This model let half of Madoff's investors profit. These investors must pay into a victims' fund for defrauded investors.

Madoff wooed investors with his philanthropy. He defrauded nonprofits, including the Elie Wiesel Foundation for Peace and Hadassah. He approached congregants through his friendship with J. Ezra Merkin, a synagogue officer. Madoff allegedly stole $1 billion to $2 billion from his investors.

Investors believed Madoff for several reasons:

• His public portfolio seemed to be blue-chip stocks.
• His returns were high (10-20%) but consistent and not outlandish. In a 1992 interview with Madoff, the Wall Street Journal reported: "[Madoff] insists the returns were nothing special, given that the S&P 500-stock index returned 16.3% annually from 1982 to 1992. 'I'd be surprised if anyone thought matching the S&P over 10 years was remarkable,' he says.
• "He said he was using a split-strike collar strategy. A collar protects underlying shares by purchasing an out-of-the-money put option.

SEC inquiry

The Securities and Exchange Commission had been investigating Madoff and his securities firm since 1999, which frustrated many after he was prosecuted because they felt the biggest damage could have been prevented if the initial investigations had been rigorous enough.

Harry Markopolos was a whistleblower. In 1999, he figured Madoff must be lying in an afternoon. The SEC ignored his first Madoff complaint in 2000.

Markopolos wrote to the SEC in 2005: "The largest Ponzi scheme is Madoff Securities. This case has no SEC reward, so I'm turning it in because it's the right thing to do."

Markopolos found irregularities using a "Mosaic Method." Madoff's firm claimed to be profitable even when the S&P fell, which made no mathematical sense given what he was investing in. Markopolos said Madoff Securities' "undisclosed commissions" were the biggest red flag (1 percent of the total plus 20 percent of the profits).

Markopolos concluded that "investors don't know Bernie Madoff manages their money." Markopolos learned Madoff was applying for large loans from European banks (seemingly unnecessary if Madoff's returns were high).

The regulator asked Madoff for trading account documentation in 2005, after he nearly went bankrupt due to redemptions. The SEC drafted letters to two of the firms on his six-page list but didn't send them. Diana Henriques, author of "The Wizard of Lies: Bernie Madoff and the Death of Trust," documents the episode.

In 2008, the SEC was criticized for its slow response to Madoff's fraud.

Confession, sentencing of Bernie Madoff

Bernard L. Madoff Investment Securities LLC reported 5.6% year-to-date returns in November 2008; the S&P 500 fell 39%. As the selling continued, Madoff couldn't keep up with redemption requests, and on Dec. 10, he confessed to his sons Mark and Andy, who worked at his firm. "After I told them, they left, went to a lawyer, who told them to turn in their father, and I never saw them again. 2008-12-11: Bernie Madoff arrested.

Madoff insists he acted alone, but several of his colleagues were jailed. Mark Madoff died two years after his father's fraud was exposed. Madoff's investors committed suicide. Andy Madoff died of cancer in 2014.

2009 saw Madoff's 150-year prison sentence and $170 billion forfeiture. Marshals sold his three homes and yacht. Prisoner 61727-054 at Butner Federal Correctional Institution in North Carolina.

Madoff's lawyers requested early release on February 5, 2020, claiming he has a terminal kidney disease that may kill him in 18 months. Ten years have passed since Madoff's sentencing.

Bernie Madoff's Ponzi scheme aftermath

The paper trail of victims' claims shows Madoff's complexity and size. Documents show Madoff's scam began in the 1960s. His final account statements show $47 billion in "profit" from fake trades and shady accounting.

Thousands of investors lost their life savings, and multiple stories detail their harrowing loss.

Irving Picard, a New York lawyer overseeing Madoff's bankruptcy, has helped investors. By December 2018, Picard had recovered $13.3 billion from Ponzi scheme profiteers.

A Madoff Victim Fund (MVF) was created in 2013 to help compensate Madoff's victims, but the DOJ didn't start paying out the $4 billion until late 2017. Richard Breeden, a former SEC chair who oversees the fund, said thousands of claims were from "indirect investors"

Breeden and his team had to reject many claims because they weren't direct victims. Breeden said he based most of his decisions on one simple rule: Did the person invest more than they withdrew? Breeden estimated 11,000 "feeder" investors.

Breeden wrote in a November 2018 update for the Madoff Victim Fund, "We've paid over 27,300 victims 56.65% of their losses, with thousands more to come." In December 2018, 37,011 Madoff victims in the U.S. and around the world received over $2.7 billion. Breeden said the fund expected to make "at least one more significant distribution in 2019"

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Instagram (IG) is officially joining NFT. With the debut of new in-app NFT functionalities, influential producers can interact with blockchain tech on the social media platform.

Meta unveiled intentions for an Instagram NFT marketplace in March, but these latest capabilities focus more on content sharing than commerce. And why shouldn’t they? IG's entry into the NFT market is overdue, given that Twitter and Discord are NFT hotspots.

The NFT marketplace/Web3 social media race has continued to expand, with the expected Coinbase NFT Beta now live and blazing a trail through the NFT ecosystem.

IG's focus is on visual art. It's unlike any NFT marketplace or platform. IG NFTs and artists: what's the deal? Let’s take a look.

What are Instagram’s NFT features anyways?

As said, not everyone has Instagram's new features. 16 artists, NFT makers, and collectors can now post NFTs on IG by integrating third-party digital wallets (like Rainbow or MetaMask) in-app. IG doesn't charge to publish or share digital collectibles.

NFTs displayed on the app have a "shimmer" aesthetic effect. NFT posts also have a "digital collectable" badge that lists metadata such as the creator and/or owner, the platform it was created on, a brief description, and a blockchain identification.

Meta's social media NFTs have launched on Instagram, but the company is also preparing to roll out digital collectibles on Facebook, with more on the way for IG. Currently, only Ethereum and Polygon are supported, but Flow and Solana will be added soon.

How will artists use these new features?

Artists are publishing NFTs they developed or own on IG by linking third-party digital wallets. These features have no NFT trading aspects built-in, but are aimed to let authors share NFTs with IG audiences.

Creators, like IG-native aerial/street photographer Natalie Amrossi (@misshattan), are discovering novel uses for IG NFTs.

Amrossi chose to not only upload his own NFTs but also encourage other artists in the field. "That's the beauty of connecting your wallet and sharing NFTs. It's not just what you make, but also what you accumulate."

Amrossi has been producing and posting Instagram art for years. With IG's NFT features, she can understand Instagram's importance in supporting artists.

Web2 offered Amrossi the tools to become an artist and make a life. "Before 'influencer' existed, I was just making art. Instagram helped me reach so many individuals and brands, giving me a living.

Even artists without millions of viewers are encouraged to share NFTs on IG. Wilson, a relatively new name in the NFT space, seems to have already gone above and beyond the scope of these new IG features. By releasing "Losing My Mind" via IG NFT posts, she has evaded the lack of IG NFT commerce by using her network to market her multi-piece collection.

"'Losing My Mind' is a long-running photo series. Wilson was preparing to release it as NFTs before IG approached him, so it was a perfect match.

Wilson says the series is about Black feminine figures and media depiction. Respectable effort, given POC artists have been underrepresented in NFT so far.

“Over the past year, I've had mental health concerns that made my emotions so severe it was impossible to function in daily life, therefore that prompted this photo series. Every Wednesday and Friday for three weeks, I'll release a new Meta photo for sale.

Wilson hopes these new IG capabilities will help develop a connection between the NFT community and other internet subcultures that thrive on Instagram.

“NFTs can look scary as an outsider, but seeing them on your daily IG feed makes it less foreign,” adds Wilson. I think Instagram might become a hub for NFT aficionados, making them more accessible to artists and collectors.

What does it all mean for the NFT space?

Meta's NFT and metaverse activities will continue to impact Instagram's NFT ecosystem. Many think it will be for the better, as IG NFT frauds are another problem hurting the NFT industry.

IG's new NFT features seem similar to Twitter's PFP NFT verifications, but Instagram's tools should help cut down on scams as users can now verify the creation and ownership of whole NFT collections included in IG posts.

Given the number of visual artists and NFT creators on IG, it might become another hub for NFT fans, as Wilson noted. If this happens, it raises questions about Instagram success. Will artists be incentivized to distribute NFTs? Or will those with a large fanbase dominate?

Elise Swopes (@swopes) believes these new features should benefit smaller artists. Swopes was one of the first profiles placed to Instagram's original suggested user list in 2012.

Swopes says she wants IG to be a magnet for discovery and understands the value of NFT artists and producers.

"I'd love to see IG become a focus of discovery for everyone, not just the Beeples and Apes and PFPs. That's terrific for them, but [IG NFT features] are more about using new technology to promote emerging artists, Swopes added.

“Especially music artists. It's everywhere. Dancers, writers, painters, sculptors, musicians. My element isn't just for digital artists; it can be anything. I'm delighted to witness people's creativity."

Swopes, Wilson, and Amrossi all believe IG's new features can help smaller artists. It remains to be seen how these new features will effect the NFT ecosystem once unlocked for the rest of the IG NFT community, but we will likely see more social media NFT integrations in the months and years ahead.