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.

[06/20/2022 update] We have now implemented a powerful text editor, but you can still use markdown.

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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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