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.

On Election Day 2020, two young entrepreneurs received a call from the CFTC chairman. Luana Lopes Lara and Tarek Mansour spent 18 months trying to start a new type of financial exchange. Instead of betting on stock prices or commodity futures, people could trade instruments tied to real-world events, such as legislation, the weather, or the Oscar winner.

Heath Tarbert, a Trump appointee, shouted "Congratulations." "You're competing with 1840s-era markets. I'm sure you'll become a powerhouse too."

Companies had tried to introduce similar event markets in the US for years, but Tarbert's agency, the CFTC, said no, arguing they were gambling and prone to cheating. Now the agency has reversed course, approving two 24-year-olds who will have first-mover advantage in what could become a huge new asset class. Kalshi Inc. raised $30 million from venture capitalists within weeks of Tarbert's call, his representative says. Mansour, 26, believes this will be bigger than crypto.

Anyone who's read The Wisdom of Crowds knows prediction markets' potential. Well-designed markets can help draw out knowledge from disparate groups, and research shows that when money is at stake, people make better predictions. Lopes Lara calls it a "bullshit tax." That's why Google, Microsoft, and even the US Department of Defense use prediction markets internally to guide decisions, and why university-linked political betting sites like PredictIt sometimes outperform polls.

Regulators feared Wall Street-scale trading would encourage investors to manipulate reality. If the stakes are high enough, traders could pressure congressional staffers to stall a bill or bet on whether Kanye West's new album will drop this week. When Lopes Lara and Mansour pitched the CFTC, senior regulators raised these issues. Politically appointed commissioners overruled their concerns, and one later joined Kalshi's board.

Kalshi's victory was due more to lobbying and legal wrangling than to Silicon Valley-style innovation. Lopes Lara and Mansour didn't invent anything; they changed a well-established concept's governance. The result could usher in a new era of market-based enlightenment or push Wall Street's destructive tendencies into the real world.

If Kalshi's founders lacked experience to bolster their CFTC application, they had comical youth success. Lopes Lara studied ballet at the Brazilian Bolshoi before coming to the US. Mansour won France's math Olympiad. They bonded over their work ethic in an MIT computer science class.

Lopes Lara had the idea for Kalshi while interning at a New York hedge fund. When the traders around her weren't working, she noticed they were betting on the news: Would Apple hit a trillion dollars? Kylie Jenner? "It was anything," she says.

Mansour saw the business potential when Lopes Lara suggested it. He interned at Goldman Sachs Group Inc., helping investors prepare for the UK leaving the EU. Goldman sold clients complex stock-and-derivative combinations. As he discussed it with Lopes Lara, they agreed that investors should hedge their risk by betting on Brexit itself rather than an imperfect proxy.

Lopes Lara and Mansour hypothesized how a marketplace might work. They settled on a "event contract," a binary-outcome instrument like "Will inflation hit 5% by the end of the month?" The contract would settle at $1 (if the event happened) or zero (if it didn't), but its price would fluctuate based on market sentiment. After a good debate, a politician's election odds may rise from 50 to 55. Kalshi would charge a commission on every trade and sell data to traders, political campaigns, businesses, and others.

In October 2018, five months after graduation, the pair flew to California to compete in a hackathon for wannabe tech founders organized by the Silicon Valley incubator Y Combinator. They built a website in a day and a night and presented it to entrepreneurs the next day. Their prototype barely worked, but they won a three-month mentorship program and $150,000. Michael Seibel, managing director of Y Combinator, said of their idea, "I had to take a chance!"

Seibel's skepticism was rooted in America's historical wariness of gambling. Roulette, poker, and other online casino games are largely illegal, and sports betting was only legal in a few states until May 2018. Kalshi as a risk-hedging platform rather than a bookmaker seemed like a good idea, but convincing the CFTC wouldn't be easy. In 2012, the CFTC said trading on politics had no "economic purpose" and was "contrary to the public interest."

Lopes Lara and Mansour cold-called 60 Googled lawyers during their time at Y Combinator. Everyone advised quitting. Mansour recalls the pain. Jeff Bandman, a former CFTC official, helped them navigate the agency and its characters.

When they weren’t busy trying to recruit lawyers, Lopes Lara and Mansour were meeting early-stage investors. Alfred Lin of Sequoia Capital Operations LLC backed Airbnb, DoorDash, and Uber Technologies. Lin told the founders their idea could capitalize on retail trading and challenge how the financial world manages risk. "Come back with regulatory approval," he said.

In the US, even small bets on most events were once illegal. Under the Commodity Exchange Act, the CFTC can stop exchanges from listing contracts relating to "terrorism, assassination, war" and "gaming" if they are "contrary to the public interest," which was often the case.

In 1988, as academic interest in the field grew, the agency allowed the University of Iowa to set up a prediction market for research purposes, as long as it didn't make a profit or advertise and limited bets to $500. PredictIt, the biggest and best-known political betting platform in the US, also got an exemption thanks to an association with Victoria University of Wellington in New Zealand. Today, it's a sprawling marketplace with its own subculture and lingo. PredictIt users call it "Rules Cuck Panther" when they lose on a technicality. Major news outlets cite PredictIt's odds on Discord and the Star Spangled Gamblers podcast.

CFTC limits PredictIt bets to $850. To keep traders happy, PredictIt will often run multiple variations of the same question, listing separate contracts for two dozen Democratic primary candidates, for example. A trader could have more than $10,000 riding on a single outcome. Some of the site's traders are current or former campaign staffers who can answer questions like "How many tweets will Donald Trump post from Nov. 20 to 27?" and "When will Anthony Scaramucci's role as White House communications director end?"

According to PredictIt co-founder John Phillips, politicians help explain the site's accuracy. "Prediction markets work well and are accurate because they attract people with superior information," he said in a 2016 podcast. “In the financial stock market, it’s called inside information.”

Trading on nonpublic information is illegal outside of academia, which presented a dilemma for Lopes Lara and Mansour. Kalshi's forecasts needed to be accurate. Kalshi must eliminate insider trading as a regulated entity. Lopes Lara and Mansour wanted to build a high-stakes PredictIt without the anarchy or blurred legal lines—a "New York Stock Exchange for Events." First, they had to convince regulators event trading was safe.

When Lopes Lara and Mansour approached the CFTC in the spring of 2019, some officials in the Division of Market Oversight were skeptical, according to interviews with people involved in the process. For all Kalshi's talk of revolutionizing finance, this was just a turbocharged version of something that had been rejected before.

The DMO couldn't see the big picture. The staff review was supposed to ensure Kalshi could complete a checklist, "23 Core Principles of a Designated Contract Market," which included keeping good records and having enough money. The five commissioners decide. With Trump as president, three of them were ideologically pro-market.

Lopes Lara, Mansour, and their lawyer Bandman, an ex-CFTC official, answered the DMO's questions while lobbying the commissioners on Zoom about the potential of event markets to mitigate risks and make better decisions. Before each meeting, they would write a script and memorize it word for word.

Several prediction markets that hadn't sought regulatory approval bolstered Kalshi's case. Polymarket let customers bet hundreds of thousands of dollars anonymously using cryptocurrencies, making it hard to track. Augur, which facilitates private wagers between parties using blockchain, couldn't regulate bets and hadn't stopped users from betting on assassinations. Kalshi, by comparison, argued it was doing everything right. (The CFTC fined Polymarket $1.4 million for operating an unlicensed exchange in January 2022. Polymarket says it's now compliant and excited to pioneer smart contract-based financial solutions with regulators.

Kalshi was approved unanimously despite some DMO members' concerns about event contracts' riskiness. "Once they check all the boxes, they're in," says a CFTC insider.

Three months after CFTC approval, Kalshi announced funding from Sequoia, Charles Schwab, and Henry Kravis. Sequoia's Lin, who joined the board, said Tarek, Luana, and team created a new way to invest and engage with the world.

The CFTC hadn't asked what markets the exchange planned to run since. After approval, Lopes Lara and Mansour had the momentum. Kalshi's March list of 30 proposed contracts caused chaos at the DMO. The division handles exchanges that create two or three new markets a year. Kalshi’s business model called for new ones practically every day.

Uncontroversial proposals included weather and GDP questions. Others, on the initial list and later, were concerning. DMO officials feared Covid-19 contracts amounted to gambling on human suffering, which is why war and terrorism markets are banned. (Similar logic doomed ex-admiral John Poindexter's Policy Analysis Market, a Bush-era plan to uncover intelligence by having security analysts bet on Middle East events.) Regulators didn't see how predicting the Grammy winners was different from betting on the Patriots to win the Super Bowl. Who, other than John Legend, would need to hedge the best R&B album winner?

Event contracts raised new questions for the DMO's product review team. Regulators could block gaming contracts that weren't in the public interest under the Commodity Exchange Act, but no one had defined gaming. It was unclear whether the CFTC had a right or an obligation to consider whether a contract was in the public interest. How was it to determine public interest? Another person familiar with the CFTC review says, "It was a mess." The agency didn't comment.

CFTC staff feared some event contracts could be cheated. Kalshi wanted to run a bee-endangerment market. The DMO pushed back, saying it saw two problems symptomatic of the asset class: traders could press government officials for information, and officials could delay adding the insects to the list to cash in.

The idea that traders might manipulate prediction markets wasn't paranoid. In 2013, academics David Rothschild and Rajiv Sethi found that an unidentified party lost $7 million buying Mitt Romney contracts on Intrade, a now-defunct, unlicensed Irish platform, in the runup to the 2012 election. The authors speculated that the trader, whom they dubbed the “Romney Whale,” may have been looking to boost morale and keep donations coming in.

Kalshi said manipulation and insider trading are risks for any market. It built a surveillance system and said it would hire a team to monitor it. "People trade on events all the time—they just use options and other instruments. This brings everything into the open, Mansour says. Kalshi didn't include election contracts, a red line for CFTC Democrats.

Lopes Lara and Mansour were ready to launch kalshi.com that summer, but the DMO blocked them. Product reviewers were frustrated by spending half their time on an exchange that represented a tiny portion of the derivatives market. Lopes Lara and Mansour pressed politically appointed commissioners during the impasse.

Tarbert, the chairman, had moved on, but Kalshi found a new supporter in Republican Brian Quintenz, a crypto-loving former hedge fund manager. He was unmoved by the DMO's concerns, arguing that speculation on Kalshi's proposed events was desirable and the agency had no legal standing to prevent it. He supported a failed bid to allow NFL futures earlier this year. Others on the commission were cautious but supportive. Given the law's ambiguity, they worried they'd be on shaky ground if Kalshi sued if they blocked a contract. Without a permanent chairman, the agency lacked leadership.

To block a contract, DMO staff needed a majority of commissioners' support, which they didn't have in all but a few cases. "We didn't have the votes," a reviewer says, paraphrasing Hamilton. By the second half of 2021, new contract requests were arriving almost daily at the DMO, and the demoralized and overrun division eventually accepted defeat and stopped fighting back. By the end of the year, three senior DMO officials had left the agency, making it easier for Kalshi to list its contracts unimpeded.

Today, Kalshi is growing. 32 employees work in a SoHo office with big windows and exposed brick. Quintenz, who left the CFTC 10 months after Kalshi was approved, is on its board. He joined because he was interested in the market's hedging and risk management opportunities.

Mid-May, the company's website had 75 markets, such as "Will Q4 GDP be negative?" Will NASA land on the moon by 2025? The exchange recently reached 2 million weekly contracts, a jump from where it started but still a small number compared to other futures exchanges. Early adopters are PredictIt and Polymarket fans. Bets on the site are currently capped at $25,000, but Kalshi hopes to increase that to $100,000 and beyond.

With the regulatory drawbridge down, Lopes Lara and Mansour must move quickly. Chicago's CME Group Inc. plans to offer index-linked event contracts. Kalshi will release a smartphone app to attract customers. After that, it hopes to partner with a big brokerage. Sequoia is a major investor in Robinhood Markets Inc. Robinhood users could have access to Kalshi so that after buying GameStop Corp. shares, they'd be prompted to bet on the Oscars or the next Fed commissioner.

Some, like Illinois Democrat Sean Casten, accuse Robinhood and its competitors of gamifying trading to encourage addiction, but Kalshi doesn't seem worried. Mansour says Kalshi's customers can't bet more than they've deposited, making debt difficult. Eventually, he may introduce leveraged bets.

Tension over event contracts recalls another CFTC episode. Brooksley Born proposed regulating the financial derivatives market in 1994. Alan Greenspan and others in the government opposed her, saying it would stifle innovation and push capital overseas. Unrestrained, derivatives grew into a trillion-dollar industry until 2008, when they sparked the financial crisis.

Today, with a midterm election looming, it seems reasonable to ask whether Kalshi plans to get involved. Elections have historically been the biggest draw in prediction markets, with 125 million shares traded on PredictIt for 2020. “We can’t discuss specifics,” Mansour says. “All I can say is, you know, we’re always working on expanding the universe of things that people can trade on.”

Any election contracts would need CFTC approval, which may be difficult with three Democratic commissioners. A Republican president would change the equation.

What are pre-sales and whitelists on NFTs?

An NFT pre-sale, as the name implies, allows community members or early supporters of an NFT project to mint before the public, usually via a whitelist or mint pass.

Coin collectors can use mint passes to claim NFTs during the public sale. Because the mint pass is executed by “burning” an NFT into a specific crypto wallet, the collector is not concerned about gas price spikes.

A whitelist is used to approve a crypto wallet address for an NFT pre-sale. In a similar way to an early access list, it guarantees a certain number of crypto wallets can mint one (or more) NFT.

New NFT projects can do a pre-sale without a whitelist, but whitelists are good practice to avoid gas wars and a fair shot at minting an NFT before launching in competitive NFT marketplaces like Opensea, Magic Eden, or CNFT.

Should NFT projects do pre-sales or whitelists? 👇

The reasons to do pre-sales or a whitelist for NFT creators:

Time the market and gain traction.

Pre-sale or whitelists can help NFT projects gauge interest early on.

Whitelist spots filling up quickly is usually a sign of a successful launch, though it does not guarantee NFT longevity (more on that later). Also, full whitelists create FOMO and momentum for the public sale among non-whitelisted NFT collectors.

If whitelist signups are low or slow, projects may need to work on their vision, community, or product. Or the market is in a bear cycle. In either case, it aids NFT projects in market timing.

Reward the early NFT Community members.

Pre-sale and whitelists can help NFT creators reward early supporters.

First, by splitting the minting process into two phases, early adopters get a chance to mint one or more NFTs from their collection at a discounted or even free price.

Did you know that BAYC started at 0.08 eth each? A serum that allowed you to mint a Mutant Ape has become as valuable as the original BAYC.

(2) Whitelists encourage early supporters to help build a project's community in exchange for a slot or status. If you invite 10 people to the NFT Discord community, you get a better ranking or even a whitelist spot.

Pre-sale and whitelisting have become popular ways for new projects to grow their communities and secure future buyers.

Prevent gas wars.

Most new NFTs are created on the Ethereum blockchain, which has the highest transaction fees (also known as gas) (Solana, Cardano, Polygon, Binance Smart Chain, etc).

An NFT public sale is a gas war when a large number of NFT collectors (or bots) try to mint an NFT at the same time.

Competing collectors are willing to pay higher gas fees to prioritize their transaction and out-price others when upcoming NFT projects are hyped and very popular.

Pre-sales and whitelisting prevent gas wars by breaking the minting process into smaller batches of members or season launches.

The reasons to do pre-sales or a whitelists for NFT collectors:

How do I get on an NFT whitelist?

1. Popular NFT collections act as a launchpad for other new or hyped NFT collections.

Example: Interfaces NFTs gives out 100 whitelist spots to Deadfellaz NFTs holders. Both NFT projects win. Interfaces benefit from Deadfellaz's success and brand equity.

In this case, to get whitelisted NFT collectors need to hold that specific NFT that is acting like a launchpad.

1. A NFT studio or collection that launches a new NFT project and rewards previous NFT holders with whitelist spots or pre-sale access.

The whitelist requires previous NFT holders or community members.

NFT Alpha Groups are closed, small, tight-knit Discord servers where members share whitelist spots or giveaways from upcoming NFTs.

The benefit of being in an alpha group is getting information about new NFTs first and getting in on pre-sale/whitelist before everyone else.

There are some entry barriers to alpha groups, but if you're active in the NFT community, you'll eventually bump into, be invited to, or form one.

1. A whitelist spot is awarded to members of an NFT community who are the most active and engaged.

This participation reward is the most democratic. To get a chance, collectors must work hard and play to their strengths.

Whitelisting participation examples:

• Raffle, games and contest: NFT Community raffles, games, and contests. To get a whitelist spot, invite 10 people to X NFT Discord community.
• Fan art: To reward those who add value and grow the community by whitelisting the best fan art and/or artists is only natural.
• Giveaways: Lucky number crypto wallet giveaways promoted by an NFT community. To grow their communities and for lucky collectors, NFT projects often offer free NFT.
• Activate your voice in the NFT Discord Community. Use voice channels to get NFT teams' attention and possibly get whitelisted.

The advantage of whitelists or NFT pre-sales.

Chainalysis's NFT stats quote is the best answer:

Sure, it's not all about cash. However, any NFT collector should feel secure in their investment by owning a piece of a valuable and thriving NFT project. These stats help collectors understand that getting in early on an NFT project (via whitelist or pre-sale) will yield a better and larger return.

The downsides of pre-sales & whitelists for NFT creators.

Pre-sales and whitelist can cause issues for NFT creators and collectors.

NFT flippers

NFT collectors who only want to profit from early minting (pre-sale) or low mint cost (via whitelist). To sell the NFT in a secondary market like Opensea or Solanart, flippers go after the discounted price.

For example, a 1000 Solana NFT collection allows 100 people to mint 1 Solana NFT at 0.25 SOL. The public sale price for the remaining 900 NFTs is 1 SOL. If an NFT collector sells their discounted NFT for 0.5 SOL, the secondary market floor price is below the public mint.

This may deter potential NFT collectors. Furthermore, without a cap in the pre-sale minting phase, flippers can get as many NFTs as possible to sell for a profit, dumping them in secondary markets and driving down the floor price.

Hijacking NFT sites, communities, and pre-sales phase

People try to scam the NFT team and their community by creating oddly similar but fake websites, whitelist links, or NFT's Discord channel.

Established and new NFT projects must be vigilant to always make sure their communities know which are the official links, how a whitelist or pre-sale rules and how the team will contact (or not) community members.

Another way to avoid the scams around the pre-sale phase, NFT projects opt to create a separate mint contract for the whitelisted crypto wallets and then another for the public sale phase.

Scam NFT projects

We've seen a lot of mid-mint or post-launch rug pulls, indicating that some bad NFT projects are trying to scam NFT communities and marketplaces for quick profit. What happened to Magic Eden's launchpad recently will help you understand the scam.

We discussed the benefits and drawbacks of NFT pre-sales and whitelists for both projects and collectors. 

Finally, some practical tools and tips for finding new NFTs 👇

Tools & resources to find new NFT on pre-sale or to get on a whitelist:

In order to never miss an update, important pre-sale dates, or a giveaway, create a Tweetdeck or Tweeten Twitter dashboard with hyped NFT project pages, hashtags ( #NFTGiveaways , #NFTCommunity), or big NFT influencers.

Search for upcoming NFT launches that have been vetted by the marketplace and try to get whitelisted before the public launch.

Save-timing discovery platforms like sealaunch.xyz for NFT pre-sales and upcoming launches. How can we help 100x NFT collectors get projects? A project's official social media links, description, pre-sale or public sale dates, price and supply. We're also working with Dune on NFT data analysis to help NFT collectors make better decisions.

Don't invest what you can't afford to lose because a) the project may fail or become rugged. Find NFTs projects that you want to be a part of and support.

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