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.

How do you DAO? Can DAOs scale?

DAO: Decentralized Autonomous. Organization.

“The whole phrase is a misnomer. They're not decentralized, autonomous, or organizations,” says Monsterplay blockchain consultant David Freuden.

As part of the DAO initiative, Freuden coauthored a 51-page report in May 2020. “We need DAOs,” he says. “‘Shareholder first' is a 1980s/90s concept. Profits became the focus, not products.”

His predictions for DAOs have come true nearly two years later. DAOs had over 1.6 million participants by the end of 2021, up from 13,000 at the start of the year. Wyoming, in the US, will recognize DAOs and the Marshall Islands in 2021. Australia may follow that example in 2022.

But what is a DAO?

Members buy (or are rewarded with) governance tokens to vote on how the DAO operates and spends its money. “DeFi spawned DAOs as an investment vehicle. So a DAO is tokenomics,” says Freuden.

DAOs are usually built around a promise or a social cause, but they still want to make money. “If you can't explain why, the DAO will fail,” he says. “A co-op without tokenomics is not a DAO.”

Operating system DAOs, protocol DAOs, investment DAOs, grant DAOs, service DAOs, social DAOs, collector DAOs, and media DAOs are now available.

Freuden liked the idea of people rallying around a good cause. Speculators and builders make up the crypto world, so it needs a DAO for them.

,Speculators and builders, or both, have mismatched expectations, causing endless, but sometimes creative friction.

Organisms that boost output

Launching a DAO with an original product such as a cryptocurrency, an IT protocol or a VC-like investment fund like FlamingoDAO is common. DAOs enable distributed open-source contributions without borders. The goal is vital. Sometimes, after a product is launched, DAOs emerge, leaving the company to eventually transition to a DAO, as Uniswap did.

Doing things together is a DAO. So it's a way to reward a distributed workforce. DAOs are essentially productivity coordination organisms.

“Those who work for the DAO make permissionless contributions and benefit from fragmented employment,” argues Freuden. DAOs are, first and foremost, a new form of cooperation.

DAO? Distributed not decentralized

In decentralized autonomous organizations, words have multiple meanings. DAOs can emphasize one aspect over another. Autonomy is a trade-off for decentralization.

DAOstack CEO Matan Field says a DAO is a distributed governance system. Power is shared. However, there are two ways to understand a DAO's decentralized nature. This clarifies the various DAO definitions.

A decentralized infrastructure allows a DAO to be decentralized. It could be created on a public permissionless blockchain to prevent a takeover.

As opposed to a company run by executives or shareholders, a DAO is distributed. Its leadership does not wield power

Option two is clearly distributed.

But not all of this is “automated.”

Think quorum, not robot.

DAOs can be autonomous in the sense that smart contracts are self-enforcing and self-executing. So every blockchain transaction is a simplified smart contract.

Dao landscape

The DAO landscape is evolving.

Consider how Ethereum's smart contracts work. They are more like self-executing computer code, which Vitalik Buterin calls “persistent scripts”.

However, a DAO is self-enforcing once its members agree on its rules. As such, a DAO is “automated upon approval by the governance committee.” This distinguishes them from traditional organizations whose rules must be interpreted and applied.

Why a DAO? They move fast

A DAO can quickly adapt to local conditions as a governance mechanism. It's a collaborative decision-making tool.

Like UkraineDAO, created in response to Putin's invasion of Ukraine by Ukrainian expat Alona Shevchenko, Nadya Tolokonnikova, Trippy Labs, and PleasrDAO. The DAO sought to support Ukrainian charities by selling Ukrainian flag NFTs. With a single mission, a DAO can quickly raise funds for a country accepting crypto where banks are distrusted.

This could be a watershed moment for DAOs.

ConstitutionDAO was another clever use case for DAOs for Freuden. In a failed but “beautiful experiment in a single-purpose DAO,” ConstitutionDAO tried to buy a copy of the US Constitution from a Sotheby's auction. In November 2021, ConstitutionDAO raised $47 million from 19,000 people, but a hedge fund manager outbid them.

Contributions were returned or lost if transactional gas fees were too high. The ConstitutionDAO, as a “beautiful experiment,” proved exceptionally fast at organizing and crowdsourcing funds for a specific purpose.

We may soon be applauding UkraineDAO's geopolitical success in support of the DAO concept.

Some of the best use cases for DAOs today, according to Adam Miller, founder of DAOplatform.io and MIDAO Directory Services, involve DAO structures.

That is, a “flat community is vital.” Prototyping by the crowd is a good example.  To succeed,  members must be enthusiastic about DAOs as an alternative to starting a company. Because DAOs require some hierarchy, he agrees that "distributed is a better acronym."

Miller sees DAOs as a “new way of organizing people and resources.” He started DAOplatform.io, a DAO tooling advisery that is currently transitioning to a DAO due to the “woeful tech options for running a DAO,” which he says mainly comprises of just “multisig admin keys and a voting system.” So today he's advising on DAO tech stacks.

Miller identifies three key elements.

How a DAO works...

They can be more than glorified Discord groups if they have a clear mission. This mission is a mix of financial speculation and utopianism. The spectrum is vast.

The founder of Dash left the cryptocurrency project in 2017. It's the story of a prophet without an heir. So creating a global tokenized evangelical missionary community via a DAO made sense.

Evan Duffield, a “libertarian/anarchist” visionary, forked Bitcoin in January 2014 to make it instant and essentially free. He went away for a while, and DASH became a DAO.

200,000 US retailers, including Walmart and Barnes & Noble, now accept Dash as payment. This payment system works like a gift card.

Arden Goldstein, Dash's head of crypto, DAO, and blockchain marketing, claims Dash is the “first successful DAO.” It was founded in 2016 and disbanded after a hack, an Ethereum hard fork and much controversy. But what are the success metrics?

Crypto success is measured differently, says Goldstein. To achieve common goals, people must participate or be motivated in a healthy DAO. People are motivated to complete tasks in a successful DAO. And, crucially, when tasks get completed.

“Yes or no, 1 or 0, voting is not a new idea. The challenge is getting people to continue to participate and keep building a community.” A DAO motivates volunteers: Nothing keeps people from building. The DAO “philosophy is old news. You need skin in the game to play.”

MasterNodes must stake 1000 Dash. Those members are rewarded with DASH for marketing (and other tasks). It uses an outsourced team to onboard new users globally.

Joining a DAO is part of the fun of meeting crazy or “very active” people on Discord. No one gets fired (usually). If your work is noticed, you may be offered a full-time job.

DAO community members worldwide are rewarded for brand building. Dash is also a great product for developing countries with high inflation and undemocratic governments. The countries with the most Dash DAO members are Russia, Brazil, Venezuela, India, China, France, Italy, and the Philippines.

Grassroots activism makes this DAO work. A DAO is local. Venezuelans can't access Dash.org, so DAO members help them use a VPN. DAO members are investors, fervent evangelicals, and local product experts.

Every month, proposals and grant applications are voted on via the Dash platform. However, the DAO may decide not to fund you. For example, the DAO once hired a PR firm, but the community complained about the lack of press coverage. This raises a great question: How are real-world contractual obligations met by a DAO?

Does the DASH DAO work?

“I see the DAO defund projects I thought were valuable,” Goldstein says. Despite working full-time, I must submit a funding proposal. “Much faster than other companies I've worked on,” he says.

Dash DAO is a headless beast. Ryan Taylor is the CEO of the company overseeing the DASH Core Group project. 

The issue is that “we don't know who has the most tokens [...] because we don't know who our customers are.” As a result, “the loudest voices usually don't have the most MasterNodes and aren't the most invested.”

Goldstein, the only female in the DAO, says she worked hard. “I was proud of the DAO when I made the logo pink for a day and got great support from the men.” This has yet to entice a major influx of female DAO members.

Many obstacles stand in the way of utopian dreams.

Governance problems remain

And what about major token holders behaving badly?

In early February, a heated crypto Twitter debate raged on about inclusion, diversity, and cancel culture in relation to decentralized projects. In this case, the question was how a DAO addresses alleged inappropriate behavior.

In a corporation, misconduct can result in termination. In a DAO, founders usually hold a large number of tokens and the keys to the blockchain (multisignature) or otherwise.

Brantly Millegan, the director of operations of Ethereum Name Service (ENS), made disparaging remarks about the LGBTQ community and other controversial topics. The screenshotted comments were made in 2016 and brought to the ENS board's attention in early 2022.

His contract with ENS has expired. But what of his large DAO governance token holdings?

Members of the DAO proposed a motion to remove Millegan from the DAO. His “delegated” votes net 370,000. He was and is the DAO's largest delegate.

What if he had refused to accept the DAO's decision?

Freuden says the answer is not so simple.

The original mission “should be dissolved” if it no longer exists. “Does a DAO fail and return the money? They must r eturn the money with interest if the marriage fails.”

Before an IPO, VCs might try to remove a problematic CEO.

While DAOs use treasury as a governance mechanism, it is usually controlled (at least initially) by the original project creators. Or, in the case of Uniswap, the venture capital firm a16z has so much voting power that it has delegated it to student-run blockchain organizations.

So, can DAOs really work at scale? How to evolve voting paradigms beyond token holdings?

The whale token holder issue has some solutions. Multiple tokens, such as a utility token on top of a governance token, and quadratic voting for whales, are now common. Other safeguards include multisignature blockchain keys and decision time locks that allow for any automated decision to be made. The structure of each DAO will depend on the assets at stake.

In reality, voter turnout is often a bigger issue.

Is DAO governance scalable?

Many DAOs have low participation. Due to a lack of understanding of technology, apathy, or busy lives. “The bigger the DAO, the fewer voters who vote,” says Freuden.

Freuden's report cites British anthropologist Dunbar's Law, who argued that people can only maintain about 150 relationships.

Freuden says a DAO requires “understanding human relationships.” He believes DAOs work best as investment funds rooted in Cryptoland and small in scale. In just three weeks, SyndicateDAO enabled the creation of 450 new investment group DAOs.

Due to SEC regulations, FlamingoDAO, a famous NFT curation investment DAO, could only have 100 investors. The “LAO” is a member-directed venture capital fund and a US LLC. To comply with US securities law, they only allow 100 members with a 120ETH minimum staking contribution.

But how did FlamingoDAO make investment decisions? How often did all 70 members vote? Art and NFTs are highly speculative.

So, investment DAOs are thought to work well in a small petri dish environment. This is due to a crypto-native club's pooled capital (maximum 7% per member) and crowdsourced knowledge.

While scalability is a concern, each DAO will operate differently depending on the goal, technology stage, and personalities. Meetups and hackathons are common ways for techies to collaborate on a cause or test an idea. But somebody still organizes the hack.

Holographic consensus voting

But clever people are working on creative solutions to every problem.

Miller of DAOplatform.io cites DXdao as a successful DAO. Decentralized product and service creator DXdao runs the DAO entirely on-chain. “You earn voting rights by contributing to the community.”

DXdao, a DAOstack fork, uses holographic consensus, a voting algorithm invented by DAOstack founder Matan Field. The system lets a random or semi-random subset make group-wide decisions.

By acting as a gatekeeper for voters, DXdao's Luke Keenan explains that “a small predictions market economy emerges around the likely outcome of a proposal as tokens are staked on it.” Also, proposals that have been financially boosted have fewer requirements to be successful, increasing system efficiency.” DXdao “makes decisions by removing voting power as an economic incentive.”

Field explains that holographic consensus “does not require a quorum to render a vote valid.”

You don't need big votes on everything. If only 5% vote, fine. To move significant value or make significant changes, you need a longer voting period (say 30 days) and a higher quorum,” says Miller.

Clearly, DAOs are maturing. The emphasis is on tools like Orca and processes that delegate power to smaller sub-DAOs, committees, and working groups.

Miller also claims that “studies in psychology show that rewarding people too much for volunteering disincentivizes them.” So, rather than giving out tokens for every activity, you may want to offer symbolic rewards like POAPs or contributor levels.

Culture and motivation

DAOs (and Web3 in general) can give early adopters a sense of ownership. In theory, they encourage early participation and bootstrapping before network effects.

"A double-edged sword," says Goldstein. In the developing world, they may not be fully scalable.

“There must always be a leader,” she says. “People won't volunteer if they don't want to.”

DAO members sometimes feel entitled. “They are not the boss, but they think they should be able to see my calendar or get a daily report,” Goldstein gripes. Say, “I own three MasterNodes and need to know X, Y, and Z.”

In most decentralized projects, strong community leaders are crucial to influencing culture.

Freuden says “the DAO's community builder is the cryptoland influencer.” They must “disseminate the DAO's culture, cause, and rally the troops” in English, not tech.

They must keep members happy.

So the community builder is vital. Building a community around a coin that promises riches is simple, but keeping DAO members motivated is difficult.

It's a human job. But tools like SourceCred or coordinate that measure contributions and allocate tokens are heavily marketed. Large growth funds/community funds/grant programs are common among DAOs.

The Future?

Onboarding, committed volunteers, and an iconic community builder may be all DAOs need.

It takes a DAO just one day to bring together a passionate (and sometimes obsessive) community. For organizations with a common goal, managing stakeholder expectations is critical.

A DAO's core values are community and cause, not scalable governance. “DAOs will work at scale like gaming communities, but we will have sub-DAOs everywhere like committees,” says Freuden.

So-called holographic consensuses “can handle, in principle, increasing rates of proposals by turning this tension between scale and resilience into an economical cost,” Field writes. Scalability is not guaranteed.

The DAO's key innovation is the fragmented workplace. “Voting is a subset of engagement,” says Freuden. DAO should allow for permissionless participation and engagement. DAOs allow for remote work.”

In 20 years, DAOs may be the AI-powered self-organizing concept. That seems far away now. But a new breed of productivity coordination organisms is maturing.