An approximate introduction to how zk-SNARKs are possible (part 2)
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.
see part 3
(Edited)
Hackernoon
4 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
4 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Ryan Weeks
3 years ago
Terra fiasco raises TRON's stablecoin backstop
After Terra's algorithmic stablecoin collapsed in May, TRON announced a plan to increase the capital backing its own stablecoin.
USDD, a near-carbon copy of Terra's UST, arrived on the TRON blockchain on May 5. TRON founder Justin Sun says USDD will be overcollateralized after initially being pegged algorithmically to the US dollar.
A reserve of cryptocurrencies and stablecoins will be kept at 130 percent of total USDD issuance, he said. TRON described the collateral ratio as "guaranteed" and said it would begin publishing real-time updates on June 5.
Currently, the reserve contains 14,040 bitcoin (around $418 million), 140 million USDT, 1.9 billion TRX, and 8.29 billion TRX in a burning contract.
Sun: "We want to hybridize USDD." We have an algorithmic stablecoin and TRON DAO Reserve.
algorithmic failure
USDD was designed to incentivize arbitrageurs to keep its price pegged to the US dollar by trading TRX, TRON's token, and USDD. Like Terra, TRON signaled its intent to establish a bitcoin and cryptocurrency reserve to support USDD in extreme market conditions.
Still, Terra's UST failed despite these safeguards. The stablecoin veered sharply away from its dollar peg in mid-May, bringing down Terra's LUNA and wiping out $40 billion in value in days. In a frantic attempt to restore the peg, billions of dollars in bitcoin were sold and unprecedented volumes of LUNA were issued.
Sun believes USDD, which has a total circulating supply of $667 million, can be backed up.
"Our reserve backing is diversified." Bitcoin and stablecoins are included. USDC will be a small part of Circle's reserve, he said.
TRON's news release lists the reserve's assets as bitcoin, TRX, USDC, USDT, TUSD, and USDJ.
All Bitcoin addresses will be signed so everyone knows they belong to us, Sun said.
Not giving in
Sun told that the crypto industry needs "decentralized" stablecoins that regulators can't touch.
Sun said the Luna Foundation Guard, a Singapore-based non-profit that raised billions in cryptocurrency to buttress UST, mismanaged the situation by trying to sell to panicked investors.
He said, "We must be ahead of the market." We want to stabilize the market and reduce volatility.
Currently, TRON finances most of its reserve directly, but Sun says the company hopes to add external capital soon.
Before its demise, UST holders could park the stablecoin in Terra's lending platform Anchor Protocol to earn 20% interest, which many deemed unsustainable. TRON's JustLend is similar. Sun hopes to raise annual interest rates from 17.67% to "around 30%."
This post is a summary. Read full article here
The Verge
3 years ago
Bored Ape Yacht Club creator raises $450 million at a $4 billion valuation.
Yuga Labs, owner of three of the biggest NFT brands on the market, announced today a $450 million funding round. The money will be used to create a media empire based on NFTs, starting with games and a metaverse project.
The team's Otherside metaverse project is an MMORPG meant to connect the larger NFT universe. They want to create “an interoperable world” that is “gamified” and “completely decentralized,” says Wylie Aronow, aka Gordon Goner, co-founder of Bored Ape Yacht Club. “We think the real Ready Player One experience will be player run.”
Just a few weeks ago, Yuga Labs announced the acquisition of CryptoPunks and Meebits from Larva Labs. The deal brought together three of the most valuable NFT collections, giving Yuga Labs more IP to work with when developing games and metaverses. Last week, ApeCoin was launched as a cryptocurrency that will be governed independently and used in Yuga Labs properties.
Otherside will be developed by “a few different game studios,” says Yuga Labs CEO Nicole Muniz. The company plans to create development tools that allow NFTs from other projects to work inside their world. “We're welcoming everyone into a walled garden.”
However, Yuga Labs believes that other companies are approaching metaverse projects incorrectly, allowing the startup to stand out. People won't bond spending time in a virtual space with nothing going on, says Yuga Labs co-founder Greg Solano, aka Gargamel. Instead, he says, people bond when forced to work together.
In order to avoid getting smacked, Solano advises making friends. “We don't think a Zoom chat and walking around saying ‘hi' creates a deep social experience.” Yuga Labs refused to provide a release date for Otherside. Later this year, a play-to-win game is planned.
The funding round was led by Andreessen Horowitz, a major investor in the Web3 space. It previously backed OpenSea and Coinbase. Animoca Brands, Coinbase, and MoonPay are among those who have invested. Andreessen Horowitz general partner Chris Lyons will join Yuga Labs' board. The Financial Times broke the story last month.
"META IS A DOMINANT DIGITAL EXPERIENCE PROVIDER IN A DYSTOPIAN FUTURE."
This emerging [Web3] ecosystem is important to me, as it is to companies like Meta,” Chris Dixon, head of Andreessen Horowitz's crypto arm, tells The Verge. “In a dystopian future, Meta is the dominant digital experience provider, and it controls all the money and power.” (Andreessen Horowitz co-founder Marc Andreessen sits on Meta's board and invested early in Facebook.)
Yuga Labs has been profitable so far. According to a leaked pitch deck, the company made $137 million last year, primarily from its NFT brands, with a 95% profit margin. (Yuga Labs declined to comment on deck figures.)
But the company has built little so far. According to OpenSea data, it has only released one game for a limited time. That means Yuga Labs gets hundreds of millions of dollars to build a gaming company from scratch, based on a hugely lucrative art project.
Investors fund Yuga Labs based on its success. That's what they did, says Dixon, “they created a culture phenomenon”. But ultimately, the company is betting on the same thing that so many others are: that a metaverse project will be the next big thing. Now they must construct it.
CNET
4 years ago
How a $300K Bored Ape Yacht Club NFT was accidentally sold for $3K
The Bored Ape Yacht Club is one of the most prestigious NFT collections in the world. A collection of 10,000 NFTs, each depicting an ape with different traits and visual attributes, Jimmy Fallon, Steph Curry and Post Malone are among their star-studded owners. Right now the price of entry is 52 ether, or $210,000.
Which is why it's so painful to see that someone accidentally sold their Bored Ape NFT for $3,066.
Unusual trades are often a sign of funny business, as in the case of the person who spent $530 million to buy an NFT from themselves. In Saturday's case, the cause was a simple, devastating "fat-finger error." That's when people make a trade online for the wrong thing, or for the wrong amount. Here the owner, real name Max or username maxnaut, meant to list his Bored Ape for 75 ether, or around $300,000. Instead he accidentally listed it for 0.75. One hundredth the intended price.
It was bought instantaneously. The buyer paid an extra $34,000 to speed up the transaction, ensuring no one could snap it up before them. The Bored Ape was then promptly listed for $248,000. The transaction appears to have been done by a bot, which can be coded to immediately buy NFTs listed below a certain price on behalf of their owners in order to take advantage of these exact situations.
"How'd it happen? A lapse of concentration I guess," Max told me. "I list a lot of items every day and just wasn't paying attention properly. I instantly saw the error as my finger clicked the mouse but a bot sent a transaction with over 8 eth [$34,000] of gas fees so it was instantly sniped before I could click cancel, and just like that, $250k was gone."
"And here within the beauty of the Blockchain you can see that it is both honest and unforgiving," he added.
Fat finger trades happen sporadically in traditional finance -- like the Japanese trader who almost bought 57% of Toyota's stock in 2014 -- but most financial institutions will stop those transactions if alerted quickly enough. Since cryptocurrency and NFTs are designed to be decentralized, you essentially have to rely on the goodwill of the buyer to reverse the transaction.
Fat finger errors in cryptocurrency trades have made many a headline over the past few years. Back in 2019, the company behind Tether, a cryptocurrency pegged to the US dollar, nearly doubled its own coin supply when it accidentally created $5 billion-worth of new coins. In March, BlockFi meant to send 700 Gemini Dollars to a set of customers, worth roughly $1 each, but mistakenly sent out millions of dollars worth of bitcoin instead. Last month a company erroneously paid a $24 million fee on a $100,000 transaction.
Similar incidents are increasingly being seen in NFTs, now that many collections have accumulated in market value over the past year. Last month someone tried selling a CryptoPunk NFT for $19 million, but accidentally listed it for $19,000 instead. Back in August, someone fat finger listed their Bored Ape for $26,000, an error that someone else immediately capitalized on. The original owner offered $50,000 to the buyer to return the Bored Ape -- but instead the opportunistic buyer sold it for the then-market price of $150,000.
"The industry is so new, bad things are going to happen whether it's your fault or the tech," Max said. "Once you no longer have control of the outcome, forget and move on."
The Bored Ape Yacht Club launched back in April 2021, with 10,000 NFTs being sold for 0.08 ether each -- about $190 at the time. While NFTs are often associated with individual digital art pieces, collections like the Bored Ape Yacht Club, which allow owners to flaunt their NFTs by using them as profile pictures on social media, are becoming increasingly prevalent. The Bored Ape Yacht Club has since become the second biggest NFT collection in the world, second only to CryptoPunks, which launched in 2017 and is considered the "original" NFT collection.
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nft now
3 years ago
Instagram NFTs Are Here… How does this affect artists?
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.
Read the full article here
Monroe Mayfield
2 years ago
CES 2023: A Third Look At Upcoming Trends
Las Vegas hosted CES 2023. This third and last look at CES 2023 previews upcoming consumer electronics trends that will be crucial for market share.
Definitely start with ICT. Qualcomm CEO Cristiano Amon spoke to CNBC from Las Vegas on China's crackdown and the company's automated driving systems for electric vehicles (EV). The business showed a concept car and its latest Snapdragon processor designs, which offer expanded digital interactions through SalesForce-partnered CRM platforms.
Electrification is reviving Michigan's automobile industry. Michigan Local News reports that $14 billion in EV and battery manufacturing investments will benefit the state. The report also revealed that the Strategic Outreach and Attraction Reserve (SOAR) fund had generated roughly $1 billion for the state's automotive sector.
Ars Technica is great for technology, society, and the future. After CES 2023, Jonathan M. Gitlin published How many electric car chargers are enough? Read about EV charging network issues and infrastructure spending. Politics aside, rapid technological advances enable EV charging network expansion in American cities and abroad.
Finally, the UNEP's The Future of Electric Vehicles and Material Resources: A Foresight Brief. Understanding how lithium-ion batteries will affect EV sales is crucial. Climate change affects EVs in various ways, but electrification and mining trends stand out because more EVs demand more energy-intensive metals and rare earths. Areas & Producers has been publishing my electrification and mining trends articles. Follow me if you wish to write for the publication.
The Weekend Brief (TWB) will routinely cover tech, industrials, and global commodities in global markets, including stock markets. Read more about the future of key areas and critical producers of the global economy in Areas & Producers.
Tech With Dom
3 years ago
6 Awesome Desk Accessories You Must Have!
I'm gadget-obsessed. So I shared my top 6 desk gadgets.
These gadgets improve my workflow and are handy for working from home.
Without further ado...
Computer light bar Xiaomi Mi
I've previously recommended the Xiaomi Mi Light Bar, and I still do. It's stylish and convenient.
The Mi bar is a monitor-mounted desk lamp. The lamp's hue and brightness can be changed with a stylish wireless remote.
Changeable hue and brightness make it ideal for late-night work.
Desk Mat 2.
I wasn't planning to include a desk surface in this article, but I find it improves computer use.
The mouse feels smoother and is a better palm rest than wood or glass.
I'm currently using the overkill Razer Goliathus Extended Chroma RGB Gaming Surface, but I like RGB.
Using a desk surface or mat makes computer use more comfortable, and it's not expensive.
Third, the Logitech MX Master 3 Mouse
The Logitech MX Master 3 or any from the MX Master series is my favorite mouse.
The side scroll wheel on these mice is a feature I've never seen on another mouse.
Side scroll wheels are great for spreadsheets and video editing. It would be hard for me to switch from my Logitech MX Master 3 to another mouse. Only gaming is off-limits.
Google Nest 4.
Without a smart assistant, my desk is useless. I'm currently using the second-generation Google Nest Hub, but I've also used the Amazon Echo Dot, Echo Spot, and Apple HomePod Mini.
As a Pixel 6 Pro user, the Nest Hub works best with my phone.
My Nest Hub plays news, music, and calendar events. It also lets me control lights and switches with my smartphone. It plays YouTube videos.
Google Pixel Stand, No. 5
A wireless charger on my desk is convenient for charging my phone and other devices while I work. My desk has two wireless chargers. I have a Satechi aluminum fast charger and a second-generation Google Pixel Stand.
If I need to charge my phone and earbuds simultaneously, I use two wireless chargers. Satechi chargers are well-made and fast. Micro-USB is my only complaint.
The Pixel Stand converts compatible devices into a smart display for adjusting charging speeds and controlling other smart devices. My Pixel 6 Pro charges quickly. Here's my video review.
6. Anker Power Bank
Anker's 65W charger is my final recommendation. This online find was a must-have. This can charge my laptop and several non-wireless devices, perfect for any techie!
The charger has two USB-A ports and two USB-C ports, one with 45W and the other with 20W, so it can charge my iPad Pro and Pixel 6 Pro simultaneously.
Summary
These are some of my favorite office gadgets. My kit page has an updated list.
Links to the products mentioned in this article are in the appropriate sections. These are affiliate links.
You're up! Share the one desk gadget you can't live without and why.