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
Isobel Asher Hamilton
3 years ago
$181 million in bitcoin buried in a dump. $11 million to get them back
James Howells lost 8,000 bitcoins. He has $11 million to get them back.
His life altered when he threw out an iPhone-sized hard drive.
Howells, from the city of Newport in southern Wales, had two identical laptop hard drives squirreled away in a drawer in 2013. One was blank; the other had 8,000 bitcoins, currently worth around $181 million.
He wanted to toss out the blank one, but the drive containing the Bitcoin went to the dump.
He's determined to reclaim his 2009 stash.
Howells, 36, wants to arrange a high-tech treasure hunt for bitcoins. He can't enter the landfill.
Newport's city council has rebuffed Howells' requests to dig for his hard drive for almost a decade, stating it would be expensive and environmentally destructive.
I got an early look at his $11 million idea to search 110,000 tons of trash. He expects submitting it to the council would convince it to let him recover the hard disk.
110,000 tons of trash, 1 hard drive
Finding a hard disk among heaps of trash may seem Herculean.
Former IT worker Howells claims it's possible with human sorters, robot dogs, and an AI-powered computer taught to find hard drives on a conveyor belt.
His idea has two versions, depending on how much of the landfill he can search.
His most elaborate solution would take three years and cost $11 million to sort 100,000 metric tons of waste. Scaled-down version costs $6 million and takes 18 months.
He's created a team of eight professionals in AI-powered sorting, landfill excavation, garbage management, and data extraction, including one who recovered Columbia's black box data.
The specialists and their companies would be paid a bonus if they successfully recovered the bitcoin stash.
Howells: "We're trying to commercialize this project."
Howells claimed rubbish would be dug up by machines and sorted near the landfill.
Human pickers and a Max-AI machine would sort it. The machine resembles a scanner on a conveyor belt.
Remi Le Grand of Max-AI told us it will train AI to recognize Howells-like hard drives. A robot arm would select candidates.
Howells has added security charges to his scheme because he fears people would steal the hard drive.
He's budgeted for 24-hour CCTV cameras and two robotic "Spot" canines from Boston Dynamics that would patrol at night and look for his hard drive by day.
Howells said his crew met in May at the Celtic Manor Resort outside Newport for a pitch rehearsal.
Richard Hammond's narrative swings from banal to epic.
Richard Hammond filmed the meeting and created a YouTube documentary on Howells.
Hammond said of Howells' squad, "They're committed and believe in him and the idea."
Hammond: "It goes from banal to gigantic." "If I were in his position, I wouldn't have the strength to answer the door."
Howells said trash would be cleaned and repurposed after excavation. Reburying the rest.
"We won't pollute," he declared. "We aim to make everything better."
After the project is finished, he hopes to develop a solar or wind farm on the dump site. The council is unlikely to accept his vision soon.
A council representative told us, "Mr. Howells can't convince us of anything." "His suggestions constitute a significant ecological danger, which we can't tolerate and are forbidden by our permit."
Will the recovered hard drive work?
The "platter" is a glass or metal disc that holds the hard drive's data. Howells estimates 80% to 90% of the data will be recoverable if the platter isn't damaged.
Phil Bridge, a data-recovery expert who consulted Howells, confirmed these numbers.
If the platter is broken, Bridge adds, data recovery is unlikely.
Bridge says he was intrigued by the proposal. "It's an intriguing case," he added. Helping him get it back and proving everyone incorrect would be a great success story.
Who'd pay?
Swiss and German venture investors Hanspeter Jaberg and Karl Wendeborn told us they would fund the project if Howells received council permission.
Jaberg: "It's a needle in a haystack and a high-risk investment."
Howells said he had no contract with potential backers but had discussed the proposal in Zoom meetings. "Until Newport City Council gives me something in writing, I can't commit," he added.
Suppose he finds the bitcoins.
Howells said he would keep 30% of the data, worth $54 million, if he could retrieve it.
A third would go to the recovery team, 30% to investors, and the remainder to local purposes, including gifting £50 ($61) in bitcoin to each of Newport's 150,000 citizens.
Howells said he opted to spend extra money on "professional firms" to help convince the council.
What if the council doesn't approve?
If Howells can't win the council's support, he'll sue, claiming its actions constitute a "illegal embargo" on the hard drive. "I've avoided that path because I didn't want to cause complications," he stated. I wanted to cooperate with Newport's council.
Howells never met with the council face-to-face. He mentioned he had a 20-minute Zoom meeting in May 2021 but thought his new business strategy would help.
He met with Jessica Morden on June 24. Morden's office confirmed meeting.
After telling the council about his proposal, he can only wait. "I've never been happier," he said. This is our most professional operation, with the best employees.
The "crypto proponent" buys bitcoin every month and sells it for cash.
Howells tries not to think about what he'd do with his part of the money if the hard disk is found functional. "Otherwise, you'll go mad," he added.
This post is a summary. Read the full article here.
Vivek Singh
3 years ago
A Warm Welcome to Web3 and the Future of the Internet
Let's take a look back at the internet's history and see where we're going — and why.
Tim Berners Lee had a problem. He was at CERN, the world's largest particle physics factory, at the time. The institute's stated goal was to study the simplest particles with the most sophisticated scientific instruments. The institute completed the LEP Tunnel in 1988, a 27 kilometer ring. This was Europe's largest civil engineering project (to study smaller particles — electrons).
The problem Tim Berners Lee found was information loss, not particle physics. CERN employed a thousand people in 1989. Due to team size and complexity, people often struggled to recall past project information. While these obstacles could be overcome, high turnover was nearly impossible. Berners Lee addressed the issue in a proposal titled ‘Information Management'.
When a typical stay is two years, data is constantly lost. The introduction of new people takes a lot of time from them and others before they understand what is going on. An emergency situation may require a detective investigation to recover technical details of past projects. Often, the data is recorded but cannot be found. — Information Management: A Proposal
He had an idea. Create an information management system that allowed users to access data in a decentralized manner using a new technology called ‘hypertext'.
To quote Berners Lee, his proposal was “vague but exciting...”. The paper eventually evolved into the internet we know today. Here are three popular W3C standards used by billions of people today:
(credit: CERN)
HTML (Hypertext Markup)
A web formatting language.
URI (Unique Resource Identifier)
Each web resource has its own “address”. Known as ‘a URL'.
HTTP (Hypertext Transfer Protocol)
Retrieves linked resources from across the web.
These technologies underpin all computer work. They were the seeds of our quest to reorganize information, a task as fruitful as particle physics.
Tim Berners-Lee would probably think the three decades from 1989 to 2018 were eventful. He'd be amazed by the billions, the inspiring, the novel. Unlocking innovation at CERN through ‘Information Management'.
The fictional character would probably need a drink, walk, and a few deep breaths to fully grasp the internet's impact. He'd be surprised to see a few big names in the mix.
Then he'd say, "Something's wrong here."
We should review the web's history before going there. Was it a success after Berners Lee made it public? Web1 and Web2: What is it about what we are doing now that so many believe we need a new one, web3?
Per Outlier Ventures' Jamie Burke:
Web 1.0 was read-only.
Web 2.0 was the writable
Web 3.0 is a direct-write web.
Let's explore.
Web1: The Read-Only Web
Web1 was the digital age. We put our books, research, and lives ‘online'. The web made information retrieval easier than any filing cabinet ever. Massive amounts of data were stored online. Encyclopedias, medical records, and entire libraries were put away into floppy disks and hard drives.
In 2015, the web had around 305,500,000,000 pages of content (280 million copies of Atlas Shrugged).
Initially, one didn't expect to contribute much to this database. Web1 was an online version of the real world, but not yet a new way of using the invention.
One gets the impression that the web has been underutilized by historians if all we can say about it is that it has become a giant global fax machine. — Daniel Cohen, The Web's Second Decade (2004)
That doesn't mean developers weren't building. The web was being advanced by great minds. Web2 was born as technology advanced.
Web2: Read-Write Web
Remember when you clicked something on a website and the whole page refreshed? Is it too early to call the mid-2000s ‘the good old days'?
Browsers improved gradually, then suddenly. AJAX calls augmented CGI scripts, and applications began sending data back and forth without disrupting the entire web page. One button to ‘digg' a post (see below). Web experiences blossomed.
In 2006, Digg was the most active ‘Web 2.0' site. (Photo: Ethereum Foundation Taylor Gerring)
Interaction was the focus of new applications. Posting, upvoting, hearting, pinning, tweeting, liking, commenting, and clapping became a lexicon of their own. It exploded in 2004. Easy ways to ‘write' on the internet grew, and continue to grow.
Facebook became a Web2 icon, where users created trillions of rows of data. Google and Amazon moved from Web1 to Web2 by better understanding users and building products and services that met their needs.
Business models based on Software-as-a-Service and then managing consumer data within them for a fee have exploded.
Web2 Emerging Issues
Unbelievably, an intriguing dilemma arose. When creating this read-write web, a non-trivial question skirted underneath the covers. Who owns it all?
You have no control over [Web 2] online SaaS. People didn't realize this because SaaS was so new. People have realized this is the real issue in recent years.
Even if these organizations have good intentions, their incentive is not on the users' side.
“You are not their customer, therefore you are their product,” they say. With Laura Shin, Vitalik Buterin, Unchained
A good plot line emerges. Many amazing, world-changing software products quietly lost users' data control.
For example: Facebook owns much of your social graph data. Even if you hate Facebook, you can't leave without giving up that data. There is no ‘export' or ‘exit'. The platform owns ownership.
While many companies can pull data on you, you cannot do so.
On the surface, this isn't an issue. These companies use my data better than I do! A complex group of stakeholders, each with their own goals. One is maximizing shareholder value for public companies. Tim Berners-Lee (and others) dislike the incentives created.
“Show me the incentive and I will show you the outcome.” — Berkshire Hathaway's CEO
It's easy to see what the read-write web has allowed in retrospect. We've been given the keys to create content instead of just consume it. On Facebook and Twitter, anyone with a laptop and internet can participate. But the engagement isn't ours. Platforms own themselves.
Web3: The ‘Unmediated’ Read-Write Web
Tim Berners Lee proposed a decade ago that ‘linked data' could solve the internet's data problem.
However, until recently, the same principles that allowed the Web of documents to thrive were not applied to data...
The Web of Data also allows for new domain-specific applications. Unlike Web 2.0 mashups, Linked Data applications work with an unbound global data space. As new data sources appear on the Web, they can provide more complete answers.
At around the same time as linked data research began, Satoshi Nakamoto created Bitcoin. After ten years, it appears that Berners Lee's ideas ‘link' spiritually with cryptocurrencies.
What should Web 3 do?
Here are some quick predictions for the web's future.
Users' data:
Users own information and provide it to corporations, businesses, or services that will benefit them.
Defying censorship:
No government, company, or institution should control your access to information (1, 2, 3)
Connect users and platforms:
Create symbiotic rather than competitive relationships between users and platform creators.
Open networks:
“First, the cryptonetwork-participant contract is enforced in open source code. Their voices and exits are used to keep them in check.” Dixon, Chris (4)
Global interactivity:
Transacting value, information, or assets with anyone with internet access, anywhere, at low cost
Self-determination:
Giving you the ability to own, see, and understand your entire digital identity.
Not pull, push:
‘Push' your data to trusted sources instead of ‘pulling' it from others.
Where Does This Leave Us?
Change incentives, change the world. Nick Babalola
People believe web3 can help build a better, fairer system. This is not the same as equal pay or outcomes, but more equal opportunity.
It should be noted that some of these advantages have been discussed previously. Will the changes work? Will they make a difference? These unanswered questions are technical, economic, political, and philosophical. Unintended consequences are likely.
We hope Web3 is a more democratic web. And we think incentives help the user. If there’s one thing that’s on our side, it’s that open has always beaten closed, given a long enough timescale.
We are at the start.
Sam Bourgi
3 years ago
NFT was used to serve a restraining order on an anonymous hacker.
The international law firm Holland & Knight used an NFT built and airdropped by its asset recovery team to serve a defendant in a hacking case.
The law firms Holland & Knight and Bluestone used a nonfungible token to serve a defendant in a hacking case with a temporary restraining order, marking the first documented legal process assisted by an NFT.
The so-called "service token" or "service NFT" was served to an unknown defendant in a hacking case involving LCX, a cryptocurrency exchange based in Liechtenstein that was hacked for over $8 million in January. The attack compromised the platform's hot wallets, resulting in the loss of Ether (ETH), USD Coin (USDC), and other cryptocurrencies, according to Cointelegraph at the time.
On June 7, LCX claimed that around 60% of the stolen cash had been frozen, with investigations ongoing in Liechtenstein, Ireland, Spain, and the United States. Based on a court judgment from the New York Supreme Court, Centre Consortium, a company created by USDC issuer Circle and crypto exchange Coinbase, has frozen around $1.3 million in USDC.
The monies were laundered through Tornado Cash, according to LCX, but were later tracked using "algorithmic forensic analysis." The organization was also able to identify wallets linked to the hacker as a result of the investigation.
In light of these findings, the law firms representing LCX, Holland & Knight and Bluestone, served the unnamed defendant with a temporary restraining order issued on-chain using an NFT. According to LCX, this system "was allowed by the New York Supreme Court and is an example of how innovation can bring legitimacy and transparency to a market that some say is ungovernable."
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Mike Meyer
3 years ago
Reality Distortion
Old power paradigm blocks new planetary paradigm
The difference between our reality and the media's reality is like a tale of two worlds. The greatest and worst of times, really.
Expanding information demands complex skills and understanding to separate important information from ignorance and crap. And that's just the start of determining the source's aim.
Trust who? We see people trust liars in public and then be destroyed by their decisions. Mistakes may be devastating.
Many give up and don't trust anyone. Reality is a choice, though. Same risks.
We must separate our needs and wants from reality. Needs and wants have rules. Greed and selfishness create an unlivable planet.
Culturally, we know this, but we ignore it as foolish. Selfish and greedy people obtain what they want, while others suffer.
We invade, plunder, rape, and burn. We establish civilizations by institutionalizing an exploitable underclass and denying its existence. These cultural lies promote greed and selfishness despite their destructiveness.
Controlling parts of society institutionalize these lies as fact. Many of each age are willing to gamble on greed because they were taught to see greed and selfishness as principles justified by prosperity.
Our cultural understanding recognizes the long-term benefits of collaboration and sharing. This older understanding generates an increasing tension between greedy people and those who see its planetary effects.
Survival requires distinguishing between global and regional realities. Simple, yet many can't do it. This is the first time human greed has had a global impact.
In the past, conflict stories focused on regional winners and losers. Losers lose, winners win, etc. Powerful people see potential decades of nuclear devastation as local, overblown, and not personally dangerous.
Mutually Assured Destruction (MAD) was a human choice that required people to acquiesce to irrational devastation. This prevented nuclear destruction. Most would refuse.
A dangerous “solution” relies on nuclear trigger-pullers not acting irrationally. Since then, we've collected case studies of sane people performing crazy things in experiments. We've been lucky, but the climate apocalypse could be different.
Climate disaster requires only continuing current behavior. These actions already cause global harm, but that's not a threat. These activities must be viewed differently.
Once grasped, denying planetary facts is hard to accept. Deniers can't think beyond regional power. Seeing planet-scale is unusual.
Decades of indoctrination defining any planetary perspective as un-American implies communal planetary assets are for plundering. The old paradigm limits any other view.
In the same way, the new paradigm sees the old regional power paradigm as a threat to planetary civilization and lifeforms. Insane!
While MAD relied on leaders not acting stupidly to trigger a nuclear holocaust, the delayed climatic holocaust needs correcting centuries of lunacy. We must stop allowing craziness in global leadership.
Nothing in our acknowledged past provides a paradigm for such. Only primitive people have failed to reach our level of sophistication.
Before European colonization, certain North American cultures built sophisticated regional nations but abandoned them owing to authoritarian cruelty and destruction. They were overrun by societies that saw no wrong in perpetual exploitation. David Graeber's The Dawn of Everything is an example of historical rediscovery, which is now crucial.
From the new paradigm's perspective, the old paradigm is irrational, yet it's too easy to see those in it as ignorant or malicious, if not both. These people are both, but the collapsing paradigm they promote is older or more ingrained than we think.
We can't shift that paradigm's view of a dead world. We must eliminate this mindset from our nations' leadership. No other way will preserve the earth.
Change is occurring. As always with tremendous transition, younger people are building the new paradigm.
The old paradigm's disintegration is insane. The ability to detect errors and abandon their sources is more important than age. This is gaining recognition.
The breakdown of the previous paradigm is not due to senile leadership, but to systemic problems that the current, conservative leadership cannot recognize.
Stop following the old paradigm.
Jim Clyde Monge
3 years ago
Can You Sell Images Created by AI?
Some AI-generated artworks sell for enormous sums of money.
But can you sell AI-Generated Artwork?
Simple answer: yes.
However, not all AI services enable allow usage and redistribution of images.
Let's check some of my favorite AI text-to-image generators:
Dall-E2 by OpenAI
The AI art generator Dall-E2 is powerful. Since it’s still in beta, you can join the waitlist here.
OpenAI DOES NOT allow the use and redistribution of any image for commercial purposes.
Here's the policy as of April 6, 2022.
Here are some images from Dall-E2’s webpage to show its art quality.
Several Reddit users reported receiving pricing surveys from OpenAI.
This suggests the company may bring out a subscription-based tier and a commercial license to sell images soon.
MidJourney
I like Midjourney's art generator. It makes great AI images. Here are some samples:
Standard Licenses are available for $10 per month.
Standard License allows you to use, copy, modify, merge, publish, distribute, and/or sell copies of the images, except for blockchain technologies.
If you utilize or distribute the Assets using blockchain technology, you must pay MidJourney 20% of revenue above $20,000 a month or engage in an alternative agreement.
Here's their copyright and trademark page.
Dream by Wombo
Dream is one of the first public AI art generators.
This AI program is free, easy to use, and Wombo gives a royalty-free license to copy or share artworks.
Users own all artworks generated by the tool. Including all related copyrights or intellectual property rights.
Here’s Wombos' intellectual property policy.
Final Reflections
AI is creating a new sort of art that's selling well. It’s becoming popular and valued, despite some skepticism.
Now that you know MidJourney and Wombo let you sell AI-generated art, you need to locate buyers. There are several ways to achieve this, but that’s for another story.
Taher Batterywala
3 years ago
Do You Have Focus Issues? Use These 5 Simple Habits
Many can't concentrate. The first 20% of the day isn't optimized.
Elon Musk, Tony Robbins, and Bill Gates share something:
Morning Routines.
A repeatable morning ritual saves time.
The result?
Time for hobbies.
I'll discuss 5 easy morning routines you can use.
1. Stop pressing snooze
Waking up starts the day. You disrupt your routine by hitting snooze.
One sleep becomes three. Your morning routine gets derailed.
Fix it:
Hide your phone. This disables snooze and wakes you up.
Once awake, staying awake is 10x easier. Simple trick, big results.
2. Drink water
Chronic dehydration is common. Mostly urban, air-conditioned workers/residents.
2% cerebral dehydration causes short-term memory loss.
Dehydration shrinks brain cells.
Drink 3-4 liters of water daily to avoid this.
3. Improve your focus
How to focus better?
Meditation.
Improve your mood
Enhance your memory
increase mental clarity
Reduce blood pressure and stress
Headspace helps with the habit.
Here's a meditation guide.
Sit comfortably
Shut your eyes.
Concentrate on your breathing
Breathe in through your nose
Breathe out your mouth.
5 in, 5 out.
Repeat for 1 to 20 minutes.
Here's a beginner's video:
4. Workout
Exercise raises:
Mental Health
Effort levels
focus and memory
15-60 minutes of fun:
Exercise Lifting
Running
Walking
Stretching and yoga
This helps you now and later.
5. Keep a journal
You have countless thoughts daily. Many quietly steal your focus.
Here’s how to clear these:
Write for 5-10 minutes.
You'll gain 2x more mental clarity.
Recap
5 morning practices for 5x more productivity:
Say no to snoozing
Hydrate
Improve your focus
Exercise
Journaling
Conclusion
One step starts a thousand-mile journey. Try these easy yet effective behaviors if you have trouble concentrating or have too many thoughts.
Start with one of these behaviors, then add the others. Its astonishing results are instant.