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
3 years ago
👏 Awesome post! When is part 3 coming?

Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto

Stephen Moore
3 years ago
Web 2 + Web 3 = Web 5.
Monkey jpegs and shitcoins have tarnished Web3's reputation. Let’s move on.
Web3 was called "the internet's future."
Well, 'crypto bros' shouted about it loudly.
As quickly as it arrived to be the next internet, it appears to be dead. It's had scandals, turbulence, and crashes galore:
Web 3.0's cryptocurrencies have crashed. Bitcoin's all-time high was $66,935. This month, Ethereum fell from $2130 to $1117. Six months ago, the cryptocurrency market peaked at $3 trillion. Worst is likely ahead.
Gas fees make even the simplest Web3 blockchain transactions unsustainable.
Terra, Luna, and other dollar pegs collapsed, hurting crypto markets. Celsius, a crypto lender backed by VCs and Canada's second-largest pension fund, and Binance, a crypto marketplace, have withheld money and coins. They're near collapse.
NFT sales are falling rapidly and losing public interest.
Web3 has few real-world uses, like most crypto/blockchain technologies. Web3's image has been tarnished by monkey profile pictures and shitcoins while failing to become decentralized (the whole concept is controlled by VCs).
The damage seems irreparable, leaving Web3 in the gutter.
Step forward our new saviour — Web5
Fear not though, as hero awaits to drag us out of the Web3 hellscape. Jack Dorsey revealed his plan to save the internet quickly.
Dorsey has long criticized Web3, believing that VC capital and silicon valley insiders have created a centralized platform. In a tweet that upset believers and VCs (he was promptly blocked by Marc Andreessen), Dorsey argued, "You don't own "Web3." VCs and LPs do. Their incentives prevent it. It's a centralized organization with a new name.
Dorsey announced Web5 on June 10 in a very Elon-like manner. Block's TBD unit will work on the project (formerly Square).
Web5's pitch is that users will control their own data and identity. Bitcoin-based. Sound familiar? The presentation pack's official definition emphasizes decentralization. Web5 is a decentralized web platform that enables developers to write decentralized web apps using decentralized identifiers, verifiable credentials, and decentralized web nodes, returning ownership and control over identity and data to individuals.
Web5 would be permission-less, open, and token-less. What that means for Earth is anyone's guess. Identity. Ownership. Blockchains. Bitcoin. Different.
Web4 appears to have been skipped, forever destined to wish it could have shown the world what it could have been. (It was probably crap.) As this iteration combines Web2 and Web3, simple math and common sense add up to 5. Or something.
Dorsey and his team have had this idea simmering for a while. Daniel Buchner, a member of Block's Decentralized Identity team, said, "We're finishing up Web5's technical components."
Web5 could be the project that decentralizes the internet. It must be useful to users and convince everyone to drop the countless Web3 projects, products, services, coins, blockchains, and websites being developed as I write this.
Web5 may be too late for Dorsey and the incoming flood of creators.
Web6 is planned!
The next months and years will be hectic and less stable than the transition from Web 1.0 to Web 2.0.
Web1 was around 1991-2004.
Web2 ran from 2004 to 2021. (though the Web3 term was first used in 2014, it only really gained traction years later.)
Web3 lasted a year.
Web4 is dead.
Silicon Valley billionaires are turning it into a startup-style race, each disrupting the next iteration until they crack it. Or destroy it completely.
Web5 won't last either.

Yusuf Ibrahim
3 years ago
How to sell 10,000 NFTs on OpenSea for FREE (Puppeteer/NodeJS)
So you've finished your NFT collection and are ready to sell it. Except you can't figure out how to mint them! Not sure about smart contracts or want to avoid rising gas prices. You've tried and failed with apps like Mini mouse macro, and you're not familiar with Selenium/Python. Worry no more, NodeJS and Puppeteer have arrived!
Learn how to automatically post and sell all 1000 of my AI-generated word NFTs (Nakahana) on OpenSea for FREE!
My NFT project — Nakahana |
NOTE: Only NFTs on the Polygon blockchain can be sold for free; Ethereum requires an initiation charge. NFTs can still be bought with (wrapped) ETH.
If you want to go right into the code, here's the GitHub link: https://github.com/Yusu-f/nftuploader
Let's start with the knowledge and tools you'll need.
What you should know
You must be able to write and run simple NodeJS programs. You must also know how to utilize a Metamask wallet.
Tools needed
- NodeJS. You'll need NodeJs to run the script and NPM to install the dependencies.
- Puppeteer – Use Puppeteer to automate your browser and go to sleep while your computer works.
- Metamask – Create a crypto wallet and sign transactions using Metamask (free). You may learn how to utilize Metamask here.
- Chrome – Puppeteer supports Chrome.
Let's get started now!
Starting Out
Clone Github Repo to your local machine. Make sure that NodeJS, Chrome, and Metamask are all installed and working. Navigate to the project folder and execute npm install. This installs all requirements.
Replace the “extension path” variable with the Metamask chrome extension path. Read this tutorial to find the path.
Substitute an array containing your NFT names and metadata for the “arr” variable and the “collection_name” variable with your collection’s name.
Run the script.
After that, run node nftuploader.js.
Open a new chrome instance (not chromium) and Metamask in it. Import your Opensea wallet using your Secret Recovery Phrase or create a new one and link it. The script will be unable to continue after this but don’t worry, it’s all part of the plan.
Next steps
Open your terminal again and copy the route that starts with “ws”, e.g. “ws:/localhost:53634/devtools/browser/c07cb303-c84d-430d-af06-dd599cf2a94f”. Replace the path in the connect function of the nftuploader.js script.
const browser = await puppeteer.connect({ browserWSEndpoint: "ws://localhost:58533/devtools/browser/d09307b4-7a75-40f6-8dff-07a71bfff9b3", defaultViewport: null });
Rerun node nftuploader.js. A second tab should open in THE SAME chrome instance, navigating to your Opensea collection. Your NFTs should now start uploading one after the other! If any errors occur, the NFTs and errors are logged in an errors.log file.
Error Handling
The errors.log file should show the name of the NFTs and the error type. The script has been changed to allow you to simply check if an NFT has already been posted. Simply set the “searchBeforeUpload” setting to true.
We're done!
If you liked it, you can buy one of my NFTs! If you have any concerns or would need a feature added, please let me know.
Thank you to everyone who has read and liked. I never expected it to be so popular.

Ren & Heinrich
2 years ago
200 DeFi Projects were examined. Here is what I learned.
I analyze the top 200 DeFi crypto projects in this article.
This isn't a study. The findings benefit crypto investors.
Let’s go!
A set of data
I analyzed data from defillama.com. In my analysis, I used the top 200 DeFis by TVL in October 2022.
Total Locked Value
The chart below shows platform-specific locked value.
14 platforms had $1B+ TVL. 65 platforms have $100M-$1B TVL. The remaining 121 platforms had TVLs below $100 million, with the lowest being $23 million.
TVLs are distributed Pareto. Top 40% of DeFis account for 80% of TVLs.
Compliant Blockchains
Ethereum's blockchain leads DeFi. 96 of the examined projects offer services on Ethereum. Behind BSC, Polygon, and Avalanche.
Five platforms used 10+ blockchains. 36 between 2-10 159 used 1 blockchain.
Use Cases for DeFi
The chart below shows platform use cases. Each platform has decentralized exchanges, liquid staking, yield farming, and lending.
These use cases are DefiLlama's main platform features.
Which use case costs the most? Chart explains. Collateralized debt, liquid staking, dexes, and lending have high TVLs.
The DeFi Industry
I compared three high-TVL platforms (Maker DAO, Balancer, AAVE). The columns show monthly TVL and token price changes. The graph shows monthly Bitcoin price changes.
Each platform's market moves similarly.
Probably because most DeFi deposits are cryptocurrencies. Since individual currencies are highly correlated with Bitcoin, it's not surprising that they move in unison.
Takeaways
This analysis shows that the most common DeFi services (decentralized exchanges, liquid staking, yield farming, and lending) also have the highest average locked value.
Some projects run on one or two blockchains, while others use 15 or 20. Our analysis shows that a project's blockchain count has no correlation with its success.
It's hard to tell if certain use cases are rising. Bitcoin's price heavily affects the entire DeFi market.
TVL seems to be a good indicator of a DeFi platform's success and quality. Higher TVL platforms are cheaper. They're a better long-term investment because they gain or lose less value than DeFis with lower TVLs.
You might also like
Jason Kottke
3 years ago
Lessons on Leadership from the Dancing Guy
This is arguably the best three-minute demonstration I've ever seen of anything. Derek Sivers turns a shaky video of a lone dancing guy at a music festival into a leadership lesson.
A leader must have the courage to stand alone and appear silly. But what he's doing is so straightforward that it's almost instructive. This is critical. You must be simple to follow!
Now comes the first follower, who plays an important role: he publicly demonstrates how to follow. The leader embraces him as an equal, so it's no longer about the leader — it's about them, plural. He's inviting his friends to join him. It takes courage to be the first follower! You stand out and dare to be mocked. Being a first follower is a style of leadership that is underappreciated. The first follower elevates a lone nut to the position of leader. If the first follower is the spark that starts the fire, the leader is the flint.
This link was sent to me by @ottmark, who noted its resemblance to Kurt Vonnegut's three categories of specialists required for revolution.
The rarest of these specialists, he claims, is an actual genius – a person capable generating seemingly wonderful ideas that are not widely known. "A genius working alone is generally dismissed as a crazy," he claims.
The second type of specialist is much easier to find: a highly intellectual person in good standing in his or her community who understands and admires the genius's new ideas and can attest that the genius is not insane. "A person like him working alone can only crave loudly for changes, but fail to say what their shapes should be," Slazinger argues.
Jeff Veen reduced the three personalities to "the inventor, the investor, and the evangelist" on Twitter.

Alex Mathers
24 years ago
400 articles later, nobody bothered to read them.
Writing for readers:
14 years of daily writing.
I post practically everything on social media. I authored hundreds of articles, thousands of tweets, and numerous volumes to almost no one.
Tens of thousands of readers regularly praise me.
I despised writing. I'm stuck now.
I've learned what readers like and what doesn't.
Here are some essential guidelines for writing with impact:
Readers won't understand your work if you can't.
Though obvious, this slipped me up. Share your truths.
Stories engage human brains.
Showing the journey of a person from worm to butterfly inspires the human spirit.
Overthinking hinders powerful writing.
The best ideas come from inner understanding in between thoughts.
Avoid writing to find it. Write.
Writing a masterpiece isn't motivating.
Write for five minutes to simplify. Step-by-step, entertaining, easy steps.
Good writing requires a willingness to make mistakes.
So write loads of garbage that you can edit into a good piece.
Courageous writing.
A courageous story will move readers. Personal experience is best.
Go where few dare.
Templates, outlines, and boundaries help.
Limitations enhance writing.
Excellent writing is straightforward and readable, removing all the unnecessary fat.
Use five words instead of nine.
Use ordinary words instead of uncommon ones.
Readers desire relatability.
Too much perfection will turn it off.
Write to solve an issue if you can't think of anything to write.
Instead, read to inspire. Best authors read.
Every tweet, thread, and novel must have a central idea.
What's its point?
This can make writing confusing.
️ Don't direct your reader.
Readers quit reading. Demonstrate, describe, and relate.
Even if no one responds, have fun. If you hate writing it, the reader will too.

Pat Vieljeux
3 years ago
Your entrepreneurial experience can either be a beautiful adventure or a living hell with just one decision.
Choose.
DNA makes us distinct.
We act alike. Most people follow the same road, ignoring differences. We remain quiet about our uniqueness for fear of exclusion (family, social background, religion). We live a more or less imposed life.
Off the beaten path, we stand out from the others. We obey without realizing we're sewing a shroud. We're told to do as everyone else and spend 40 years dreaming of a golden retirement and regretting not living.
“One of the greatest regrets in life is being what others would want you to be, rather than being yourself.” - Shannon L. Alder
Others dare. Again, few are creative; most follow the example of those who establish a business for the sake of entrepreneurship. To live.
They pick a potential market and model their MVP on an existing solution. Most mimic others, alter a few things, appear to be original, and end up with bland products, adding to an already crowded market.
SaaS, PaaS, etc. followed suit. It's reduced pricing, profitability, and product lifespan.
As competitors become more aggressive, their profitability diminishes, making life horrible for them and their employees. They fail to innovate, cut costs, and close their company.
Few of them look happy and fulfilled.
How did they do it?
The answer is unsettlingly simple.
They are themselves.
They start their company, propelled at first by a passion or maybe a calling.
Then, at their own pace, they create it with the intention of resolving a dilemma.
They assess what others are doing and consider how they might improve it.
In contrast to them, they respond to it in their own way by adding a unique personal touch. Therefore, it is obvious.
Originals, like their DNA, can't be copied. Or if they are, they're poorly printed. Originals are unmatched. Artist-like. True collectors only buy Picasso paintings by the master, not forgeries, no matter how good.
Imaginative people are constantly ahead. Copycats fall behind unless they innovate. They watch their competition continuously. Their solution or product isn't sexy. They hope to cash in on their copied product by flooding the market.
They're mostly pirates. They're short-sighted, unlike creators.
Creators see further ahead and have no rivals. They use copiers to confirm a necessity. To maintain their individuality, creators avoid copying others. They find copying boring. It's boring. They oppose plagiarism.
It's thrilling and inspiring.
It will also make them more able to withstand their opponents' tension. Not to mention roadblocks. For creators, impediments are games.
Others fear it. They race against the clock and fear threats that could interrupt their momentum since they lack inventiveness and their product has a short life cycle.
Creators have time on their side. They're dedicated. Clearly. Passionate booksellers will have their own bookstore. Their passion shows in their book choices. Only the ones they love.
The copier wants to display as many as possible, including mediocre authors, and will cut costs. All this to dominate the market. They're digging their own grave.
The bookseller is just one example. I could give you tons of them.
Closing remarks
Entrepreneurs might follow others or be themselves. They risk exhaustion trying to predict what their followers will do.
It's true.
Life offers choices.
Being oneself or doing as others do, with the possibility of regretting not expressing our uniqueness and not having lived.
“Be yourself; everyone else is already taken”. Oscar Wilde
The choice is yours.
