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.

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.

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.

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."

The dictator's playbook.

Stalin's successor, Nikita Khrushchev, delivered a speech titled "On The Cult Of Personality And Its Consequences" in 1956, three years after Stalin’s death.

The speech, which was never made public, shook the Soviet Union and the Soviet Bloc. After Stalin's "cult of personality" was exposed as a lie, only reality remained.

As I've watched the nightmare unfold in Ukraine, I'm reminded of that question. Primarily by Putin's repeated denials.

His odd claim that Ukraine is run by drug addicts and Nazis (especially strange given that Volodymyr Zelenskyy, the Ukrainian president, is Jewish). Others attempt to portray Russia as liberators rather than occupiers. For example, he portrays Luhansk and Donetsk as plucky, newly independent states when they have been totalitarian statelets for 8 years.

Putin seemed to have lost all sense of reality.

Maybe that's why his remarks to an oligarchs' gathering stood out:

This is almost certainly true from Putin's perspective. Even for Putin, a military invasion seems unlikely. So, what exactly is putting Russia's security in jeopardy? How could Ukraine's independence endanger Russia's existence?

The truth is the only thing that truly terrifies leaders like these.

Trump, the president of “alternative facts,” "and “fake news” praised Putin's fabricated justifications for the Ukraine invasion. Russia tightened news censorship as news of their losses came in. It's no accident that modern dictatorships like Russia (and China and North Korea) restrict citizens' access to information.

Controlling what people see, hear, and think is the simplest method. And Ukraine's recent efforts to join the European Union showed a country whose thoughts Putin couldn't control. With the Russian and Ukrainian peoples so close, he could not control their reality.
He appears to think this is a threat worth fighting NATO over.

It's easy to disown history's great dictators. By the magnitude of their harm. But the strategy they used is still in use today, albeit not to the same devastating effect.

The Kim dynasty in North Korea has ruled for 74 years, Putin has ruled Russia for 19 years (using loopholes and even rewriting the constitution).

When their egos are threatened, they sabre-rattle, as in Kim Jong-un and Donald Trump's famous spat about the size of their...ahem, “nuclear buttons”." Or Putin's threats of mutual destruction this weekend.

Most importantly, they have cult-like control over their followers.

When a leader whose power is built on lies feels he is losing control of the narrative, things like Trump's Jan. 6 meltdown and Putin's current actions in Ukraine are unavoidable.

Leaders who try to control their people's reality will have to die to keep the illusion alive.

Long version of this post available here