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.

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.

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%."

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This post is a summary. Read full article here

As a tech-savvy person (and software engineer) or just a casual technology user, I'm sure you've had this same question countless times: How do I make money online? and how do I make money with my PC/Mac?
You're in luck! Today, I will list the top 5 easiest ways to make money online. Maybe a top ten in the future? Top 5 tips for 2022.

1. Using the gig economy

There are many websites on the internet that allow you to earn extra money using skills and equipment that you already own.
I'm referring to the gig economy. It's a great way to earn a steady passive income from the comfort of your own home. For some sites, premium subscriptions are available to increase sales and access features like bidding on more proposals.
Some of these are:

• Freelancer
• Upwork
• Fiverr (⭐ my personal favorite)
• TaskRabbit

2. Mineprize

MINEPRIZE is a great way to make money online. What's more, You need not do anything! You earn money by lending your idle CPU power to MINEPRIZE.
To register with MINEPRIZE, all you need is an email address and a password. Let MINEPRIZE use your resources, and watch the money roll in! You can earn up to $100 per month by letting your computer calculate. That's insane.

3. Writing

“O Romeo, Romeo, why art thou Romeo?” Okay, I admit that not all writing is Shakespearean. To be a copywriter, you'll need to be fluent in English. Thankfully, we don't have to use typewriters anymore.

Writing is a skill that can earn you a lot of money (claps for the rhyme).
Here are a few ways you can make money typing on your fancy keyboard:
Self-publish a book
Write scripts for video creators
Write for social media
Book-checking
Content marketing help
What a list within a list!

4. Coding

Yes, kids. You've probably coded before if you understand 
You've probably coded before if you understand 

Computational thinking (or coding) is one of the most lucrative ways to earn extra money, or even as a main source of income.
Of course, there are hardcode coders (like me) who write everything line by line, binary di — okay, that last part is a bit exaggerated.
But you can also make money by writing websites or apps or creating low code or no code platforms.
But you can also make money by writing websites or apps or creating low code or no code platforms.
Some low-code platforms
Sheet : spreadsheets to apps :
Loading... We'll install your new app... No-Code Your team can create apps and automate tasks. Agile…
www.appsheet.com

Low-code platform | Business app creator - Zoho Creator
Work is going digital, and businesses of all sizes must adapt quickly. Zoho Creator is a...
www.zoho.com

Sell your data with TrueSource. NO CODE NEEDED
Upload data, configure your product, and earn in minutes.
www.truesource.io

Cool, huh?

5. Created Content

If we use the internet correctly, we can gain unfathomable wealth and extra money. But this one is a bit more difficult. Unlike some of the other items on this list, it takes a lot of time up front.
I'm referring to sites like YouTube and Medium. It's a great way to earn money both passively and actively. With the likes of Jake- and Logan Paul, PewDiePie (a.k.a. Felix Kjellberg) and others, it's never too late to become a millionaire on YouTube. YouTubers are always rising to the top with great content.

6. NFTs and Cryptocurrency

It is now possible to amass large sums of money by buying and selling digital assets on NFTs and cryptocurrency exchanges. Binance's Initial Game Offer rewards early investors who produce the best results.
One awesome game sold a piece of its plot for US$7.2 million! It's Axie Infinity. It's free and available on Google Play and Apple Store.

7. Affiliate Marketing

Affiliate marketing is a form of advertising where businesses pay others (like bloggers) to promote their goods and services. Here's an example. I write a blog (like this one) and post an affiliate link to an item I recommend buying — say, a camera — and if you buy the camera, I get a commission!
These programs pay well:

• Elementor
• AWeber
• Sendinblue
• ConvertKit\sLeadpages
• GetResponse
• SEMRush\sFiverr
• Pabbly

8. Start a blog

Now, if you're a writer or just really passionate about something or a niche, blogging could potentially monetize that passion!
Create a blog about anything you can think of. It's okay to start right here on Medium, as I did.

9. Dropshipping

And I mean that in the best possible way — drop shopping is ridiculously easy to set up, but difficult to maintain for some.
Luckily, Shopify has made setting up an online store a breeze. Drop-shipping from Alibaba and DHGate is quite common. You've got a winner if you can find a local distributor willing to let you drop ship their product!

10. Set up an Online Course

If you have a skill and can articulate it, online education is for you.
Skillshare, Pluralsight, and Coursera have all made inroads in recent years, upskilling people with courses that YOU can create and earn from.

That's it for today! Please share if you liked this post. If not, well —

Democracy's demise

Our Ukrainian brothers and sisters are being attacked by a far superior force.
It's the biggest invasion since WWII.

43.3 million peaceful Ukrainians awoke this morning to tanks, mortars, and missiles. Russia is already 15 miles away.

America and the West will not deploy troops.
They're sanctioning. Except railways. And luxuries. And energy. Diamonds. Their dependence on Russian energy exports means they won't even cut Russia off from SWIFT.

Ukraine is desperate enough to hand out guns on the street.

France, Austria, Turkey, and the EU are considering military aid, but Ukraine will fall without America or NATO.

The Russian goal is likely to encircle Kyiv and topple Zelenskyy's government. A proxy power will be reinstated once Russia has total control.

Also, Putin has never lost a war.

Why is Russia attacking Ukraine?

Putin, like a snowflake college student, “feels unsafe.”
Why?

Because Ukraine is full of “Nazi ideas.”

Putin claims he has felt threatened by Ukraine since the country's pro-Putin leader was ousted and replaced by a popular Jewish comedian.

Hee hee

He fears a full-scale enemy on his doorstep if Ukraine joins NATO. But he refuses to see it both ways. NATO has never invaded Russia, but Russia has always stolen land from its neighbors. Can you blame them for joining a mutual defense alliance when a real threat exists?
Nations that feel threatened can join NATO. That doesn't justify an attack by Russia. It allows them to defend themselves. But NATO isn't attacking Moscow. They aren't.
Russian President Putin's "special operation" aims to de-Nazify the Jewish-led nation.
To keep Crimea and the other two regions he has already stolen, he wants Ukraine undefended by NATO.

(Warlords have fought for control of the strategically important Crimea for over 2,000 years.)
Putin wants to own all of Ukraine.

Why?

The Black Sea is his goal.

Ports bring money and power, and Ukraine pipelines transport Russian energy products.
Putin wants their wheat, too — with 70% crop coverage, Ukraine would be their southern breadbasket, and Russia has no qualms about starving millions of Ukrainians to death to feed its people.

In the end, it's all about greed and power.
Putin wants to own everything Russia has ever owned. This year he turns 70, and he wants to be remembered like his hero Peter the Great.
In order to get it, he's willing to kill thousands of Ukrainians

Art imitates life

This story began when a Jewish TV comedian portrayed a teacher elected President after ranting about corruption.
Servant of the People, the hit sitcom, is now the leading centrist political party.
Right, President Zelenskyy won the hearts and minds of Ukrainians by imagining a fairer world.
A fair fight is something dictators, corporatists, monopolists, and warlords despise.
Now Zelenskyy and his people will die, allowing one of history's most corrupt leaders to amass even more power.

The poor always lose

Meanwhile, the West will impose economic sanctions on Russia.

China is likely to step in to help Russia — or at least the wealthy.

The poor and working class in Russia will suffer greatly if there is a hard crash or long-term depression.
Putin's friends will continue to drink champagne and eat caviar.

Russia cutting off oil, gas, and fertilizer could cause more inflation and possibly a recession if it cuts off supplies to the West. This causes more suffering and hardship for the Western poor and working class.

Why? a billionaire sociopath gets his dirt.

Yes, Russia is simply copying America. Some of us think all war is morally wrong, regardless of who does it.

But let's not kid ourselves right now.

The markets rallied after the biggest invasion in Europe since WWII.
Investors hope Ukraine collapses and Russian oil flows.
Unbridled capitalists value lifeless.

What we can do about Ukraine

When the Russian army invaded eastern Finland, my wife's grandmother fled as a child. 80 years later, Russia still has Karelia.
Russia invaded Ukraine today to retake two eastern provinces.
History has taught us nothing.
Past mistakes won't fix the future.

Instead, we should try:

• Pray and/or meditate on our actions with our families.
• Stop buying Russian products (vodka, obviously, but also pay more for hydro/solar/geothermal/etc.)
• Stop wasting money on frivolous items and donate it to Ukrainian charities.

Here are 35+ places to donate.

• To protest, gather a few friends, contact the media, and shake signs in front of the Russian embassy.
• Prepare to welcome refugees.

More war won't save the planet or change hearts.

Only love can work.