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Gill Pratt

Gill Pratt

3 years ago

War's Human Cost

War's Human Cost
I didn't start crying until I was outside a McDonald's in an Olempin, Poland rest area on highway S17.


Children pick toys at a refugee center, Olempin, Poland, March 4, 2022.

Refugee children, mostly alone with their mothers, but occasionally with a gray-haired grandfather or non-Ukrainian father, were coaxed into picking a toy from boxes provided by a kind-hearted company and volunteers.
I went to Warsaw to continue my research on my family's history during the Holocaust. In light of the ongoing Ukrainian conflict, I asked former colleagues in the US Department of Defense and Intelligence Community if it was safe to travel there. They said yes, as Poland was a NATO member.
I stayed in a hotel in the Warsaw Ghetto, where 90% of my mother's family was murdered in the Holocaust. Across the street was the first Warsaw Judenrat. It was two blocks away from the apartment building my mother's family had owned and lived in, now dilapidated and empty.


Building of my great-grandfather, December 2021.

A mass grave of thousands of rocks for those killed in the Warsaw Ghetto, I didn't cry when I touched its cold walls.


Warsaw Jewish Cemetery, 200,000–300,000 graves.


Mass grave, Warsaw Jewish Cemetery.

My mother's family had two homes, one in Warszawa and the rural one was a forest and sawmill complex in Western Ukraine. For the past half-year, a local Ukrainian historian had been helping me discover faint traces of her family’s life there — in fact, he had found some people still alive who remembered the sawmill and that it belonged to my mother’s grandfather. The historian was good at his job, and we had become close.


My historian friend, December 2021, talking to a Ukrainian.

With war raging, my second trip to Warsaw took on a different mission. To see his daughter and one-year-old grandson, I drove east instead of to Ukraine. They had crossed the border shortly after the war began, leaving men behind, and were now staying with a friend on Poland's eastern border.
I entered after walking up to the house and settling with the dog. The grandson greeted me with a huge smile and the Ukrainian word for “daddy,” “Tato!” But it was clear he was awaiting his real father's arrival, and any man he met would be so tentatively named.
After a few moments, the boy realized I was only a stranger. He had musical talent, like his mother and grandfather, both piano teachers, as he danced to YouTube videos of American children's songs dubbed in Ukrainian, picking the ones he liked and crying when he didn't.


Songs chosen by my historian friend's grandson, March 4, 2022

He had enough music and began crying regardless of the song. His mother picked him up and started nursing him, saying she was worried about him. She had no idea where she would live or how she would survive outside Ukraine. She showed me her father's family history of losses in the Holocaust, which matched my own research.
After an hour of drinking tea and trying to speak of hope, I left for the 3.5-hour drive west to Warsaw.
It was unlike my drive east. It was reminiscent of the household goods-filled carts pulled by horses and people fleeing war 80 years ago.


Jewish refugees relocating, USHMM Holocaust Encyclopaedia, 1939.

The carefully chosen trinkets by children to distract them from awareness of what is really happening and the anxiety of what lies ahead, made me cry despite all my research on the Holocaust. There is no way for them to communicate with their mothers, who are worried, absent, and without their fathers.
It's easy to see war as a contest of nations' armies, weapons, and land. The most costly aspect of war is its psychological toll. My father screamed in his sleep from nightmares of his own adolescent trauma in Warsaw 80 years ago.


Survivor father studying engineering, 1961.

In the airport, I waited to return home while Ukrainian public address systems announced refugee assistance. Like at McDonald's, many mothers were alone with their children, waiting for a flight to distant relatives.
That's when I had my worst trip experience.
A woman near me, clearly a refugee, answered her phone, cried out, and began wailing.
The human cost of war descended like a hammer, and I realized that while I was going home, she never would

Full article

More on Current Events

MartinEdic

MartinEdic

3 years ago

Russia Through the Windows: It's Very Bad

And why we must keep arming Ukraine

Photo by Alexander Smagin on Unsplash

Russian expatriates write about horrific news from home.

Read this from Nadin Brzezinski. She's not a native English speaker, so there are grammar errors, but her tale smells true.

Terrible truth.

There's much more that reveals Russia's grim reality.

Non-leadership. Millions of missing supplies are presumably sold for profit, leaving untrained troops without food or gear. Missile attacks pause because they run out. Fake schemes to hold talks as a way of stalling while they scramble for solutions.

Street men were mobilized. Millions will be ground up to please a crazed despot. Fear, wrath, and hunger pull apart civilization.

It's the most dystopian story, but Ukraine is worse. Destruction of a society, country, and civilization. Only the invaders' corruption and incompetence save the Ukrainians.

Rochester, NY. My suburb had many Soviet-era Ukrainian refugees. Their kids were my classmates. Fifty years later, many are still my friends. I loved their food and culture. My town has 20,000 Ukrainians.

Grieving but determined. They don't quit. They won't quit. Russians are eternal enemies.

It's the Russian people's willingness to tolerate corruption, abuse, and stupidity by their leaders. They are paying. 65000 dead. Ruined economy. No freedom to speak. Americans do not appreciate that freedom as we should.

It lets me write/publish.

Russian friends are shocked. Many are here because their parents escaped Russian anti-semitism and authoritarian oppression. A Russian cultural legacy says a strongman's methods are admirable.

A legacy of a slavery history disguised as serfdom. Peasants and Princes.

Read Tolstoy. Then Anna Karenina. The main characters are princes and counts, whose leaders are incompetent idiots with wealth and power.

Peasants who die in their wars due to incompetence are nameless ciphers.

Sound familiar?

Scott Galloway

Scott Galloway

2 years ago

Text-ure

While we played checkers, we thought billionaires played 3D chess. They're playing the same game on a fancier board.

Every medium has nuances and norms. Texting is authentic and casual. A smaller circle has access, creating intimacy and immediacy. Most people read all their texts, but not all their email and mail. Many of us no longer listen to our voicemails, and calling your kids ages you.

Live interviews and testimony under oath inspire real moments, rare in a world where communications departments sanitize everything powerful people say. When (some of) Elon's text messages became public in Twitter v. Musk, we got a glimpse into tech power. It's bowels.

These texts illuminate the tech community's upper caste.

Checkers, Not Chess

Elon texts with Larry Ellison, Joe Rogan, Sam Bankman-Fried, Satya Nadella, and Jack Dorsey. They reveal astounding logic, prose, and discourse. The world's richest man and his followers are unsophisticated, obtuse, and petty. Possibly. While we played checkers, we thought billionaires played 3D chess. They're playing the same game on a fancier board.

They fumble with their computers.

They lean on others to get jobs for their kids (no surprise).

No matter how rich, they always could use more (money).

Differences A social hierarchy exists. Among this circle, the currency of deference is... currency. Money increases sycophantry. Oculus and Elon's "friends'" texts induce nausea.

Autocorrect frustrates everyone.

Elon doesn't stand out to me in these texts; he comes off mostly OK in my view. It’s the people around him. It seems our idolatry of innovators has infected the uber-wealthy, giving them an uncontrollable urge to kill the cool kid for a seat at his cafeteria table. "I'd grenade for you." If someone says this and they're not fighting you, they're a fan, not a friend.

Many powerful people are undone by their fake friends. Facilitators, not well-wishers. When Elon-Twitter started, I wrote about power. Unchecked power is intoxicating. This is a scientific fact, not a thesis. Power causes us to downplay risk, magnify rewards, and act on instincts more quickly. You lose self-control and must rely on others.

You'd hope the world's richest person has advisers who push back when necessary (i.e., not yes men). Elon's reckless, childish behavior and these texts show there is no truth-teller. I found just one pushback in the 151-page document. It came from Twitter CEO Parag Agrawal, who, in response to Elon’s unhelpful “Is Twitter dying?” tweet, let Elon know what he thought: It was unhelpful. Elon’s response? A childish, terse insult.

Scale

The texts are mostly unremarkable. There are some, however, that do remind us the (super-)rich are different. Specifically, the discussions of possible equity investments from crypto-billionaire Sam Bankman-Fried (“Does he have huge amounts of money?”) and this exchange with Larry Ellison:

Ellison, who co-founded $175 billion Oracle, is wealthy. Less clear is whether he can text a billion dollars. Who hasn't been texted $1 billion? Ellison offered 8,000 times the median American's net worth, enough to buy 3,000 Ferraris or the Chicago Blackhawks. It's a bedrock principle of capitalism to have incredibly successful people who are exponentially wealthier than the rest of us. It creates an incentive structure that inspires productivity and prosperity. When people offer billions over text to help a billionaire's vanity project in a country where 1 in 5 children are food insecure, isn't America messed up?

Elon's Morgan Stanley banker, Michael Grimes, tells him that Web3 ventures investor Bankman-Fried can invest $5 billion in the deal: “could do $5bn if everything vision lock... Believes in your mission." The message bothers Elon. In Elon's world, $5 billion doesn't warrant a worded response. $5 billion is more than many small nations' GDP, twice the SEC budget, and five times the NRC budget.

If income inequality worries you after reading this, trust your gut.

Billionaires aren't like the rich.

As an entrepreneur, academic, and investor, I've met modest-income people, rich people, and billionaires. Rich people seem different to me. They're smarter and harder working than most Americans. Monty Burns from The Simpsons is a cartoon about rich people. Rich people have character and know how to make friends. Success requires supporters.

I've never noticed a talent or intelligence gap between wealthy and ultra-wealthy people. Conflating talent and luck infects the tech elite. Timing is more important than incremental intelligence when going from millions to hundreds of millions or billions. Proof? Elon's texting. Any man who electrifies the auto industry and lands two rockets on barges is a genius. His mega-billions come from a well-regulated capital market, enforceable contracts, thousands of workers, and billions of dollars in government subsidies, including a $465 million DOE loan that allowed Tesla to produce the Model S. So, is Mr. Musk a genius or an impressive man in a unique time and place?

The Point

Elon's texts taught us more? He can't "fix" Twitter. For two weeks in April, he was all in on blockchain Twitter, brainstorming Dogecoin payments for tweets with his brother — i.e., paid speech — while telling Twitter's board he was going to make a hostile tender offer. Kimbal approved. By May, he was over crypto and "laborious blockchain debates." (Mood.)

Elon asked the Twitter CEO for "an update from the Twitter engineering team" No record shows if he got the meeting. It doesn't "fix" Twitter either. And this is Elon's problem. He's a grown-up child with all the toys and no boundaries. His yes-men encourage his most facile thoughts, and shitposts and errant behavior diminish his genius and ours.

Post-Apocalyptic

The universe's titans have a sense of humor.

Every day, we must ask: Who keeps me real? Who will disagree with me? Who will save me from my psychosis, which has brought down so many successful people? Elon Musk doesn't need anyone to jump on a grenade for him; he needs to stop throwing them because one will explode in his hand.

B Kean

B Kean

2 years ago

To prove his point, Putin is prepared to add 200,000 more dead soldiers.

What does Ukraine's murderous craziness mean?

Photo by Anastasiya Romanova on Unsplash

Vladimir Putin expressed his patience to Israeli Prime Minister Naftali Bennet. Thousands, even hundreds of thousands of young and middle-aged males in his country have no meaning to him.

During a meeting in March with Prime Minister Naftali Bennett of Israel, Mr. Putin admitted that the Ukrainians were tougher “than I was told,” according to two people familiar with the exchange. “This will probably be much more difficult than we thought. But the war is on their territory, not ours. We are a big country and we have patience (The Inside Story of a Catastrophe).”

Putin should explain to Russian mothers how patient he is with his invasion of Ukraine.

Putin is rich. Even while sanctions have certainly limited Putin's access to his fortune, he has access to everything in Russia. Unlimited wealth.

The Russian leader's infrastructure was designed with his whims in mind. Vladimir Putin is one of the wealthiest and most catered-to people alive. He's also all-powerful, as his lack of opposition shows. His incredible wealth and power have isolated him from average people so much that he doesn't mind turning lives upside down to prove a point.

For many, losing a Russian spouse or son is painful. Whether the soldier was a big breadwinner or unemployed, the loss of a male figure leaves many families bewildered and anxious. Putin, Russia's revered president, seems unfazed.

People who know Mr. Putin say he is ready to sacrifice untold lives and treasure for as long as it takes, and in a rare face-to-face meeting with the Americans last month the Russians wanted to deliver a stark message to President Biden: No matter how many Russian soldiers are killed or wounded on the battlefield, Russia will not give up (The Inside Story of a Catastrophe).

Imagine a country's leader publicly admitting a mistake he's made. Imagine getting Putin's undivided attention.

So, I underestimated Ukrainians. I can't allow them make me appear terrible, so I'll utilize as many drunken dopes as possible to cover up my error. They'll die fulfilled and heroic.

Russia's human resources are limited, but its willingness to cause suffering is not. How many Russian families must die before the curse is broken? If mass protests started tomorrow, Russia's authorities couldn't stop them.

When Moscovites faced down tanks in August 1991, the Gorbachev coup ended in three days. Even though few city residents showed up, everything collapsed. This wicked disaster won't require many Russians.

One NATO member is warning allies that Mr. Putin is ready to accept the deaths or injuries of as many as 300,000 Russian troops — roughly three times his estimated losses so far.

If 100,000 Russians have died in Ukraine and Putin doesn't mind another 200,000 dying, why don't these 200,000 ghosts stand up and save themselves? Putin plays the role of concerned and benevolent leader effectively, but things aren't going well for Russia.

What would 300,000 or more missing men signify for Russia's future? How many kids will have broken homes? How many families won't form, and what will the economy do?

Putin reportedly cared about his legacy. His place in Russian history Putin's invasion of Ukraine settled his legacy. He has single-handedly weakened and despaired Russia since the 1980s.

Putin will be viewed by sensible people as one of Russia's worst adversaries, but Russians will think he was fantastic despite Ukraine.

The more setbacks Mr. Putin endures on the battlefield, the more fears grow over how far he is willing to go. He has killed tens of thousands in Ukraine, leveled cities, and targeted civilians for maximum pain — obliterating hospitals, schools, and apartment buildings while cutting off power and water to millions before winter. Each time Ukrainian forces score a major blow against Russia, the bombing of their country intensifies. And Mr. Putin has repeatedly reminded the world that he can use anything at his disposal, including nuclear arms, to pursue his notion of victory.

How much death and damage will there be in Ukraine if Putin sends 200,000 more Russians to the front? It's scary, sad, and sick.

Monster.

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Vitalik

Vitalik

3 years ago

An approximate introduction to how zk-SNARKs are possible (part 1)

You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.

In the context of blockchains, this has 2 very powerful applications: Perhaps the most powerful cryptographic technology to come out of the last decade is general-purpose succinct zero knowledge proofs, usually called zk-SNARKs ("zero knowledge succinct arguments of knowledge"). A zk-SNARK allows you to generate a proof that some computation has some particular output, in such a way that the proof can be verified extremely quickly even if the underlying computation takes a very long time to run. The "ZK" part adds an additional feature: the proof can keep some of the inputs to the computation hidden.

You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.

In the context of blockchains, this has two very powerful applications:

  1. Scalability: if a block takes a long time to verify, one person can verify it and generate a proof, and everyone else can just quickly verify the proof instead
  2. Privacy: you can prove that you have the right to transfer some asset (you received it, and you didn't already transfer it) without revealing the link to which asset you received. This ensures security without unduly leaking information about who is transacting with whom to the public.

But zk-SNARKs are quite complex; indeed, as recently as in 2014-17 they were still frequently called "moon math". The good news is that since then, the protocols have become simpler and our understanding of them has become much better. This post will try to explain how ZK-SNARKs work, in a way that should be understandable to someone with a medium level of understanding of mathematics.

Why ZK-SNARKs "should" be hard

Let us take the example that we started with: we have a number (we can encode "cow" followed by the secret input as an integer), we take the SHA256 hash of that number, then we do that again another 99,999,999 times, we get the output, and we check what its starting digits are. This is a huge computation.

A "succinct" proof is one where both the size of the proof and the time required to verify it grow much more slowly than the computation to be verified. If we want a "succinct" proof, we cannot require the verifier to do some work per round of hashing (because then the verification time would be proportional to the computation). Instead, the verifier must somehow check the whole computation without peeking into each individual piece of the computation.

One natural technique is random sampling: how about we just have the verifier peek into the computation in 500 different places, check that those parts are correct, and if all 500 checks pass then assume that the rest of the computation must with high probability be fine, too?

Such a procedure could even be turned into a non-interactive proof using the Fiat-Shamir heuristic: the prover computes a Merkle root of the computation, uses the Merkle root to pseudorandomly choose 500 indices, and provides the 500 corresponding Merkle branches of the data. The key idea is that the prover does not know which branches they will need to reveal until they have already "committed to" the data. If a malicious prover tries to fudge the data after learning which indices are going to be checked, that would change the Merkle root, which would result in a new set of random indices, which would require fudging the data again... trapping the malicious prover in an endless cycle.

But unfortunately there is a fatal flaw in naively applying random sampling to spot-check a computation in this way: computation is inherently fragile. If a malicious prover flips one bit somewhere in the middle of a computation, they can make it give a completely different result, and a random sampling verifier would almost never find out.


It only takes one deliberately inserted error, that a random check would almost never catch, to make a computation give a completely incorrect result.

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? There is a clever solution.

see part 2

Enrique Dans

Enrique Dans

3 years ago

You may not know about The Merge, yet it could change society

IMAGE: Ethereum.org

Ethereum is the second-largest cryptocurrency. The Merge, a mid-September event that will convert Ethereum's consensus process from proof-of-work to proof-of-stake if all goes according to plan, will be a game changer.

Why is Ethereum ditching proof-of-work? Because it can. We're talking about a fully functioning, open-source ecosystem with a capacity for evolution that other cryptocurrencies lack, a change that would allow it to scale up its performance from 15 transactions per second to 100,000 as its blockchain is used for more and more things. It would reduce its energy consumption by 99.95%. Vitalik Buterin, the system's founder, would play a less active role due to decentralization, and miners, who validated transactions through proof of work, would be far less important.

Why has this conversion taken so long and been so cautious? Because it involves modifying a core process while it's running to boost its performance. It requires running the new mechanism in test chains on an ever-increasing scale, assessing participant reactions, and checking for issues or restrictions. The last big test was in early June and was successful. All that's left is to converge the mechanism with the Ethereum blockchain to conclude the switch.

What's stopping Bitcoin, the leader in market capitalization and the cryptocurrency that began blockchain's appeal, from doing the same? Satoshi Nakamoto, whoever he or she is, departed from public life long ago, therefore there's no community leadership. Changing it takes a level of consensus that is impossible to achieve without strong leadership, which is why Bitcoin's evolution has been sluggish and conservative, with few modifications.

Secondly, The Merge will balance the consensus mechanism (proof-of-work or proof-of-stake) and the system decentralization or centralization. Proof-of-work prevents double-spending, thus validators must buy hardware. The system works, but it requires a lot of electricity and, as it scales up, tends to re-centralize as validators acquire more hardware and the entire network activity gets focused in a few nodes. Larger operations save more money, which increases profitability and market share. This evolution runs opposed to the concept of decentralization, and some anticipate that any system that uses proof of work as a consensus mechanism will evolve towards centralization, with fewer large firms able to invest in efficient network nodes.

Yet radical bitcoin enthusiasts share an opposite argument. In proof-of-stake, transaction validators put their funds at stake to attest that transactions are valid. The algorithm chooses who validates each transaction, giving more possibilities to nodes that put more coins at stake, which could open the door to centralization and government control.

In both cases, we're talking about long-term changes, but Bitcoin's proof-of-work has been evolving longer and seems to confirm those fears, while proof-of-stake is only employed in coins with a minuscule volume compared to Ethereum and has no predictive value.

As of mid-September, we will have two significant cryptocurrencies, each with a different consensus mechanisms and equally different characteristics: one is intrinsically conservative and used only for economic transactions, while the other has been evolving in open source mode, and can be used for other types of assets, smart contracts, or decentralized finance systems. Some even see it as the foundation of Web3.

Many things could change before September 15, but The Merge is likely to be a turning point. We'll have to follow this closely.

Vitalik

Vitalik

3 years ago

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