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.

Pink or blue AR-15s?

A teacher teaches; a gun kills. Killing isn't teaching. Killing is opposite of teaching.

Without 27 school shootings this year, we wouldn't be talking about arming teachers. Gun makers, distributors, and the NRA cause most school shootings. Gun makers, distributors, and the NRA wouldn't be huge business if weapons weren't profitable.

Guns, ammo, body armor, holsters, concealed carriers, bore sights, cleaner kits, spare magazines and speed loaders, gun safes, and ear protection are sold. And more guns.

And lots more profit.

Guns aren't bread. You eat a loaf of bread in a week or so and then must buy more. Bread makers will make money. Winchester 94.30–30 1899 Lever Action Rifle from 1894 still kills. (For safety, I won't link to the ad.) Gun makers don't object if you collect antique weapons, but they need you to buy the latest, in-style killing machine. The youngster who killed 19 students and 2 teachers at Robb Elementary School in Uvalde, Texas, used an AR-15. Better yet, two.

Salvador Ramos, the Robb Elementary shooter, is a "killing influencer" He pushes consumers to buy items, which benefits manufacturers and distributors. Like every previous AR-15 influencer, he profits Colt, the rifle's manufacturer, and 52,779 gun dealers in the U.S. Ramos and other AR-15 influences make us fear for our safety and our children's. Fearing for our safety, we acquire 20 million firearms a year and live in a gun culture.

So now at school, we want to arm teachers.

Consider. Which of your teachers would you have preferred in body armor with a gun drawn?

Miss Summers? Remember her bringing daisies from her yard to second grade? She handed each student a beautiful flower. Miss Summers loved everyone, even those with AR-15s. She can't shoot.

Frasier? Mr. Frasier turned a youngster over down to explain "invert." Mr. Frasier's hands shook when he wasn't flipping fifth-graders and fractions. He may have shot wrong.

Mrs. Barkley barked in high school English class when anyone started an essay with "But." Mrs. Barkley dubbed Abie a "Jewboy" and gave him terrible grades. Arming Miss Barkley is like poisoning the chef.

Think back. Do you remember a teacher with a gun? No. Arming teachers so the gun industry can make more money is the craziest idea ever.

Or maybe you agree with Ted Cruz, the gun lobby-bought senator, that more guns reduce gun violence. After the next school shooting, you'll undoubtedly talk about arming teachers and pupils. Colt will likely develop a backpack-sized, lighter version of its popular killing machine in pink and blue for kids and boys. The MAR-15? (M for mini).

---

This post is a summary. Read the full one here.

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