More on Web3 & Crypto
Scott Hickmann
3 years ago
YouTube
This is a YouTube video:
Scott Hickmann
3 years ago
Welcome
Welcome to Integrity's Web3 community!
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
You might also like
Aaron Dinin, PhD
3 years ago
I'll Never Forget the Day a Venture Capitalist Made Me Feel Like a Dunce
Are you an idiot at fundraising?
Humans undervalue what they don't grasp. Consider NASCAR. How is that a sport? ask uneducated observers. Circular traffic. Driving near a car's physical limits is different from daily driving. When driving at 200 mph, seemingly simple things like changing gas weight or asphalt temperature might be life-or-death.
Venture investors do something similar in entrepreneurship. Most entrepreneurs don't realize how complex venture finance is.
In my early startup days, I didn't comprehend venture capital's intricacy. I thought VCs were rich folks looking for the next Mark Zuckerberg. I was meant to be a sleek, enthusiastic young entrepreneur who could razzle-dazzle investors.
Finally, one of the VCs I was trying to woo set me straight. He insulted me.
How I learned that I was approaching the wrong investor
I was constructing a consumer-facing, pre-revenue marketplace firm. I looked for investors in my old university's alumni database. My city had one. After some research, I learned he was a partner at a growth-stage, energy-focused VC company with billions under management.
Billions? I thought. Surely he can write a million-dollar cheque. He'd hardly notice.
I emailed the VC about our shared alumni status, explaining that I was building a startup in the area and wanted advice. When he agreed to meet the next week, I prepared my pitch deck.
First error.
The meeting seemed like a funding request. Imagine the awkwardness.
His assistant walked me to the firm's conference room and told me her boss was running late. While waiting, I prepared my pitch. I connected my computer to the projector, queued up my PowerPoint slides, and waited for the VC.
He didn't say hello or apologize when he entered a few minutes later. What are you doing?
Hi! I said, Confused but confident. Dinin Aaron. My startup's pitch.
Who? Suspicious, he replied. Your email says otherwise. You wanted help.
I said, "Isn't that a euphemism for contacting investors?" Fundraising I figured I should pitch you.
As he sat down, he smiled and said, "Put away your computer." You need to study venture capital.
Recognizing the business aspects of venture capital
The VC taught me venture capital in an hour. Young entrepreneur me needed this lesson. I assume you need it, so I'm sharing it.
Most people view venture money from an entrepreneur's perspective, he said. They envision a world where venture capital serves entrepreneurs and startups.
As my VC indicated, VCs perceive their work differently. Venture investors don't serve entrepreneurs. Instead, they run businesses. Their product doesn't look like most products. Instead, the VCs you're proposing have recognized an undervalued market segment. By investing in undervalued companies, they hope to profit. It's their investment thesis.
Your company doesn't fit my investment thesis, the venture capitalist told me. Your pitch won't beat my investing theory. I invest in multimillion-dollar clean energy companies. Asking me to invest in you is like ordering a breakfast burrito at a fancy steakhouse. They could, but why? They don't do that.
Yeah, I’m not a fine steak yet, I laughed, feeling like a fool for pitching a growth-stage VC used to looking at energy businesses with millions in revenues on my pre-revenue, consumer startup.
He stressed that it's not necessary. There are investors targeting your company. Not me. Find investors and pitch them.
Remember this when fundraising. Your investors aren't philanthropists who want to help entrepreneurs realize their company goals. Venture capital is a sophisticated investment strategy, and VC firm managers are industry experts. They're looking for companies that meet their investment criteria. As a young entrepreneur, I didn't grasp this, which is why I struggled to raise money. In retrospect, I probably seemed like an idiot. Hopefully, you won't after reading this.
The Velocipede
2 years ago
Stolen wallet
How a misplaced item may change your outlook
Losing your wallet means life stops. Money vanishes. No credit. Your identity is unverifiable. As you check your pockets for the missing object, you can't drive. You can't borrow a library book.
Last seen? intuitively. Every kid asks this, including yours. However, you know where you lost it: On the Providence River cycling trail. While pedaling vigorously, the wallet dropped out of your back pocket and onto the pavement.
A woman you know—your son's art teacher—says it will be returned. Faith.
You want that faith. Losing a wallet is all-consuming. You must presume it has been stolen and is being used to buy every diamond and non-fungible token on the market. Your identity may have been used to open bank accounts and fake passports. Because he used your license address, a ski mask-wearing man may be driving slowly past your house.
As you delete yourself by canceling cards, these images run through your head. You wait in limbo for replacements. Digital text on the DMV website promises your new license will come within 60 days and be approved by local and state law enforcement. In the following two months, your only defense is a screenshot.
Your wallet was ordinary. A worn, overstuffed leather rectangle. You understand how tenuous your existence has always been since you've never lost a wallet. You barely breathe without your documents.
Ironically, you wore a wallet-belt chain. You adored being a 1993 slacker for 15 years. Your wife just convinced you last year that your office job wasn't professional. You nodded and hid the chain.
Never lost your wallet. Until now.
Angry. Feeling stupid. How could you drop something vital? Why? Is the world cruel? No more dumb luck. You're always one pedal-stroke from death.
Then you get a call: We have your wallet.
Local post office, not cops.
The clerk said someone returned it. Due to trying to identify you, it's a chaos. It has your cards but no cash.
Your automobile screeches down the highway. You yell at the windshield, amazed. Submitted. Art teacher was right. Have some trust.
You thank the postmaster. You ramble through the story. The clerk doesn't know the customer, simply a neighborhood Good Samaritan. You wish you could thank that person for lifting your spirits.
You get home, beaming with gratitude. You thumb through your wallet, amazed that it’s all intact. Then you dig out your chain and reattach it.
Because even faith could use a little help.
Anton Franzen
3 years ago
This is the driving force for my use of NFTs, which will completely transform the world.
Its not a fuc*ing fad.
It's not about boring monkeys or photos as nfts; that's just what's been pushed up and made a lot of money. The technology underlying those ridiculous nft photos will one day prove your house and automobile ownership and tell you where your banana came from. Are you ready for web3? Soar!
People don't realize that absolutely anything can and will be part of the blockchain and smart contracts, making them even better. I'll tell you a secret: it will and is happening.
Why?
Why is something blockchain-based a good idea? So let’s speak about cars!
So a new Tesla car is manufactured, and when you buy it, it is bound to an NFT on the blockchain that proves current ownership. The NFT in the smart contract can contain some data about the current owner of the car and some data about the car's status, such as the number of miles driven, the car's overall quality, and so on, as well as a reference to a digital document bound to the NFT that has more information.
Now, 40 years from now, if you want to buy a used automobile, you can scan the car's serial number to view its NFT and see all of its history, each owner, how long they owned it, if it had damages, and more. Since it's on the blockchain, it can't be tampered with.
When you're ready to buy it, the owner posts it for sale, you buy it, and it's sent to your wallet. 5 seconds to change owner, 100% safe and verifiable.
Incorporate insurance logic into the car contract. If you crashed, your car's smart contract would take money from your insurance contract and deposit it in an insurance company wallet.
It's limitless. Your funds may be used by investors to provide insurance as they profit from everyone's investments.
Or suppose all car owners in a country deposit a fixed amount of money into an insurance smart contract that promises if something happens, we'll take care of it. It could be as little as $100-$500 per year, and in a country with 10 million people, maybe 3 million would do that, which would be $500 000 000 in that smart contract and it would be used by the insurance company to invest in assets or take a cut, literally endless possibilities.
Instead of $300 per month, you may pay $300 per year to be covered if something goes wrong, and that may include multiple insurances.
What about your grocery store banana, though?
Yes that too.
You can scan a banana to learn its complete history. You'll be able to see where it was cultivated, every middleman in the supply chain, and hopefully the banana's quality, farm, and ingredients used.
If you want locally decent bananas, you can only buy them, offering you transparency and options. I believe it will be an online marketplace where farmers publish their farms and products for trust and transparency. You might also buy bananas from the farmer.
And? Food security to finish the article. If an order of bananas included a toxin, you could easily track down every banana from the same origin and supply chain and uncover the root cause. This is a tremendous thing that will save lives and have a big impact; did you realize that 1 in 6 Americans gets poisoned by food every year? This could lower the number.
To summarize:
Smart contracts can issue nfts as proof of ownership and include functionality.