More on Web3 & Crypto
Scott Hickmann
4 years ago
Welcome
Welcome to Integrity's Web3 community!
Vitalik
4 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
Sam Hickmann
4 years ago
A quick guide to formatting your text on INTΞGRITY
[06/20/2022 update] We have now implemented a powerful text editor, but you can still use markdown.
Markdown:
Headers
SYNTAX:
# This is a heading 1
## This is a heading 2
### This is a heading 3
#### This is a heading 4
RESULT:
This is a heading 1
This is a heading 2
This is a heading 3
This is a heading 4
Emphasis
SYNTAX:
**This text will be bold**
~~Strikethrough~~
*You **can** combine them*
RESULT:
This text will be italic
This text will be bold
You can combine them
Images
SYNTAX:
RESULT:
Videos
SYNTAX:
https://www.youtube.com/watch?v=7KXGZAEWzn0
RESULT:
Links
SYNTAX:
[Int3grity website](https://www.int3grity.com)
RESULT:
Tweets
SYNTAX:
https://twitter.com/samhickmann/status/1503800505864130561
RESULT:
Blockquotes
SYNTAX:
> Human beings face ever more complex and urgent problems, and their effectiveness in dealing with these problems is a matter that is critical to the stability and continued progress of society. \- Doug Engelbart, 1961
RESULT:
Human beings face ever more complex and urgent problems, and their effectiveness in dealing with these problems is a matter that is critical to the stability and continued progress of society. - Doug Engelbart, 1961
Inline code
SYNTAX:
Text inside `backticks` on a line will be formatted like code.
RESULT:
Text inside backticks on a line will be formatted like code.
Code blocks
SYNTAX:
'''js
function fancyAlert(arg) {
if(arg) {
$.facebox({div:'#foo'})
}
}
'''
RESULT:
function fancyAlert(arg) {
if(arg) {
$.facebox({div:'#foo'})
}
}
Maths
We support LaTex to typeset math. We recommend reading the full documentation on the official website
SYNTAX:
$$[x^n+y^n=z^n]$$
RESULT:
[x^n+y^n=z^n]
Tables
SYNTAX:
| header a | header b |
| ---- | ---- |
| row 1 col 1 | row 1 col 2 |
RESULT:
| header a | header b | header c |
|---|---|---|
| row 1 col 1 | row 1 col 2 | row 1 col 3 |
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Jano le Roux
3 years ago
Apple Quietly Introduces A Revolutionary Savings Account That Kills Banks
Would you abandon your bank for Apple?
Banks are struggling.
not as a result of inflation
not due to the economic downturn.
not due to the conflict in Ukraine.
But because they’re underestimating Apple.
Slowly but surely, Apple is looking more like a bank.
An easy new savings account like Apple
Apple has a new savings account.
Apple says Apple Card users may set up and manage savings straight in Wallet.
No more charges
Colorfully high yields
With no minimum balance
No minimal down payments
Most consumer-facing banks will have to match Apple's offer or suffer disruption.
Users may set it up from their iPhones without traveling to a bank or filling out paperwork.
It’s built into the iPhone in your pocket.
So now more waiting for slow approval processes.
Once the savings account is set up, Apple will automatically transfer all future Daily Cash into it. Users may also add these cash to an Apple Cash card in their Apple Wallet app and adjust where Daily Cash is paid at any time.
Apple Pay and Apple Wallet VP Jennifer Bailey:
Savings enables Apple Card users to grow their Daily Cash rewards over time, while also saving for the future.
Bailey says Savings adds value to Apple Card's Daily Cash benefit and offers another easy-to-use tool to help people lead healthier financial lives.
Transfer money from a linked bank account or Apple Cash to a Savings account. Users can withdraw monies to a connected bank account or Apple Cash card without costs.
Once set up, Apple Card customers can track their earnings via Wallet's Savings dashboard. This dashboard shows their account balance and interest.
This product targets younger people as the easiest way to start a savings account on the iPhone.
Why would a Gen Z account holder travel to the bank if their iPhone could be their bank?
Using this concept, Apple will transform the way we think about banking by 2030.
Two other nightmares keep bankers awake at night
Apple revealed two new features in early 2022 that banks and payment gateways hated.
Tap to Pay with Apple
Late Apple Pay
They startled the industry.
Tap To Pay converts iPhones into mobile POS card readers. Apple Pay Later is pushing the BNPL business in a consumer-friendly direction, hopefully ending dodgy lending practices.
Tap to Pay with Apple
iPhone POS
Millions of US merchants, from tiny shops to huge establishments, will be able to accept Apple Pay, contactless credit and debit cards, and other digital wallets with a tap.
No hardware or payment terminal is needed.
Revolutionary!
Stripe has previously launched this feature.
Tap to Pay on iPhone will provide companies with a secure, private, and quick option to take contactless payments and unleash new checkout experiences, said Bailey.
Apple's solution is ingenious. Brilliant!
Bailey says that payment platforms, app developers, and payment networks are making it easier than ever for businesses of all sizes to accept contactless payments and thrive.
I admire that Apple is offering this up to third-party services instead of closing off other functionalities.
Slow POS terminals, farewell.
Late Apple Pay
Pay Apple later.
Apple Pay Later enables US consumers split Apple Pay purchases into four equal payments over six weeks with no interest or fees.
The Apple ecosystem integration makes this BNPL scheme unique. Nonstick. No dumb forms.
Frictionless.
Just double-tap the button.
Apple Pay Later was designed with users' financial well-being in mind. Apple makes it easy to use, track, and pay back Apple Pay Later from Wallet.
Apple Pay Later can be signed up in Wallet or when using Apple Pay. Apple Pay Later can be used online or in an app that takes Apple Pay and leverages the Mastercard network.
Apple Pay Order Tracking helps consumers access detailed receipts and order tracking in Wallet for Apple Pay purchases at participating stores.
Bad BNPL suppliers, goodbye.
Most bankers will be caught in Apple's eye playing mini golf in high-rise offices.
The big problem:
Banks still think about features and big numbers just like other smartphone makers did not too long ago.
Apple thinks about effortlessness, seamlessness, and frictionlessness that just work through integrated hardware and software.
Let me know what you think Apple’s next power moves in the banking industry could be.
Nick Nolan
3 years ago
In five years, starting a business won't be hip.
People are slowly recognizing entrepreneurship's downside.
Growing up, entrepreneurship wasn't common. High school class of 2012 had no entrepreneurs.
Businesses were different.
They had staff and a lengthy history of achievement.
I never wanted a business. It felt unattainable. My friends didn't care.
Weird.
People desired degrees to attain good jobs at big companies.
When graduated high school:
9 out of 10 people attend college
Earn minimum wage (7%) working in a restaurant or retail establishment
Or join the military (3%)
Later, entrepreneurship became a thing.
2014-ish
I was in the military and most of my high school friends were in college, so I didn't hear anything.
Entrepreneurship soared in 2015, according to Google Trends.
Then more individuals were interested. Entrepreneurship went from unusual to cool.
In 2015, it was easier than ever to build a website, run Facebook advertisements, and achieve organic social media reach.
There were several online business tools.
You didn't need to spend years or money figuring it out. Most entry barriers were gone.
Everyone wanted a side gig to escape the 95.
Small company applications have increased during the previous 10 years.
2011-2014 trend continues.
2015 adds 150,000 applications. 2016 adds 200,000. Plus 300,000 in 2017.
The graph makes it look little, but that's a considerable annual spike with no indications of stopping.
By 2021, new business apps had doubled.
Entrepreneurship will return to its early 2010s level.
I think we'll go backward in 5 years.
Entrepreneurship is half as popular as it was in 2015.
In the late 2020s and 30s, entrepreneurship will again be obscure.
Entrepreneurship's decade-long splendor is fading. People will cease escaping 9-5 and launch fewer companies.
That’s not a bad thing.
I think people have a rose-colored vision of entrepreneurship. It's fashionable. People feel that they're missing out if they're not entrepreneurial.
Reality is showing up.
People say on social media, "I knew starting a business would be hard, but not this hard."
More negative posts on entrepreneurship:
Luke adds:
Is being an entrepreneur ‘healthy’? I don’t really think so. Many like Gary V, are not role models for a well-balanced life. Despite what feel-good LinkedIn tells you the odds are against you as an entrepreneur. You have to work your face off. It’s a tough but rewarding lifestyle. So maybe let’s stop glorifying it because it takes a lot of (bleepin) work to survive a pandemic, mental health battles, and a competitive market.
Entrepreneurship is no longer a pipe dream.
It’s hard.
I went full-time in March 2020. I was done by April 2021. I had a good-paying job with perks.
When that fell through (on my start date), I had to continue my entrepreneurial path. I needed money by May 1 to pay rent.
Entrepreneurship isn't as great as many think.
Entrepreneurship is a serious business.
If you have a 9-5, the grass isn't greener here. Most people aren't telling the whole story when they post on social media or quote successful entrepreneurs.
People prefer to communicate their victories than their defeats.
Is this a bad thing?
I don’t think so.
Over the previous decade, entrepreneurship went from impossible to the finest thing ever.
It peaked in 2020-21 and is returning to reality.
Startups aren't for everyone.
If you like your job, don't quit.
Entrepreneurship won't amaze people if you quit your job.
It's irrelevant.
You're doomed.
And you'll probably make less money.
If you hate your job, quit. Change jobs and bosses. Changing jobs could net you a greater pay or better perks.
When you go solo, your paycheck and perks vanish. Did I mention you'll fail, sleep less, and stress more?
Nobody will stop you from pursuing entrepreneurship. You'll face several challenges.
Possibly.
Entrepreneurship may be romanticized for years.
Based on what I see from entrepreneurs on social media and trends, entrepreneurship is challenging and few will succeed.
Francesca Furchtgott
3 years ago
Giving customers what they want or betraying the values of the brand?
A J.Crew collaboration for fashion label Eveliina Vintage is not a paradox; it is a solution.
Eveliina Vintage's capsule collection debuted yesterday at J.Crew. This J.Crew partnership stopped me in my tracks.
Eveliina Vintage sells vintage goods. Eeva Musacchia founded the shop in Finland in the 1970s. It's recognized for its one-of-a-kind slip dresses from the 1930s and 1940s.
I wondered why a vintage brand would partner with a mass shop. Fast fashion against vintage shopping? Will Eveliina Vintages customers be turned off?
But Eveliina Vintages customers don't care about sustainability. They want Eveliina's Instagram look. Eveliina Vintage collaborated with J.Crew to give customers what they wanted: more Eveliina at a lower price.
Vintage: A Fashion Option That Is Eco-Conscious
Secondhand shopping is a trendy response to quick fashion. J.Crew releases hundreds of styles annually. Waste and environmental damage have been criticized. A pair of jeans requires 1,800 gallons of water. J.Crew's limited-time deals promote more purchases. J.Crew items are likely among those Americans wear 7 times before discarding.
Consumers and designers have emphasized sustainability in recent years. Stella McCartney and Eileen Fisher are popular eco-friendly brands. They've also flocked to ThredUp and similar sites.
Gap, Levis, and Allbirds have listened to consumer requests. They promote recycling, ethical sourcing, and secondhand shopping.
Secondhand shoppers feel good about reusing and recycling clothing that might have ended up in a landfill.
Eco-conscious fashionistas shop vintage. These shoppers enjoy the thrill of the hunt (that limited-edition Chanel bag!) and showing off a unique piece (nobody will have my look!). They also reduce their environmental impact.
Is Eveliina Vintage capitalizing on an aesthetic or is it a sustainable brand?
Eveliina Vintage emphasizes environmental responsibility. Vogue's Amanda Musacchia emphasized sustainability. Amanda, founder Eeva's daughter, is a company leader.
But Eveliina's press message doesn't address sustainability, unlike Instagram. Scarcity and fame rule.
Eveliina Vintages Instagram has see-through dresses and lace-trimmed slip dresses. Celebrities and influencers are often photographed in Eveliina's apparel, which has 53,000+ followers. Vogue appreciates Eveliina's style. Multiple publications discuss Alexa Chung's Eveliina dress.
Eveliina Vintage markets its one-of-a-kind goods. It teases future content, encouraging visitors to return. Scarcity drives demand and raises clothing prices. One dress is $1,600+, but most are $500-$1,000.
The catch: Eveliina can't monetize its expanding popularity due to exorbitant prices and limited quantity. Why?
Most people struggle to pay for their clothing. But Eveliina Vintage lacks those more affordable entry-level products, in contrast to other luxury labels that sell accessories or perfume.
Many people have trouble fitting into their clothing. The bodies of most women in the past were different from those for which vintage clothing was designed. Each Eveliina dress's specific measurements are mentioned alongside it. Be careful, you can fall in love with an ill-fitting dress.
No matter how many people can afford it and fit into it, there is only one item to sell. To get the item before someone else does, those people must be on the Eveliina Vintage website as soon as it becomes available.
A Way for Eveliina Vintage to Make Money (and Expand) with J.Crew Its following
Eveliina Vintages' cooperation with J.Crew makes commercial sense.
This partnership spreads Eveliina's style. Slightly better pricing The $390 outfits have multicolored slips and gauzy cotton gowns. Sizes range from 00 to 24, which is wider than vintage racks.
Eveliina Vintage customers like the combination. Excited comments flood the brand's Instagram launch post. Nobody is mocking the 50-year-old vintage brand's fast-fashion partnership.
Vintage may be a sustainable fashion trend, but that's not why Eveliina's clients love the brand. They only care about the old look.
And that is a tale as old as fashion.