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
(Edited)
Hackernoon
3 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Dylan Smyth
3 years ago
10 Ways to Make Money Online in 2022
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
print("hello world");
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 —
Jeff John Roberts
3 years ago
Jack Dorsey and Jay-Z Launch 'Bitcoin Academy' in Brooklyn rapper's home
The new Bitcoin Academy will teach Jay-Marcy Z's Houses neighbors "What is Cryptocurrency."
Jay-Z grew up in Brooklyn's Marcy Houses. The rapper and Block CEO Jack Dorsey are giving back to his hometown by creating the Bitcoin Academy.
The Bitcoin Academy will offer online and in-person classes, including "What is Money?" and "What is Blockchain?"
The program will provide participants with a mobile hotspot and a small amount of Bitcoin for hands-on learning.
Students will receive dinner and two evenings of instruction until early September. The Shawn Carter Foundation will help with on-the-ground instruction.
Jay-Z and Dorsey announced the program Thursday morning. It will begin at Marcy Houses but may be expanded.
Crypto Blockchain Plug and Black Bitcoin Billionaire, which has received a grant from Block, will teach the classes.
Jay-Z, Dorsey reunite
Jay-Z and Dorsey have previously worked together to promote a Bitcoin and crypto-based future.
In 2021, Dorsey's Block (then Square) acquired the rapper's streaming music service Tidal, which they propose using for NFT distribution.
Dorsey and Jay-Z launched an endowment in 2021 to fund Bitcoin development in Africa and India.
Dorsey is funding the new Bitcoin Academy out of his own pocket (as is Jay-Z), but he's also pushed crypto-related charitable endeavors at Block, including a $5 million fund backed by corporate Bitcoin interest.
This post is a summary. Read full article here
Scott Hickmann
3 years ago
YouTube
This is a YouTube video:
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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.
Florian Wahl
3 years ago
An Approach to Product Strategy
I've been pondering product strategy and how to articulate it. Frameworks helped guide our thinking.
If your teams aren't working together or there's no clear path to victory, your product strategy may not be well-articulated or communicated (if you have one).
Before diving into a product strategy's details, it's important to understand its role in the bigger picture — the pieces that move your organization forward.
the overall picture
A product strategy is crucial, in my opinion. It's part of a successful product or business. It's the showpiece.
To simplify, we'll discuss four main components:
Vision
Product Management
Goals
Roadmap
Vision
Your company's mission? Your company/product in 35 years? Which headlines?
The vision defines everything your organization will do in the long term. It shows how your company impacted the world. It's your organization's rallying cry.
An ambitious but realistic vision is needed.
Without a clear vision, your product strategy may be inconsistent.
Product Management
Our main subject. Product strategy connects everything. It fulfills the vision.
In Part 2, we'll discuss product strategy.
Goals
This component can be goals, objectives, key results, targets, milestones, or whatever goal-tracking framework works best for your organization.
These product strategy metrics will help your team prioritize strategies and roadmaps.
Your company's goals should be unified. This fuels success.
Roadmap
The roadmap is your product strategy's timeline. It provides a prioritized view of your team's upcoming deliverables.
A roadmap is time-bound and includes measurable goals for your company. Your team's steps and capabilities for executing product strategy.
If your team has trouble prioritizing or defining a roadmap, your product strategy or vision is likely unclear.
Formulation of a Product Strategy
Now that we've discussed where your product strategy fits in the big picture, let's look at a framework.
A product strategy should include challenges, an approach, and actions.
Challenges
First, analyze the problems/situations you're solving. It can be customer- or company-focused.
The analysis should explain the problems and why they're important. Try to simplify the situation and identify critical aspects.
Some questions:
What issues are we attempting to resolve?
What obstacles—internal or otherwise—are we attempting to overcome?
What is the opportunity, and why should we pursue it, in your opinion?
Decided Method
Second, describe your approach. This can be a set of company policies for handling the challenge. It's the overall approach to the first part's analysis.
The approach can be your company's bets, the solutions you've found, or how you'll solve the problems you've identified.
Again, these questions can help:
What is the value that we hope to offer to our clients?
Which market are we focusing on first?
What makes us stand out? Our benefit over rivals?
Actions
Third, identify actions that result from your approach. Second-part actions should be these.
Coordinate these actions. You may need to add products or features to your roadmap, acquire new capabilities through partnerships, or launch new marketing campaigns. Whatever fits your challenges and strategy.
Final questions:
What skills do we need to develop or obtain?
What is the chosen remedy? What are the main outputs?
What else ought to be added to our road map?
Put everything together
… and iterate!
Strategy isn't one-and-done. Changes occur. Economies change. Competitors emerge. Customer expectations change.
One unexpected event can make strategies obsolete quickly. Muscle it. Review, evaluate, and course-correct your strategies with your teams. Quarterly works. In a new or unstable industry, more often.
Sneaker News
3 years ago
This Month Will See The Release Of Travis Scott x Nike Footwear
Following the catastrophes at Astroworld, Travis Scott was swiftly vilified by both media outlets and fans alike, and the names who had previously supported him were quickly abandoned. Nike, on the other hand, remained silent, only delaying the release of La Flame's planned collaborations, such as the Air Max 1 and Air Trainer 1, indefinitely. While some may believe it is too soon for the artist to return to the spotlight, the Swoosh has other ideas, as Nice Kicks reveals that these exact sneakers will be released in May.
Both the Travis Scott x Nike Air Max 1 and the Travis Scott x Nike Air Trainer 1 are set to come in two colorways this month. Tinker Hatfield's renowned runner will meet La Flame's "Baroque Brown" and "Saturn Gold" make-ups, which have been altered with backwards Swooshes and outdoors-themed webbing. The high-top trainer is being customized with Hatfield's "Wheat" and "Grey Haze" palettes, both of which include zippers across the heel, co-branded patches, and other details.
See below for a closer look at the four footwear. TravisScott.com is expected to release the shoes on May 20th, according to Nice Kicks. Following that, on May 27th, Nike SNKRS will release the shoe.
Travis Scott x Nike Air Max 1 "Baroque Brown"
Release Date: 2022
Color: Baroque Brown/Lemon Drop/Wheat/Chile Red
Mens: $160
Style Code: DO9392-200
Pre-School: $85
Style Code: DN4169-200
Infant & Toddler: $70
Style Code: DN4170-200
Travis Scott x Nike Air Max 1 "Saturn Gold"
Release Date: 2022
Color: N/A
Mens: $160
Style Code: DO9392-700
Travis Scott x Nike Air Trainer 1 "Wheat"
Restock Date: May 27th, 2022 (Friday)
Original Release Date: May 20th, 2022 (Friday)
Color: N/A
Mens: $140
Style Code: DR7515-200
Travis Scott x Nike Air Trainer 1 "Grey Haze"
Restock Date: May 27th, 2022 (Friday)
Original Release Date: May 20th, 2022 (Friday)
Color: N/A
Mens: $140
Style Code: DR7515-001