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 —
Rishi Dean
3 years ago
Coinbase's web3 app
Use popular Ethereum dapps with Coinbase’s new dapp wallet and browser
Tl;dr: This post highlights the ability to access web3 directly from your Coinbase app using our new dapp wallet and browser.
Decentralized autonomous organizations (DAOs) and decentralized finance (DeFi) have gained popularity in the last year (DAOs). The total value locked (TVL) of DeFi investments on the Ethereum blockchain has grown to over $110B USD, while NFTs sales have grown to over $30B USD in the last 12 months (LTM). New innovative real-world applications are emerging every day.
Today, a small group of Coinbase app users can access Ethereum-based dapps. Buying NFTs on Coinbase NFT and OpenSea, trading on Uniswap and Sushiswap, and borrowing and lending on Curve and Compound are examples.
Our new dapp wallet and dapp browser enable you to access and explore web3 directly from your Coinbase app.
Web3 in the Coinbase app
Users can now access dapps without a recovery phrase. This innovative dapp wallet experience uses Multi-Party Computation (MPC) technology to secure your on-chain wallet. This wallet's design allows you and Coinbase to share the 'key.' If you lose access to your device, the key to your dapp wallet is still safe and Coinbase can help recover it.
Set up your new dapp wallet by clicking the "Browser" tab in the Android app's navigation bar. Once set up, the Coinbase app's new dapp browser lets you search, discover, and use Ethereum-based dapps.
Looking forward
We want to enable everyone to seamlessly and safely participate in web3, and today’s launch is another step on that journey. We're rolling out the new dapp wallet and browser in the US on Android first to a small subset of users and plan to expand soon. Stay tuned!
CyberPunkMetalHead
3 years ago
Developed an automated cryptocurrency trading tool for nearly a year before unveiling it this month.
Overview
I'm happy to provide this important update. We've worked on this for a year and a half, so I'm glad to finally write it. We named the application AESIR because we’ve love Norse Mythology. AESIR automates and runs trading strategies.
Volatility, technical analysis, oscillators, and other signals are currently supported by AESIR.
Additionally, we enhanced AESIR's ability to create distinctive bespoke signals by allowing it to analyze many indicators and produce a single signal.
AESIR has a significant social component that allows you to copy the best-performing public setups and use them right away.
Enter your email here to be notified when AEISR launches.
Views on algorithmic trading
First, let me clarify. Anyone who claims algorithmic trading platforms are money-printing plug-and-play devices is a liar. Algorithmic trading platforms are a collection of tools.
A trading algorithm won't make you a competent trader if you lack a trading strategy and yolo your funds without testing. It may hurt your trade. Test and alter your plans to account for market swings, but comprehend market signals and trends.
Status Report
Throughout closed beta testing, we've communicated closely with users to design a platform they want to use.
To celebrate, we're giving you free Aesir Viking NFTs and we cover gas fees.
Why use a trading Algorithm?
Automating a successful manual approach
experimenting with and developing solutions that are impossible to execute manually
One AESIR strategy lets you buy any cryptocurrency that rose by more than x% in y seconds.
AESIR can scan an exchange for coins that have gained more than 3% in 5 minutes. It's impossible to manually analyze over 1000 trading pairings every 5 minutes. Auto buy dips or DCA around a Dip
Sneak Preview
Here's the Leaderboard, where you can clone the best public settings.
As a tiny, self-funded team, we're excited to unveil our product. It's a beta release, so there's still more to accomplish, but we know where we stand.
If this sounds like a project that you might want to learn more about, you can sign up to our newsletter and be notified when AESIR launches.
Useful Links:
Join the Discord | Join our subreddit | Newsletter | Mint Free NFT
You might also like
Victoria Kurichenko
3 years ago
What Happened After I Posted an AI-Generated Post on My Website
This could cost you.
Content creators may have heard about Google's "Helpful content upgrade."
This change is another Google effort to remove low-quality, repetitive, and AI-generated content.
Why should content creators care?
Because too much content manipulates search results.
My experience includes the following.
Website admins seek high-quality guest posts from me. They send me AI-generated text after I say "yes." My readers are irrelevant. Backlinks are needed.
Companies copy high-ranking content to boost their Google rankings. Unfortunately, it's common.
What does this content offer?
Nothing.
Despite Google's updates and efforts to clean search results, webmasters create manipulative content.
As a marketer, I knew about AI-powered content generation tools. However, I've never tried them.
I use old-fashioned content creation methods to grow my website from 0 to 3,000 monthly views in one year.
Last year, I launched a niche website.
I do keyword research, analyze search intent and competitors' content, write an article, proofread it, and then optimize it.
This strategy is time-consuming.
But it yields results!
Here's proof from Google Analytics:
Proven strategies yield promising results.
To validate my assumptions and find new strategies, I run many experiments.
I tested an AI-powered content generator.
I used a tool to write this Google-optimized article about SEO for startups.
I wanted to analyze AI-generated content's Google performance.
Here are the outcomes of my test.
First, quality.
I dislike "meh" content. I expect articles to answer my questions. If not, I've wasted my time.
My essays usually include research, personal anecdotes, and what I accomplished and achieved.
AI-generated articles aren't as good because they lack individuality.
Read my AI-generated article about startup SEO to see what I mean.
It's dry and shallow, IMO.
It seems robotic.
I'd use quotes and personal experience to show how SEO for startups is different.
My article paraphrases top-ranked articles on a certain topic.
It's readable but useless. Similar articles abound online. Why read it?
AI-generated content is low-quality.
Let me show you how this content ranks on Google.
The Google Search Console report shows impressions, clicks, and average position.
Low numbers.
No one opens the 5th Google search result page to read the article. Too far!
You may say the new article will improve.
Marketing-wise, I doubt it.
This article is shorter and less comprehensive than top-ranking pages. It's unlikely to win because of this.
AI-generated content's terrible reality.
I'll compare how this content I wrote for readers and SEO performs.
Both the AI and my article are fresh, but trends are emerging.
My article's CTR and average position are higher.
I spent a week researching and producing that piece, unlike AI-generated content. My expert perspective and unique consequences make it interesting to read.
Human-made.
In summary
No content generator can duplicate a human's tone, writing style, or creativity. Artificial content is always inferior.
Not "bad," but inferior.
Demand for content production tools will rise despite Google's efforts to eradicate thin content.
Most won't spend hours producing link-building articles. Costly.
As guest and sponsored posts, artificial content will thrive.
Before accepting a new arrangement, content creators and website owners should consider this.
Sam Hickmann
3 years ago
What is this Fed interest rate everybody is talking about that makes or breaks the stock market?
The Federal Funds Rate (FFR) is the target interest rate set by the Federal Reserve System (Fed)'s policy-making body (FOMC). This target is the rate at which the Fed suggests commercial banks borrow and lend their excess reserves overnight to each other.
The FOMC meets 8 times a year to set the target FFR. This is supposed to promote economic growth. The overnight lending market sets the actual rate based on commercial banks' short-term reserves. If the market strays too far, the Fed intervenes.
Banks must keep a certain percentage of their deposits in a Federal Reserve account. A bank's reserve requirement is a percentage of its total deposits. End-of-day bank account balances averaged over two-week reserve maintenance periods are used to determine reserve requirements.
If a bank expects to have end-of-day balances above what's needed, it can lend the excess to another institution.
The FOMC adjusts interest rates based on economic indicators that show inflation, recession, or other issues that affect economic growth. Core inflation and durable goods orders are indicators.
In response to economic conditions, the FFR target has changed over time. In the early 1980s, inflation pushed it to 20%. During the Great Recession of 2007-2009, the rate was slashed to 0.15 percent to encourage growth.
Inflation picked up in May 2022 despite earlier rate hikes, prompting today's 0.75 percent point increase. The largest increase since 1994. It might rise to around 3.375% this year and 3.1% by the end of 2024.
Sammy Abdullah
3 years ago
SaaS payback period data
It's ok and even desired to be unprofitable if you're gaining revenue at a reasonable cost and have 100%+ net dollar retention, meaning you never lose customers and expand them. To estimate the acceptable cost of new SaaS revenue, we compare new revenue to operating loss and payback period. If you pay back the customer acquisition cost in 1.5 years and never lose them (100%+ NDR), you're doing well.
To evaluate payback period, we compared new revenue to net operating loss for the last 73 SaaS companies to IPO since October 2017. (55 out of 73). Here's the data. 1/(new revenue/operating loss) equals payback period. New revenue/operating loss equals cost of new revenue.
Payback averages a year. 55 SaaS companies that weren't profitable at IPO got a 1-year payback. Outstanding. If you pay for a customer in a year and never lose them (100%+ NDR), you're establishing a valuable business. The average was 1.3 years, which is within the 1.5-year range.
New revenue costs $0.96 on average. These SaaS companies lost $0.96 every $1 of new revenue last year. Again, impressive. Average new revenue per operating loss was $1.59.
Loss-in-operations definition. Operating loss revenue COGS S&M R&D G&A (technical point: be sure to use the absolute value of operating loss). It's wrong to only consider S&M costs and ignore other business costs. Operating loss and new revenue are measured over one year to eliminate seasonality.
Operating losses are desirable if you never lose a customer and have a quick payback period, especially when SaaS enterprises are valued on ARR. The payback period should be under 1.5 years, the cost of new income < $1, and net dollar retention 100%.