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
4 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
4 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Protos
3 years ago
StableGains lost $42M in Anchor Protocol.
StableGains lost millions of dollars in customer funds in Anchor Protocol without telling its users. The Anchor Protocol offered depositors 19-20% APY before its parent ecosystem, Terra LUNA, lost tens of billions of dollars in market capitalization as LUNA fell below $0.01 and its stablecoin (UST) collapsed.
A Terra Research Forum member raised the alarm. StableGains changed its homepage and Terms and Conditions to reflect how it mitigates risk, a tacit admission that it should have done so from the start.
StableGains raised $600,000 in YCombinator's W22 batch. Moonfire, Broom Ventures, and Goodwater Capital invested $3 million more.
StableGains' 15% yield product attracted $42 million in deposits. StableGains kept most of its deposits in Anchor's UST pool earning 19-20% APY, kept one-quarter of the interest as a management fee, and then gave customers their promised 15% APY. It lost almost all customer funds when UST melted down. It changed withdrawal times, hurting customers.
- StableGains said de-pegging was unlikely. According to its website, 1 UST can be bought and sold for $1 of LUNA. LUNA became worthless, and Terra shut down its blockchain.
- It promised to diversify assets across several stablecoins to reduce the risk of one losing its $1 peg, but instead kept almost all of them in one basket.
- StableGains promised withdrawals in three business days, even if a stablecoin needed time to regain its peg. StableGains uses Coinbase for deposits and withdrawals, and customers receive the exact amount of USDC requested.
StableGains scrubs its website squeaky clean
StableGains later edited its website to say it only uses the "most trusted and tested stablecoins" and extended withdrawal times from three days to indefinite time "in extreme cases."
Previously, USDC, TerraUST (UST), and Dai were used (DAI). StableGains changed UST-related website content after the meltdown. It also removed most references to DAI.
Customers noticed a new clause in the Terms and Conditions denying StableGains liability for withdrawal losses. This new clause would have required customers to agree not to sue before withdrawing funds, avoiding a class-action lawsuit.
Customers must sign a waiver to receive a refund.
Erickson Kramer & Osborne law firm has asked StableGains to preserve all internal documents on customer accounts, marketing, and TerraUSD communications. The firm has not yet filed a lawsuit.
Thousands of StableGains customers lost an estimated $42 million.
Celsius Network customers also affected
CEL used Terra LUNA's Anchor Protocol. Celsius users lost money in the crypto market crash and UST meltdown. Many held CEL and LUNA as yielding deposits.
CEO Alex Mashinsky accused "unknown malefactors" of targeting Celsius Network without evidence. Celsius has not publicly investigated this claim as of this article's publication.
CEL fell before UST de-pegged. On June 2, 2021, it reached $8.01. May 19's close: $0.82.
When some Celsius Network users threatened to leave over token losses, Mashinsky replied, "Leave if you don't think I'm sincere and working harder than you, seven days a week."
Celsius Network withdrew $500 million from Anchor Protocol, but smaller holders had trouble.
Read original article here
Tim Denning
3 years ago
The Dogecoin millionaire mysteriously disappeared.
The American who bought a meme cryptocurrency.
Cryptocurrency is the financial underground.
I love it. But there’s one thing I hate: scams. Over the last few years the Dogecoin cryptocurrency saw massive gains.
Glauber Contessoto overreacted. He shared his rags-to-riches cryptocurrency with the media.
He's only wealthy on paper. No longer Dogecoin millionaire.
Here's what he's doing now. It'll make you rethink cryptocurrency investing.
Strange beginnings
Glauber once had a $36,000-a-year job.
He grew up poor and wanted to make his mother proud. Tesla was his first investment. He bought GameStop stock after Reddit boosted it.
He bought whatever was hot.
He was a young investor. Memes, not research, influenced his decisions.
Elon Musk (aka Papa Elon) began tweeting about Dogecoin.
Doge is a 2013 cryptocurrency. One founder is Australian. He insists it's funny.
He was shocked anyone bought it LOL.
Doge is a Shiba Inu-themed meme. Now whenever I see a Shiba Inu, I think of Doge.
Elon helped drive up the price of Doge by talking about it in 2020 and 2021 (don't take investment advice from Elon; he's joking and gaslighting you).
Glauber caved. He invested everything in Doge. He borrowed from family and friends. He maxed out his credit card to buy more Doge. Yuck.
Internet dubbed him a genius. Slumdog millionaire and The Dogefather were nicknames. Elon pumped Doge on social media.
Good times.
From $180,000 to $1,000,000+
TikTok skyrocketed Doge's price.
Reddit fueled up. Influencers recommended buying Doge because of its popularity. Glauber's motto:
Scared money doesn't earn.
Glauber was no broke ass anymore.
His $180,000 Dogecoin investment became $1M. He championed investing. He quit his dumb job like a rebellious millennial.
A puppy dog meme captivated the internet.
Rise and fall
Whenever I invest in anything I ask myself “what utility does this have?”
Dogecoin is useless.
You buy it for the cute puppy face and hope others will too, driving up the price. All cryptocurrencies fell in 2021's second half.
Central banks raised interest rates, and inflation became a pain.
Dogecoin fell more than others. 90% decline.
Glauber’s Dogecoin is now worth $323K. Still no sales. His dog god is unshakeable. Confidence rocks. Dogecoin millionaire recently said...
“I should have sold some.”
Yes, sir.
He now avoids speculative cryptocurrencies like Dogecoin and focuses on Bitcoin and Ethereum.
I've long said this. Starbucks is building on Ethereum.
It's useful. Useful. Developers use Ethereum daily. Investing makes you wiser over time, like the Dogecoin millionaire.
When risk b*tch slaps you, humility follows, as it did for me when I lost money.
You have to lose money to make money. Few understand.
Dogecoin's omissions
You might be thinking Dogecoin is crap.
I'll take a contrarian stance. Dogecoin does nothing, but it has a strong community. Dogecoin dominates internet memes.
It's silly.
Not quite. The message of crypto that many people forget is that it’s a change in business model.
Businesses create products and services, then advertise to find customers. Crypto Web3 works backwards. A company builds a fanbase but sells them nothing.
Once the community reaches MVC (minimum viable community), a business can be formed.
Community members are relational versus transactional. They're invested in a cause and care about it (typically ownership in the business via crypto).
In this new world, Dogecoin has the most important feature.
Summary
While Dogecoin does have a community I still dislike it.
It's all shady. Anything Elon Musk recommends is a bad investment (except SpaceX & Tesla are great companies).
Dogecoin Millionaire has wised up and isn't YOLOing into more dog memes.
Don't follow the crowd or the hype. Investing is a long-term sport based on fundamentals and research.
Since Ethereum's inception, I've spent 10,000 hours researching.
Dogecoin will be the foundation of something new, like Pets.com at the start of the dot-com revolution. But I doubt Doge will boom.
Be safe!
Jayden Levitt
3 years ago
The country of El Salvador's Bitcoin-obsessed president lost $61.6 million.
It’s only a loss if you sell, right?
Nayib Bukele proclaimed himself “the world’s coolest dictator”.
His jokes aren't clear.
El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.
His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.
He's popular, though.
Bukele won 53% of the vote by fighting violent crime and opposition party corruption.
El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.
They were powerless.
Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.
He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.
Transfer and deposit costs.
This makes logical sense when the president’s theatrics don’t blind you.
El Salvador's Bukele revealed plans to make bitcoin legal tender.
Remittances total $5.9 billion (23%) of the country's expenses.
Anything that reduces costs could boost the economy.
The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).
According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.
In 2021, 36% of El Salvadorans had bank accounts.
Large rural countries like Kenya seem to have resolved their unbanked dilemma.
An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.
Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.
96% of Kenyans utilize M-PESA, which doesn't require a bank account.
The software involves human agents who hang out with cash and a phone.
These people are like ATMs.
You offer them cash to deposit money in your mobile money account or withdraw cash.
In a country with a faulty banking system, cash availability and a safe place to deposit it are important.
William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.
Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.
The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.
Currently, El Salvador has 2,301 Bitcoin.
At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.
Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.
It's still falling.
This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.
The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.
You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.
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Matthew Royse
3 years ago
Ten words and phrases to avoid in presentations
Don't say this in public!
Want to wow your audience? Want to deliver a successful presentation? Do you want practical takeaways from your presentation?
Then avoid these phrases.
Public speaking is difficult. People fear public speaking, according to research.
"Public speaking is people's biggest fear, according to studies. Number two is death. "Sounds right?" — Comedian Jerry Seinfeld
Yes, public speaking is scary. These words and phrases will make your presentation harder.
Using unnecessary words can weaken your message.
You may have prepared well for your presentation and feel confident. During your presentation, you may freeze up. You may blank or forget.
Effective delivery is even more important than skillful public speaking.
Here are 10 presentation pitfalls.
1. I or Me
Presentations are about the audience, not you. Replace "I or me" with "you, we, or us." Focus on your audience. Reward them with expertise and intriguing views about your issue.
Serve your audience actionable items during your presentation, and you'll do well. Your audience will have a harder time listening and engaging if you're self-centered.
2. Sorry if/for
Your presentation is fine. These phrases make you sound insecure and unprepared. Don't pressure the audience to tell you not to apologize. Your audience should focus on your presentation and essential messages.
3. Excuse the Eye Chart, or This slide's busy
Why add this slide if you're utilizing these phrases? If you don't like this slide, change it before presenting. After the presentation, extra data can be provided.
Don't apologize for unclear slides. Hide or delete a broken PowerPoint slide. If so, divide your message into multiple slides or remove the "business" slide.
4. Sorry I'm Nervous
Some think expressing yourself will win over the audience. Nerves are horrible. Even public speakers are nervous.
Nerves aren't noticeable. What's the point? Let the audience judge your nervousness. Please don't make this obvious.
5. I'm not a speaker or I've never done this before.
These phrases destroy credibility. People won't listen and will check their phones or computers.
Why present if you use these phrases?
Good speakers aren't necessarily public speakers. Be confident in what you say. When you're confident, many people will like your presentation.
6. Our Key Differentiators Are
Overused term. It's widely utilized. This seems "salesy," and your "important differentiators" are probably like a competitor's.
This statement has been diluted; say, "what makes us different is..."
7. Next Slide
Many slides or stories? Your presentation needs transitions. They help your viewers understand your argument.
You didn't transition well when you said "next slide." Think about organic transitions.
8. I Didn’t Have Enough Time, or I’m Running Out of Time
The phrase "I didn't have enough time" implies that you didn't care about your presentation. This shows the viewers you rushed and didn't care.
Saying "I'm out of time" shows poor time management. It means you didn't rehearse enough and plan your time well.
9. I've been asked to speak on
This phrase is used to emphasize your importance. This phrase conveys conceit.
When you say this sentence, you tell others you're intelligent, skilled, and appealing. Don't utilize this term; focus on your topic.
10. Moving On, or All I Have
These phrases don't consider your transitions or presentation's end. People recall a presentation's beginning and end.
How you end your discussion affects how people remember it. You must end your presentation strongly and use natural transitions.
Conclusion
10 phrases to avoid in a presentation. I or me, sorry if or sorry for, pardon the Eye Chart or this busy slide, forgive me if I appear worried, or I'm really nervous, and I'm not good at public speaking, I'm not a speaker, or I've never done this before.
Please don't use these phrases: next slide, I didn't have enough time, I've been asked to speak about, or that's all I have.
We shouldn't make public speaking more difficult than it is. We shouldn't exacerbate a difficult issue. Better public speakers avoid these words and phrases.
“Remember not only to say the right thing in the right place, but far more difficult still, to leave unsaid the wrong thing at the tempting moment.” — Benjamin Franklin, Founding Father
This is a summary. See the original post here.
Wayne Duggan
3 years ago
What An Inverted Yield Curve Means For Investors
The yield spread between 10-year and 2-year US Treasury bonds has fallen below 0.2 percent, its lowest level since March 2020. A flattening or negative yield curve can be a bad sign for the economy.
What Is An Inverted Yield Curve?
In the yield curve, bonds of equal credit quality but different maturities are plotted. The most commonly used yield curve for US investors is a plot of 2-year and 10-year Treasury yields, which have yet to invert.
A typical yield curve has higher interest rates for future maturities. In a flat yield curve, short-term and long-term yields are similar. Inverted yield curves occur when short-term yields exceed long-term yields. Inversions of yield curves have historically occurred during recessions.
Inverted yield curves have preceded each of the past eight US recessions. The good news is they're far leading indicators, meaning a recession is likely not imminent.
Every US recession since 1955 has occurred between six and 24 months after an inversion of the two-year and 10-year Treasury yield curves, according to the San Francisco Fed. So, six months before COVID-19, the yield curve inverted in August 2019.
Looking Ahead
The spread between two-year and 10-year Treasury yields was 0.18 percent on Tuesday, the smallest since before the last US recession. If the graph above continues, a two-year/10-year yield curve inversion could occur within the next few months.
According to Bank of America analyst Stephen Suttmeier, the S&P 500 typically peaks six to seven months after the 2s-10s yield curve inverts, and the US economy enters recession six to seven months later.
Investors appear unconcerned about the flattening yield curve. This is in contrast to the iShares 20+ Year Treasury Bond ETF TLT +2.19% which was down 1% on Tuesday.
Inversion of the yield curve and rising interest rates have historically harmed stocks. Recessions in the US have historically coincided with or followed the end of a Federal Reserve rate hike cycle, not the start.
Tim Denning
3 years ago
One of the biggest publishers in the world offered me a book deal, but I don't feel deserving of it.
My ego is so huge it won't fit through the door.
I don't know how I feel about it. I should be excited. Many of you have this exact dream to publish a book with a well-known book publisher and get a juicy advance.
Let me dissect how I'm thinking about it to help you.
How it happened
An email comes in. A generic "can we put a backlink on your website and get a freebie" email.
Almost deleted it.
Then I noticed the logo. It seemed shady. I found the URL. Check. I searched the employee's LinkedIn. Legit. I avoided middlemen. Check.
Mixed feelings. LinkedIn hasn't valued my writing for years. I'm just a guy in an unironed t-shirt whose content they sell advertising against.
They get big dollars. I get $0 and a few likes, plus some email subscribers.
Still, I felt adrenaline for hours.
I texted a few friends to see how they felt. I wrapped them.
Messages like "No shocker. You're entertaining online." I didn't like praises, so I blushed.
The thrill faded after hours. Who knows?
Most authors desire this chance.
"You entitled piece of crap, Denning!"
You may think so. Okay. My job is to stand on the internet and get bananas thrown at me.
I approached writing backwards. More important than a book deal was a social media audience converted to an email list.
Romantic authors think backward. They hope a fantastic book will land them a deal and an audience.
Rarely occurs. So I never pursued it. It's like permission-seeking or the lottery.
Not being a professional writer, I've never written a good book. I post online for fun and to express my opinions.
Writing is therapeutic. I overcome mental illness and rebuilt my life this way. Without blogging, I'd be dead.
I've always dreamed of staying alive and doing something I love, not getting a book contract. Writing is my passion. I'm a winner without a book deal.
Why I was given a book deal
You may assume I received a book contract because of my views or follows. Nope.
They gave me a deal because they like my writing style. I've heard this for eight years.
Several authors agree. One asked me to improve their writer's voice.
Takeaway: highlight your writer's voice.
What if they discover I'm writing incompetently?
An edited book is published. It's edited.
I need to master writing mechanics, thus this concerns me. I need help with commas and sentence construction.
I must learn verb, noun, and adjective. Seriously.
Writing a book may reveal my imposter status to a famous publisher. Imagine the email
"It happened again. He doesn't even know how to spell. He thinks 'less' is the correct word, not 'fewer.' Are you sure we should publish his book?"
Fears stink.
I'm capable of blogging. Even listicles. So what?
Writing for a major publisher feels advanced.
I only blog. I'm good at listicles. Digital media executives have criticized me for this.
It is allegedly clickbait.
Or it is following trends.
Alternately, growth hacking.
Never. I learned copywriting to improve my writing.
Apple, Amazon, and Tesla utilize copywriting to woo customers. Whoever thinks otherwise is the wisest person in the room.
Old-schoolers loathe copywriters.
Their novels sell nothing.
They assume their elitist version of writing is better and that the TikTok generation will invest time in random writing with no subheadings and massive walls of text they can't read on their phones.
I'm terrified of book proposals.
My friend's book proposal suggestion was contradictory and made no sense.
They told him to compose another genre. This book got three Amazon reviews. Is that a good model?
The process disappointed him. I've heard other book proposal horror stories. Tim Ferriss' book "The 4-Hour Workweek" was criticized.
Because he has thick skin, his book came out. He wouldn't be known without that.
I hate book proposals.
An ongoing commitment
Writing a book is time-consuming.
I appreciate time most. I want to focus on my daughter for the next few years. I can't recreate her childhood because of a book.
No idea how parents balance kids' goals.
My silly face in a bookstore. Really?
Genuine thought.
I don't want my face in bookstores. I fear fame. I prefer anonymity.
I want to purchase a property in a bad Australian area, then piss off and play drums. Is bookselling worth it?
Are there even bookstores anymore?
(Except for Ryan Holiday's legendary Painted Porch Bookshop in Texas.)
What's most important about books
Many were duped.
Tweets and TikTok hopscotch vids are their future. Short-form content creates devoted audiences that buy newsletter subscriptions.
Books=depth.
Depth wins (if you can get people to buy your book). Creating a book will strengthen my reader relationships.
It's cheaper than my classes, so more people can benefit from my life lessons.
A deeper justification for writing a book
Mind wandered.
If I write this book, my daughter will follow it. "Look what you can do, love, when you ignore critics."
That's my favorite.
I'll be her best leader and teacher. If her dad can accomplish this, she can too.
My kid can read my book when I'm gone to remember her loving father.
Last paragraph made me cry.
The positive
This book thing might make me sound like Karen.
The upside is... Building in public, like I have with online writing, attracts the right people.
Proof-of-work over proposals, beautiful words, or huge aspirations. If you want a book deal, try writing online instead of the old manner.
Next steps
No idea.
I'm a rural Aussie. Writing a book in the big city is intimidating. Will I do it? Lots to think about. Right now, some level of reflection and gratitude feels most appropriate.
Sometimes when you don't feel worthy, it gives you the greatest lessons. That's how I feel about getting offered this book deal.
Perhaps you can relate.