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Scott Galloway

Scott Galloway

7 months ago

Don't underestimate the foolish

More on Society & Culture

The woman

The woman

6 months ago

The renowned and highest-paid Google software engineer

His story will inspire you.

Made by me with Midjourney

“Google search went down for a few hours in 2002; Jeff Dean handled all the queries by hand and checked quality doubled.”- Jeff Dean Facts.

One of many Jeff Dean jokes, but you get the idea.

Google's top six engineers met in a war room in mid-2000. Google's crawling system, which indexed the Web, stopped working. Users could still enter queries, but results were five months old.

Google just signed a deal with Yahoo to power a ten-times-larger search engine. Tension rose. It was crucial. If they failed, the Yahoo agreement would likely fall through, risking bankruptcy for the firm. Their efforts could be lost.

A rangy, tall, energetic thirty-one-year-old man named Jeff dean was among those six brilliant engineers in the makeshift room. He had just left D. E. C. a couple of months ago and started his career in a relatively new firm Google, which was about to change the world. He rolled his chair over his colleague Sanjay and sat right next to him, cajoling his code like a movie director. The history started from there.

When you think of people who shaped the World Wide Web, you probably picture founders and CEOs like Larry Page and Sergey Brin, Marc Andreesen, Tim Berners-Lee, Bill Gates, and Mark Zuckerberg. They’re undoubtedly the brightest people on earth.

Under these giants, legions of anonymous coders work at keyboards to create the systems and products we use. These computer workers are irreplaceable.

Let's get to know him better.

It's possible you've never heard of Jeff Dean. He's American. Dean created many behind-the-scenes Google products. Jeff, co-founder and head of Google's deep learning research engineering team, is a popular technology, innovation, and AI keynote speaker.

While earning an MS and Ph.D. in computer science at the University of Washington, he was a teaching assistant, instructor, and research assistant. Dean joined the Compaq Computer Corporation Western Research Laboratory research team after graduating.

Jeff co-created ProfileMe and the Continuous Profiling Infrastructure for Digital at Compaq. He co-designed and implemented Swift, one of the fastest Java implementations. He was a senior technical staff member at mySimon Inc., retrieving and caching electronic commerce content.

Dean, a top young computer scientist, joined Google in mid-1999. He was always trying to maximize a computer's potential as a child.

An expert

His high school program for processing massive epidemiological data was 26 times faster than professionals'. Epi Info, in 13 languages, is used by the CDC. He worked on compilers as a computer science Ph.D. These apps make source code computer-readable.

Dean never wanted to work on compilers forever. He left Academia for Google, which had less than 20 employees. Dean helped found Google News and AdSense, which transformed the internet economy. He then addressed Google's biggest issue, scaling.

Growing Google faced a huge computing challenge. They developed PageRank in the late 1990s to return the most relevant search results. Google's popularity slowed machine deployment.

Dean solved problems, his specialty. He and fellow great programmer Sanjay Ghemawat created the Google File System, which distributed large data over thousands of cheap machines.

These two also created MapReduce, which let programmers handle massive data quantities on parallel machines. They could also add calculations to the search algorithm. A 2004 research article explained MapReduce, which became an industry sensation.

Several revolutionary inventions

Dean's other initiatives were also game-changers. BigTable, a petabyte-capable distributed data storage system, was based on Google File. The first global database, Spanner, stores data on millions of servers in dozens of data centers worldwide.

It underpins Gmail and AdWords. Google Translate co-founder Jeff Dean is surprising. He contributes heavily to Google News. Dean is Senior Fellow of Google Research and Health and leads Google AI.

Recognitions

The National Academy of Engineering elected Dean in 2009. He received the 2009 Association for Computing Machinery fellowship and the 2016 American Academy of Arts and Science fellowship. He received the 2007 ACM-SIGOPS Mark Weiser Award and the 2012 ACM-Infosys Foundation Award. Lists could continue.

A sneaky question may arrive in your mind: How much does this big brain earn? Well, most believe he is one of the highest-paid employees at Google. According to a survey, he is paid $3 million a year.

He makes espresso and chats with a small group of Googlers most mornings. Dean steams milk, another grinds, and another brews espresso. They discuss families and technology while making coffee. He thinks this little collaboration and idea-sharing keeps Google going.

“Some of us have been working together for more than 15 years,” Dean said. “We estimate that we’ve collectively made more than 20,000 cappuccinos together.”

We all know great developers and software engineers. It may inspire many.

Michelle Teheux

Michelle Teheux

5 months ago

Get Real, All You Grateful Laid-Off LinkedIn Users

WTF is wrong with you people?

She looks so happy. She was probably just fired. Photo by Michael Dam on Unsplash

When I was laid off as editor of my town's daily newspaper, I went silent on social media. I knew it was coming and had been quietly removing personal items each day, but the pain was intense.

I posted a day later. I didn't bad-mouth GateHouse Media but expressed my sadness at leaving the newspaper industry, pride in my accomplishments, and hope for success in another industry.

Normal job-loss response.

What do you recognize as abnormal?

The bullshit I’ve been reading from laid-off folks on LinkedIn.

If you're there, you know. Many Twitter or Facebook/Meta employees recently lost their jobs.

Well, many of them did not “lose their job,” actually. They were “impacted by the layoffs” at their former employer. I keep seeing that phrase.

Why don’t they want to actually say it? Why the euphemism?

Many are excited about the opportunities ahead. The jobless deny being sad.

They're ecstatic! They have big plans.

Hope so. Sincerely! Being laid off stinks, especially if, like me, your skills are obsolete. It's worse if, like me, you're too old to start a new career. Ageism exists despite denials.

Nowadays, professionalism seems to demand psychotic levels of fake optimism.

Why? Life is unpredictable. That's indisputable. You shouldn't constantly complain or cry in public, but you also shouldn't pretend everything's great.

It makes you look psychotic, not positive. It's like saying at work:

“I was impacted by the death of my spouse of 20 years this week, and many of you have reached out to me, expressing your sympathy. However, I’m choosing to remember the amazing things we shared. I feel confident that there is another marriage out there for me, and after taking a quiet weekend trip to reset myself, I’ll be out there looking for the next great marital adventure! #staypositive #available #opentolove

Also:

“Now looking for our next #dreamhome after our entire neighborhood was demolished by a wildfire last night. We feel so lucky to have lived near so many amazing and inspirational neighbors, all of whom we will miss as we go on our next housing adventure. The best house for us is yet to come! If you have a great neighborhood you’d recommend, please feel free to reach out and touch base with us! #newhouse #newneighborhood #newlife

Admit it. That’s creepy.

The constant optimism makes me feel sick to my stomach.

Viscerally.

I hate fakes.

Imagine a fake wood grain desk. Wouldn't it be better if the designer accepted that it's plastic and went with that?

Real is better but not always nice. When something isn't nice, you don't have to go into detail, but you also shouldn't pretend it's great.

How to announce your job loss to the world.

Do not pretend to be happy, but don't cry and drink vodka all afternoon.

Say you loved your job, and that you're looking for new opportunities.

Yes, if you'll miss your coworkers. Otherwise, don't badmouth. No bridge-burning!

Please specify the job you want. You may want to pivot.

Alternatively, try this.

You could always flame out.

If you've pushed yourself too far into toxic positivity, you may be ready to burn it all down. If so, make it worthwhile by writing something like this:

Well, I was shitcanned by the losers at #Acme today. That bitch Linda in HR threw me under the bus just because she saw that one of my “friends” tagged me in some beach pics on social media after I called in sick with Covid. The good thing is I will no longer have to watch my ass around that #asspincher Ron in accounting, but I’m sad that I will no longer have a cushy job with high pay or access to the primo office supplies I’ve been sneaking home for the last five years. (Those gel pens were the best!) I am going to be taking some time off to enjoy my unemployment and hammer down shots of Jägermeister but in about five months I’ll be looking for anything easy with high pay and great benefits. Reach out if you can help! #officesupplies #unemploymentrocks #drinkinglikeagirlboss #acmesucks

It beats the fake positivity.

Hudson Rennie

Hudson Rennie

6 months ago

Meet the $5 million monthly controversy-selling King of Toxic Masculinity.

Trigger warning — Andrew Tate is running a genius marketing campaign

Image via Instagram: @cobratate

Andrew Tate is a 2022 internet celebrity.

Kickboxing world champion became rich playboy with controversial views on gender roles.

Andrew's get-rich-quick scheme isn't new. His social media popularity is impressive.

He’s currently running one of the most genius marketing campaigns in history.

He pulls society's pendulum away from diversity and inclusion and toward diversion and exclusion. He's unstoppable.

Here’s everything you need to know about Andrew Tate. And how he’s playing chess while the world plays checkers.

Cobra Tate is the name he goes by.

American-born, English-raised entrepreneur Andrew Tate lives in Romania.

Romania? Says Andrew,

“I prefer a country in which corruption is available to everyone.”

Andrew was a professional kickboxer with the ring moniker Cobra before starting Hustlers University.

Before that, he liked chess and worshipped his father.

Emory Andrew Tate III is named after his grandmaster chess player father.

Emory was the first black-American chess champion. He was military, martial arts-trained, and multilingual. A superhuman.

He lived in his car to make ends meet.

Andrew and Tristan relocated to England with their mother when their parents split.

It was there that Andrew began his climb toward becoming one of the internet’s greatest villains.

Andrew fell in love with kickboxing.

Andrew spent his 20s as a professional kickboxer and reality TV star, featuring on Big Brother UK and The Ultimate Traveller.

These 3 incidents, along with a chip on his shoulder, foreshadowed Andrews' social media breakthrough.

  • Chess

  • Combat sports

  • Reality television

A dangerous trio.

Andrew started making money online after quitting kickboxing in 2017 due to an eye issue.

Andrew didn't suddenly become popular.

Andrew's web work started going viral in 2022.

Due to his contentious views on patriarchy and gender norms, he's labeled the King of Toxic Masculinity. His most contentious views (trigger warning):

  • “Women are intrinsically lazy.”

  • “Female promiscuity is disgusting.”

  • “Women shouldn’t drive cars or fly planes.”

  • “A lot of the world’s problems would be solved if women had their body count tattooed on their foreheads.”

Andrew's two main beliefs are:

  1. “These are my personal opinions based on my experiences.”

2. “I believe men are better at some things and women are better at some things. We are not equal.”

Andrew intentionally offends.

Andrew's thoughts began circulating online in 2022.

Image from Google Trends

In July 2022, he was one of the most Googled humans, surpassing:

  • Joe Biden

  • Donald Trump

  • Kim Kardashian

Andrews' rise is a mystery since no one can censure or suppress him. This is largely because Andrew nor his team post his clips.

But more on that later.

Andrew's path to wealth.

Andrew Tate is a self-made millionaire. His morality is uncertain.

Andrew and Tristan needed money soon after retiring from kickboxing.

“I owed some money to some dangerous people. I had $70K and needed $100K to stay alive.”

Andrews lost $20K on roulette at a local casino.

Andrew had one week to make $50,000, so he started planning. Andrew locked himself in a chamber like Thomas Edison to solve an energy dilemma.

He listed his assets.

  • Physical strength (but couldn’t fight)

  • a BMW (worth around $20K)

  • Intelligence (but no outlet)

A lightbulb.

He had an epiphany after viewing a webcam ad. He sought aid from women, ironically. His 5 international girlfriends are assets.

Then, a lightbulb.

Andrew and Tristan messaged and flew 7 women to a posh restaurant. Selling desperation masked as opportunity, Andrew pitched his master plan:

A webcam business — with a 50/50 revenue split.

5 women left.

2 stayed.

Andrew Tate, a broke kickboxer, became Top G, Cobra Tate.

The business model was simple — yet sad.

Andrew's girlfriends moved in with him and spoke online for 15+ hours a day. Andrew handled ads and equipment as the women posed.

Andrew eventually took over their keyboards, believing he knew what men wanted more than women.

Andrew detailed on the Full Send Podcast how he emotionally manipulated men for millions. They sold houses, automobiles, and life savings to fuel their companionship addiction.

When asked if he felt bad, Andrew said,

“F*ck no.“

Andrew and Tristan wiped off debts, hired workers, and diversified.

Tristan supervised OnlyFans models.

Andrew bought Romanian casinos and MMA league RXF (Real Xtreme Fighting).

Pandemic struck suddenly.

Andrew couldn't run his 2 businesses without a plan. Another easy moneymaker.

He banked on Hustlers University.

The actual cause of Andrew's ubiquity.

On a Your Mom’s House episode Andrew's 4 main revenue sources:

  1. Hustler’s University

2. Owning casinos in Romania

3. Owning 10% of the Romanian MMA league “RXF

4. “The War Room” — a society of rich and powerful men

When the pandemic hit, 3/4 became inoperable.

So he expanded Hustlers University.

But what is Hustler’s University?

Andrew says Hustlers University teaches 18 wealth-building tactics online. Examples:

  • Real estate

  • Copywriting

  • Amazon FBA

  • Dropshipping

  • Flipping Cryptos

How to swiftly become wealthy.

Lessons are imprecise, rudimentary, and macro-focused, say reviews. Invest wisely, etc. Everything is free online.

You pay for community. One unique income stream.

The only money-making mechanism that keeps the course from being a scam.

The truth is, many of Andrew’s students are actually making money. Maybe not from the free YouTube knowledge Andrew and his professors teach in the course, but through Hustler’s University’s affiliate program.

Affiliates earn 10% commission for each new student = $5.

Students can earn $10 for each new referral in the first two months.

Andrew earns $50 per membership per month.

This affiliate program isn’t anything special — in fact, it’s on the lower end of affiliate payouts. Normally, it wouldn’t be very lucrative.

But it has one secret weapon— Andrew and his viral opinions.

Andrew is viral. Andrew went on a media tour in January 2022 after appearing on Your Mom's House.

And many, many more…

He chatted with Twitch streamers. Hustlers University wanted more controversy (and clips).

Here’s the strategy behind Hustler’s University that has (allegedly) earned students upwards of $10K per month:

  1. Make a social media profile with Andrew Tates' name and photo.

  2. Post any of the online videos of Andrews that have gone viral.

  3. Include a referral link in your bio.

Effectively simple.

Andrew's controversy attracts additional students. More student clips circulate as more join. Andrew's students earn more and promote the product as he goes viral.

A brilliant plan that's functioning.

At the beginning of his media tour, Hustler’s University had 5,000 students. 6 months in, and he now has over 100,000.

One income stream generates $5 million every month.

Andrew's approach is not new.

But it is different.

In the early 2010s, Tai Lopez dominated the internet.

His viral video showed his house.

“Here in my garage. Just bought this new Lamborghini.”

Tais' marketing focused on intellect, not strength, power, and wealth to attract women.

How reading quicker leads to financial freedom in 67 steps.

Years later, it was revealed that Tai Lopez rented the mansion and Lamborghini as a marketing ploy to build social proof. Meanwhile, he was living in his friend’s trailer.

Faked success is an old tactic.

Andrew is doing something similar. But with one major distinction.

Andrew outsources his virality — making him nearly impossible to cancel.

In 2022, authorities searched Andrews' estate over human trafficking suspicions. Investigation continues despite withdrawn charges.

Andrew's divisive nature would normally get him fired. Andrew's enterprises and celebrity don't rely on social media.

He doesn't promote or pay for ads. Instead, he encourages his students and anyone wishing to get rich quick to advertise his work.

Because everything goes through his affiliate program. Old saying:

“All publicity is good publicity.”

Final thoughts: it’s ok to feel triggered.

Tate is divisive.

His emotionally charged words are human nature. Andrews created the controversy.

It's non-personal.

His opinions are those of one person. Not world nor generational opinion.

Briefly:

  • It's easy to understand why Andrews' face is ubiquitous. Money.

  • The world wide web is a chessboard. Misdirection is part of it.

  • It’s not personal, it’s business.

  • Controversy sells

Sometimes understanding the ‘why’, can help you deal with the ‘what.’

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Matt Nutsch

Matt Nutsch

6 months ago

Most people are unaware of how artificial intelligence (A.I.) is changing the world.

Image created by MidjourneyAI user Dreamland3K

Recently, I saw an interesting social media post. In an entrepreneurship forum. A blogger asked for help because he/she couldn't find customers. I now suspect that the writer’s occupation is being disrupted by A.I.

Introduction

Artificial Intelligence (A.I.) has been a hot topic since the 1950s. With recent advances in machine learning, A.I. will touch almost every aspect of our lives. This article will discuss A.I. technology and its social and economic implications.

What's AI?

A computer program or machine with A.I. can think and learn. In general, it's a way to make a computer smart. Able to understand and execute complex tasks. Machine learning, NLP, and robotics are common types of A.I.

AI's global impact

MidjourneyAI image generated by user Desmesne

AI will change the world, but probably faster than you think. A.I. already affects our daily lives. It improves our decision-making, efficiency, and productivity.

A.I. is transforming our lives and the global economy. It will create new business and job opportunities but eliminate others. Affected workers may face financial hardship.

AI examples:

OpenAI's GPT-3 text-generation

MidjourneyAI generated image of robot typing

Developers can train, deploy, and manage models on GPT-3. It handles data preparation, model training, deployment, and inference for machine learning workloads. GPT-3 is easy to use for both experienced and new data scientists.

My team conducted an experiment. We needed to generate some blog posts for a website. We hired a blogger on Upwork. OpenAI created a blog post. The A.I.-generated blog post was of higher quality and lower cost.

MidjourneyAI's Art Contests

Théâtre D’opéra Spatial by Jason M. Allen via MidjourneyAI

AI already affects artists. Artists use A.I. to create realistic 3D images and videos for digital art. A.I. is also used to generate new art ideas and methods.

MidjourneyAI and GigapixelAI won a contest last month. It's AI. created a beautiful piece of art that captured the contest's spirit. AI triumphs. It could open future doors.

After the art contest win, I registered to try out these new image generating A.I.s. In the MidjourneyAI chat forum, I noticed an artist's plea. The artist begged others to stop flooding RedBubble with AI-generated art.

Shutterstock and Getty Images have halted user uploads. AI-generated images flooded online marketplaces.

Imagining Videos with Meta

AI generated video example from Meta AI

Meta released Make-a-Video this week. It's an A.I. app that creates videos from text. What you type creates a video.

This technology will impact TV, movies, and video games greatly. Imagine a movie or game that's personalized to your tastes. It's closer than you think.

Uses and Abuses of Deepfakes

Carrie Fischer’s likeness in the movie The Rise of Skywalker

Deepfake videos are computer-generated images of people. AI creates realistic images and videos of people.

Deepfakes are entertaining but have social implications. Porn introduced deepfakes in 2017. People put famous faces on porn actors and actresses without permission.

Soon, deepfakes were used to show dead actors/actresses or make them look younger. Carrie Fischer was included in films after her death using deepfake technology.

Deepfakes can be used to create fake news or manipulate public opinion, according to an AI.

Voices for Darth Vader and Iceman

James Earl Jones, who voiced Darth Vader, sold his voice rights this week. Aged actor won't be in those movies. Respeecher will use AI to mimic Jones's voice. This technology could change the entertainment industry. One actor can now voice many characters.

Val Kilmer in Top Gun as imagined by MidjourneyAI

AI can generate realistic voice audio from text. Top Gun 2 actor Val Kilmer can't speak for medical reasons. Sonantic created Kilmer's voice from the movie script. This entertaining technology has social implications. It blurs authentic recordings and fake media.

Medical A.I. fights viruses

MidjourneyAI generated image of virus

A team of Chinese scientists used machine learning to predict effective antiviral drugs last year. They started with a large dataset of virus-drug interactions. Researchers combined that with medication and virus information. Finally, they used machine learning to predict effective anti-virus medicines. This technology could solve medical problems.

AI ideas AI-generated Itself

MidjourneyAI image generated by user SubjectChunchunmaru

OpenAI's GPT-3 predicted future A.I. uses. Here's what it told me:

AI will affect the economy. Businesses can operate more efficiently and reinvest resources with A.I.-enabled automation. AI can automate customer service tasks, reducing costs and improving satisfaction.

A.I. makes better pricing, inventory, and marketing decisions. AI automates tasks and makes decisions. A.I.-powered robots could help the elderly or disabled. Self-driving cars could reduce accidents.

A.I. predictive analytics can predict stock market or consumer behavior trends and patterns. A.I. also personalizes recommendations. sways. A.I. recommends products and movies. AI can generate new ideas based on data analysis.

Conclusion

Image generated from MidjourneyAI by user PuddingPants.”

A.I. will change business as it becomes more common. It will change how we live and work by creating growth and prosperity.

Exciting times,  but also one which should give us all pause. Technology can be good or evil. We must use new technologies ethically, fairly, and honestly.

“The author generated some sentences in this text in part with GPT-3, OpenAI’s large-scale language-generation model. Upon generating draft language, the author reviewed, edited, and revised the language to their own liking and takes ultimate responsibility for the content of this publication. The text of this post was further edited using HemingWayApp. Many of the images used were generated using A.I. as described in the captions.”

Ben "The Hosk" Hosking

Ben "The Hosk" Hosking

7 months ago

The Yellow Cat Test Is Typically Failed by Software Developers.

Believe what you see, what people say

Photo by Артем from Pexels

It’s sad that we never get trained to leave assumptions behind. - Sebastian Thrun

Many problems in software development are not because of code but because developers create the wrong software. This isn't rare because software is emergent and most individuals only realize what they want after it's built.

Inquisitive developers who pass the yellow cat test can improve the process.

Carpenters measure twice and cut the wood once. Developers are rarely so careful.

The Yellow Cat Test

Game of Thrones made dragons cool again, so I am reading The Game of Thrones book.

The yellow cat exam is from Syrio Forel, Arya Stark's fencing instructor.

Syrio tells Arya he'll strike left when fencing. He hits her after she dodges left. Arya says “you lied”. Syrio says his words lied, but his eyes and arm told the truth.

Arya learns how Syrio became Bravos' first sword.

“On the day I am speaking of, the first sword was newly dead, and the Sealord sent for me. Many bravos had come to him, and as many had been sent away, none could say why. When I came into his presence, he was seated, and in his lap was a fat yellow cat. He told me that one of his captains had brought the beast to him, from an island beyond the sunrise. ‘Have you ever seen her like?’ he asked of me.

“And to him I said, ‘Each night in the alleys of Braavos I see a thousand like him,’ and the Sealord laughed, and that day I was named the first sword.”

Arya screwed up her face. “I don’t understand.”

Syrio clicked his teeth together. “The cat was an ordinary cat, no more. The others expected a fabulous beast, so that is what they saw. How large it was, they said. It was no larger than any other cat, only fat from indolence, for the Sealord fed it from his own table. What curious small ears, they said. Its ears had been chewed away in kitten fights. And it was plainly a tomcat, yet the Sealord said ‘her,’ and that is what the others saw. Are you hearing?” Reddit discussion.

Development teams should not believe what they are told.

We created an appointment booking system. We thought it was an appointment-booking system. Later, we realized the software's purpose was to book the right people for appointments and discourage the unneeded ones.

The first 3 months of the project had half-correct requirements and software understanding.

Open your eyes

“Open your eyes is all that is needed. The heart lies and the head plays tricks with us, but the eyes see true. Look with your eyes, hear with your ears. Taste with your mouth. Smell with your nose. Feel with your skin. Then comes the thinking afterwards, and in that way, knowing the truth” Syrio Ferel

We must see what exists, not what individuals tell the development team or how developers think the software should work. Initial criteria cover 50/70% and change.

Developers build assumptions problems by assuming how software should work. Developers must quickly explain assumptions.

When a development team's assumptions are inaccurate, they must alter the code, DevOps, documentation, and tests.

It’s always faster and easier to fix requirements before code is written.

First-draft requirements can be based on old software. Development teams must grasp corporate goals and consider needs from many angles.

Testers help rethink requirements. They look at how software requirements shouldn't operate.

Technical features and benefits might misdirect software projects.

The initiatives that focused on technological possibilities developed hard-to-use software that needed extensive rewriting following user testing.

Software development

High-level criteria are different from detailed ones.

  • The interpretation of words determines their meaning.

  • Presentations are lofty, upbeat, and prejudiced.

  • People's perceptions may be unclear, incorrect, or just based on one perspective (half the story)

  • Developers can be misled by requirements, circumstances, people, plans, diagrams, designs, documentation, and many other things.

Developers receive misinformation, misunderstandings, and wrong assumptions. The development team must avoid building software with erroneous specifications.

Once code and software are written, the development team changes and fixes them.

Developers create software with incomplete information, they need to fill in the blanks to create the complete picture.

Conclusion

Yellow cats are often inaccurate when communicating requirements.

Before writing code, clarify requirements, assumptions, etc.

Everyone will pressure the development team to generate code rapidly, but this will slow down development.

Code changes are harder than requirements.

Vitalik

Vitalik

1 year 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