More on Society & Culture

Isaiah McCall
3 years ago
Is TikTok slowly destroying a new generation?
It's kids' digital crack

TikTok is a destructive social media platform.
The interface shortens attention spans and dopamine receptors.
TikTok shares more data than other apps.
Seeing an endless stream of dancing teens on my glowing box makes me feel like a Blade Runner extra.
TikTok did in one year what MTV, Hollywood, and Warner Music tried to do in 20 years. TikTok has psychotized the two-thirds of society Aldous Huxley said were hypnotizable.
Millions of people, mostly kids, are addicted to learning a new dance, lip-sync, or prank, and those who best dramatize this collective improvisation get likes, comments, and shares.
TikTok is a great app. So what?
The Commercial Magnifying Glass TikTok made me realize my generation's time was up and the teenage Zoomers were the target.
I told my 14-year-old sister, "Enjoy your time under the commercial magnifying glass."
TikTok sells your every move, gesture, and thought. Data is the new oil. If you tell someone, they'll say, "Yeah, they collect data, but who cares? I have nothing to hide."
It's a George Orwell novel's beginning. Look up Big Brother Award winners to see if TikTok won.

TikTok shares your data more than any other social media app, and where it goes is unclear. TikTok uses third-party trackers to monitor your activity after you leave the app.
Consumers can't see what data is shared or how it will be used. — Genius URL
32.5 percent of Tiktok's users are 10 to 19 and 29.5% are 20 to 29.
TikTok is the greatest digital marketing opportunity in history, and they'll use it to sell you things, track you, and control your thoughts. Any of its users will tell you, "I don't care, I just want to be famous."
TikTok manufactures mental illness
TikTok's effect on dopamine and the brain is absurd. Dopamine controls the brain's pleasure and reward centers. It's like a switch that tells your brain "this feels good, repeat."
Dr. Julie Albright, a digital culture and communication sociologist, said TikTok users are "carried away by dopamine." It's hypnotic, you'll keep watching."
TikTok constantly releases dopamine. A guy on TikTok recently said he didn't like books because they were slow and boring.
The US didn't ban Tiktok.
Biden and Trump agree on bad things. Both agree that TikTok threatens national security and children's mental health.
The Chinese Communist Party owns and operates TikTok, but that's not its only problem.
There’s borderline child porn on TikTok
It's unsafe for children and violated COPPA.
It's also Chinese spyware. I'm not a Trump supporter, but I was glad he wanted TikTok regulated and disappointed when he failed.
Full-on internet censorship is rare outside of China, so banning it may be excessive. US should regulate TikTok more.
We must reject a low-quality present for a high-quality future.
TikTok vs YouTube
People got mad when I wrote about YouTube's death.
They didn't like when I said TikTok was YouTube's first real challenger.
Indeed. TikTok is the fastest-growing social network. In three years, the Chinese social media app TikTok has gained over 1 billion active users. In the first quarter of 2020, it had the most downloads of any app in a single quarter.
TikTok is the perfect social media app in many ways. It's brief and direct.

Can you believe they had a YouTube vs TikTok boxing match? We are doomed as a species.
YouTube hosts my favorite videos. That’s why I use it. That’s why you use it. New users expect more. They want something quicker, more addictive.
TikTok's impact on other social media platforms frustrates me. YouTube copied TikTok to compete.
It's all about short, addictive content.
I'll admit I'm probably wrong about TikTok. My friend says his feed is full of videos about food, cute animals, book recommendations, and hot lesbians.
Whatever.
TikTok makes us bad
TikTok is the opposite of what the Ancient Greeks believed about wisdom.
It encourages people to be fake. It's like a never-ending costume party where everyone competes.
It does not mean that Gen Z is doomed.
They could be the saviors of the world for all I know.
TikTok feels like a step towards Mike Judge's "Idiocracy," where the average person is a pleasure-seeking moron.

Max Chafkin
3 years ago
Elon Musk Bets $44 Billion on Free Speech's Future
Musk’s purchase of Twitter has sealed his bond with the American right—whether the platform’s left-leaning employees and users like it or not.
Elon Musk's pursuit of Twitter Inc. began earlier this month as a joke. It started slowly, then spiraled out of control, culminating on April 25 with the world's richest man agreeing to spend $44 billion on one of the most politically significant technology companies ever. There have been bigger financial acquisitions, but Twitter's significance has always outpaced its balance sheet. This is a unique Silicon Valley deal.
To recap: Musk announced in early April that he had bought a stake in Twitter, citing the company's alleged suppression of free speech. His complaints were vague, relying heavily on the dog whistles of the ultra-right. A week later, he announced he'd buy the company for $54.20 per share, four days after initially pledging to join Twitter's board. Twitter's directors noticed the 420 reference as well, and responded with a “shareholder rights” plan (i.e., a poison pill) that included a 420 joke.
Musk - Patrick Pleul/Getty Images
No one knew if the bid was genuine. Musk's Twitter plans seemed implausible or insincere. In a tweet, he referred to automated accounts that use his name to promote cryptocurrency. He enraged his prospective employees by suggesting that Twitter's San Francisco headquarters be turned into a homeless shelter, renaming the company Titter, and expressing solidarity with his growing conservative fan base. “The woke mind virus is making Netflix unwatchable,” he tweeted on April 19.
But Musk got funding, and after a frantic weekend of negotiations, Twitter said yes. Unlike most buyouts, Musk will personally fund the deal, putting up up to $21 billion in cash and borrowing another $12.5 billion against his Tesla stock.
Free Speech and Partisanship
Percentage of respondents who agree with the following
The deal is expected to replatform accounts that were banned by Twitter for harassing others, spreading misinformation, or inciting violence, such as former President Donald Trump's account. As a result, Musk is at odds with his own left-leaning employees, users, and advertisers, who would prefer more content moderation rather than less.
Dorsey - Photographer: Joe Raedle/Getty Images
Previously, the company's leadership had similar issues. Founder Jack Dorsey stepped down last year amid concerns about slowing growth and product development, as well as his dual role as CEO of payments processor Block Inc. Compared to Musk, a father of seven who already runs four companies (besides Tesla and SpaceX), Dorsey is laser-focused.
Musk's motivation to buy Twitter may be political. Affirming the American far right with $44 billion spent on “free speech” Right-wing activists have promoted a series of competing upstart Twitter competitors—Parler, Gettr, and Trump's own effort, Truth Social—since Trump was banned from major social media platforms for encouraging rioters at the US Capitol on Jan. 6, 2021. But Musk can give them a social network with lax content moderation and a real user base. Trump said he wouldn't return to Twitter after the deal was announced, but he wouldn't be the first to do so.
Trump - Eli Hiller/Bloomberg
Conservative activists and lawmakers are already ecstatic. “A great day for free speech in America,” said Missouri Republican Josh Hawley. The day the deal was announced, Tucker Carlson opened his nightly Fox show with a 10-minute laudatory monologue. “The single biggest political development since Donald Trump's election in 2016,” he gushed over Musk.
But Musk's supporters and detractors misunderstand how much his business interests influence his political ideology. He marketed Tesla's cars as carbon-saving machines that were faster and cooler than gas-powered luxury cars during George W. Bush's presidency. Musk gained a huge following among wealthy environmentalists who reserved hundreds of thousands of Tesla sedans years before they were made during Barack Obama's presidency. Musk in the Trump era advocated for a carbon tax, but he also fought local officials (and his own workers) over Covid rules that slowed the reopening of his Bay Area factory.
Teslas at the Las Vegas Convention Center Loop Central Station in April 2021. The Las Vegas Convention Center Loop was Musk's first commercial project. Ethan Miller/Getty Images
Musk's rightward shift matched the rise of the nationalist-populist right and the desire to serve a growing EV market. In 2019, he unveiled the Cybertruck, a Tesla pickup, and in 2018, he announced plans to manufacture it at a new plant outside Austin. In 2021, he decided to move Tesla's headquarters there, citing California's "land of over-regulation." After Ford and General Motors beat him to the electric truck market, Musk reframed Tesla as a company for pickup-driving dudes.
Similarly, his purchase of Twitter will be entwined with his other business interests. Tesla has a factory in China and is friendly with Beijing. This could be seen as a conflict of interest when Musk's Twitter decides how to treat Chinese-backed disinformation, as Amazon.com Inc. founder Jeff Bezos noted.
Musk has focused on Twitter's product and social impact, but the company's biggest challenges are financial: Either increase cash flow or cut costs to comfortably service his new debt. Even if Musk can't do that, he can still benefit from the deal. He has recently used the increased attention to promote other business interests: Boring has hyperloops and Neuralink brain implants on the way, Musk tweeted. Remember Tesla's long-promised robotaxis!
Musk may be comfortable saying he has no expectation of profit because it benefits his other businesses. At the TED conference on April 14, Musk insisted that his interest in Twitter was solely charitable. “I don't care about money.”
The rockets and weed jokes make it easy to see Musk as unique—and his crazy buyout will undoubtedly add to that narrative. However, he is a megabillionaire who is risking a small amount of money (approximately 13% of his net worth) to gain potentially enormous influence. Musk makes everything seem new, but this is a rehash of an old media story.

Sam Warain
2 years ago
The Brilliant Idea Behind Kim Kardashian's New Private Equity Fund
Kim Kardashian created Skky Partners. Consumer products, internet & e-commerce, consumer media, hospitality, and luxury are company targets.
Some call this another Kardashian publicity gimmick.
This maneuver is brilliance upon closer inspection. Why?
1) Kim has amassed a sizable social media fan base:
Over 320 million Instagram and 70 million Twitter users follow Kim Kardashian.
Kim Kardashian's Instagram account ranks 8th. Three Kardashians in top 10 is ridiculous.
This gives her access to consumer data. She knows what people are discussing. Investment firms need this data.
Quality, not quantity, of her followers matters. Studies suggest that her following are more engaged than Selena Gomez and Beyonce's.
Kim's followers are worth roughly $500 million to her brand, according to a research. They trust her and buy what she recommends.
2) She has a special aptitude for identifying trends.
Kim Kardashian can sense trends.
She's always ahead of fashion and beauty trends. She's always trying new things, too. She doesn't mind making mistakes when trying anything new. Her desire to experiment makes her a good business prospector.
Kim has also created a lifestyle brand that followers love. Kim is an entrepreneur, mom, and role model, not just a reality TV star or model. She's established a brand around her appearance, so people want to buy her things.
Her fragrance collection has sold over $100 million since its 2009 introduction, and her Sears apparel line did over $200 million in its first year.
SKIMS is her latest $3.2bn brand. She can establish multibillion-dollar firms with her enormous distribution platform.
Early founders would kill for Kim Kardashian's network.
Making great products is hard, but distribution is more difficult. — David Sacks, All-in-Podcast
3) She can delegate the financial choices to Jay Sammons, one of the greatest in the industry.
Jay Sammons is well-suited to develop Kim Kardashian's new private equity fund.
Sammons has 16 years of consumer investing experience at Carlyle. This will help Kardashian invest in consumer-facing enterprises.
Sammons has invested in Supreme and Beats Electronics, both of which have grown significantly. Sammons' track record and competence make him the obvious choice.
Kim Kardashian and Jay Sammons have joined forces to create a new business endeavor. The agreement will increase Kardashian's commercial empire. Sammons can leverage one of the world's most famous celebrities.
“Together we hope to leverage our complementary expertise to build the next generation consumer and media private equity firm” — Kim Kardashian
Kim Kardashian is a successful businesswoman. She developed an empire by leveraging social media to connect with fans. By developing a global lifestyle brand, she has sold things and experiences that have made her one of the world's richest celebrities.
She's a shrewd entrepreneur who knows how to maximize on herself and her image.
Imagine how much interest Kim K will bring to private equity and venture capital.
I'm curious about the company's growth.
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Victoria Kurichenko
3 years ago
My Blog Is in Google's Top 10—Here's How to Compete
"Competition" is beautiful and hateful.
Some people bury their dreams because they are afraid of competition. Others challenge themselves, shaping our world.
Competition is normal.
It spurs innovation and progress.
I wish more people agreed.
As a marketer, content writer, and solopreneur, my readers often ask:
"I want to create a niche website, but I have no ideas. Everything's done"
"Is a website worthwhile?"
I can't count how many times I said, "Yes, it makes sense, and you can succeed in a competitive market."
I encourage and share examples, but it's not enough to overcome competition anxiety.
I launched an SEO writing website for content creators a year ago, knowing it wouldn't beat Ahrefs, Semrush, Backlinko, etc.
Not needed.
Many of my website's pages rank highly on Google.
Everyone can eat the pie.
In a competitive niche, I took a different approach.
Look farther
When chatting with bloggers that want a website, I discovered something fascinating.
They want to launch a website but have no ideas. As a next step, they start listing the interests they believe they should work on, like wellness, lifestyle, investments, etc. I could keep going.
Too many generalists who claim to know everything confuse many.
Generalists aren't trusted.
We want someone to fix our problems immediately.
I don't think broad-spectrum experts are undervalued. People have many demands that go beyond generalists' work. Narrow-niche experts can help.
I've done SEO for three years. I learned from experts and courses. I couldn't find a comprehensive SEO writing resource.
I read tons of articles before realizing that wasn't it. I took courses that covered SEO basics eventually.
I had a demand for learning SEO writing, but there was no solution on the market. My website fills this micro-niche.
Have you ever had trouble online?
Professional courses too general, boring, etc.?
You've bought off-topic books, right?
You're not alone.
Niche ideas!
Big players often disregard new opportunities. Too small. Individual content creators can succeed here.
In a competitive market:
Never choose wide subjects
Think about issues you can relate to and have direct experience with.
Be a consumer to discover both the positive and negative aspects of a good or service.
Merchandise your annoyances.
Consider ways to transform your frustrations into opportunities.
The right niche is half-success. Here is what else I did to hit the Google front page with my website.
An innovative method for choosing subjects
Why publish on social media and websites?
Want likes, shares, followers, or fame?
Some people do it for fun. No judgment.
I bet you want more.
You want to make decent money from blogging.
Writing about random topics, even if they are related to your niche, won’t help you attract an audience from organic search. I'm a marketer and writer.
I worked at companies with dead blogs because they posted for themselves, not readers. They did not follow SEO writing rules; that’s why most of their content flopped.
I learned these hard lessons and grew my website from 0 to 3,000+ visitors per month while working on it a few hours a week only. Evidence:
I choose website topics using these criteria:
- Business potential. The information should benefit my audience and generate revenue. There would be no use in having it otherwise.
My topics should help me:
Attract organic search traffic with my "fluff-free" content -> Subscribers > SEO ebook sales.
Simple and effective.
- traffic on search engines. The number of monthly searches reveals how popular my topic is all across the world. If I find that no one is interested in my suggested topic, I don't write a blog article.
- Competition. Every search term is up against rivals. Some are more popular (thus competitive) since more websites target them in organic search. A new website won't score highly for keywords that are too competitive. On the other side, keywords with moderate to light competition can help you rank higher on Google more quickly.
- Search purpose. The "why" underlying users' search requests is revealed. I analyze search intent to understand what users need when they plug various queries in the search bar and what content can perfectly meet their needs.
My specialty website produces money, ranks well, and attracts the target audience because I handpick high-traffic themes.
Following these guidelines, even a new website can stand out.
I wrote a 50-page SEO writing guide where I detailed topic selection and share my front-page Google strategy.
My guide can help you run a successful niche website.
In summary
You're not late to the niche-website party.
The Internet offers many untapped opportunities.
We need new solutions and are willing to listen.
There are unexplored niches in any topic.
Don't fight giants. They have their piece of the pie. They might overlook new opportunities while trying to keep that piece of the pie. You should act now.

Koji Mochizuki
3 years ago
How to Launch an NFT Project by Yourself
Creating 10,000 auto-generated artworks, deploying a smart contract to the Ethereum / Polygon blockchain, setting up some tools, etc.
There is so much to do from launching to running an NFT project. Creating parts for artworks, generating 10,000 unique artworks and metadata, creating a smart contract and deploying it to a blockchain network, creating a website, creating a Twitter account, setting up a Discord server, setting up an OpenSea collection. In addition, you need to have MetaMask installed in your browser and have some ETH / MATIC. Did you get tired of doing all this? Don’t worry, once you know what you need to do, all you have to do is do it one by one.
To be honest, it’s best to run an NFT project in a team of three or more, including artists, developers, and marketers. However, depending on your motivation, you can do it by yourself. Some people might come later to offer help with your project. The most important thing is to take a step as soon as possible.
Creating Parts for Artworks
There are lots of free/paid software for drawing, but after all, I think Adobe Illustrator or Photoshop is the best. The images of Skulls In Love are a composite of 48x48 pixel parts created using Photoshop.
The most important thing in creating parts for generative art is to repeatedly test what your artworks will look like after each layer has been combined. The generated artworks should not be too unnatural.
How Many Parts Should You Create?
Are you wondering how many parts you should create to avoid duplication as much as possible when generating your artworks? My friend Stephane, a developer, has created a great tool to help with that.
Generating 10,000 Unique Artworks and Metadata
I highly recommend using the HashLips Art Engine to generate your artworks and metadata. Perhaps there is no better artworks generation tool at the moment.
GitHub: https://github.com/HashLips/hashlips_art_engine
YouTube:
Storing Artworks and Metadata
Ideally, the generated artworks and metadata should be stored on-chain, but if you want to store them off-chain, you should use IPFS. Do not store in centralized storage. This is because data will be lost if the server goes down or if the company goes down. On the other hand, IPFS is a more secure way to find data because it utilizes a distributed, decentralized system.
Storing to IPFS is easy with Pinata, NFT.Storage, and so on. The Skulls In Love uses Pinata. It’s very easy to use, just upload the folder containing your artworks.
Creating and Deploying a Smart Contract
You don’t have to create a smart contract from scratch. There are many great NFT projects, many of which publish their contract source code on Etherscan / PolygonScan. You can choose the contract you like and reuse it. Of course, that requires some knowledge of Solidity, but it depends on your efforts. If you don’t know which contract to choose, use the HashLips smart contract. It’s very simple, but it has almost all the functions you need.
GitHub: https://github.com/HashLips/hashlips_nft_contract
Note: Later on, you may want to change the cost value. You can change it on Remix or Etherscan / PolygonScan. But in this case, enter the Wei value instead of the Ether value. For example, if you want to sell for 1 MATIC, you have to enter “1000000000000000000”. If you set this value to “1”, you will have a nightmare. I recommend using Simple Unit Converter as a tool to calculate the Wei value.
Creating a Website
The website here is not just a static site to showcase your project, it’s a so-called dApp that allows you to access your smart contract and mint NFTs. In fact, this level of dApp is not too difficult for anyone who has ever created a website. Because the ethers.js / web3.js libraries make it easy to interact with your smart contract. There’s also no problem connecting wallets, as MetaMask has great documentation.
The Skulls In Love uses a simple, fast, and modern dApp that I built from scratch using Next.js. It is published on GitHub, so feel free to use it.
Why do people mint NFTs on a website?
Ethereum’s gas fees are high, so if you mint all your NFTs, there will be a huge initial cost. So it makes sense to get the buyers to help with the gas fees for minting.
What about Polygon? Polygon’s gas fees are super cheap, so even if you mint 10,000 NFTs, it’s not a big deal. But we don’t do that. Since NFT projects are a kind of game, it involves the fun of not knowing what will come out after minting.
Creating a Twitter Account
I highly recommend creating a Twitter account. Twitter is an indispensable tool for announcing giveaways and reaching more people. It’s better to announce your project and your artworks little by little, 1–2 weeks before launching your project.
Creating and Setting Up a Discord Server
I highly recommend creating a Discord server as well as a Twitter account. The Discord server is a community and its home. Fans of your NFT project will want to join your community and interact with many other members. So, carefully create each channel on your Discord server to make it a cozy place for your community members.
If you are unfamiliar with Discord, you may be particularly confused by the following:
What bots should I use?
How should I set roles and permissions?
But don’t worry. There are lots of great YouTube videos and blog posts about these.
It’s also a good idea to join the Discord servers of some NFT projects and see how they’re made. Our Discord server is so simple that even beginners will find it easy to understand. Please join us and see it!
Note: First, create a test account and a test server to make sure your bots and permissions work properly. It is better to verify the behavior on the test server before setting up your production server.
UPDATED: As your Discord server grows, you cannot manage it on your own. In this case, you will be hiring several moderators, but choose carefully before hiring. And don’t give them important role permissions right after hiring. Initially, the same permissions as other members are sufficient. After a while, you can add permissions as needed, such as kicking/banning, using the “@every” tag, and adding roles. Again, don’t immediately give significant permissions to your Mod role. Your server can be messed up by fake moderators.
Setting Up Your OpenSea Collection
Before you start selling your NFTs, you need to reserve some for airdrops, giveaways, staff, and more. It’s up to you whether it’s 100, 500, or how many.
After minting some of your NFTs, your account and collection should have been created in OpenSea. Go to OpenSea, connect to your wallet, and set up your collection. Just set your logo, banner image, description, links, royalties, and more. It’s not that difficult.
Promoting Your Project
After all, promotion is the most important thing. In fact, almost every successful NFT project spends a lot of time and effort on it.
In addition to Twitter and Discord, it’s even better to use Instagram, Reddit, and Medium. Also, register your project in NFTCalendar and DISBOARD
DISBOARD is the public Discord server listing community.
About Promoters
You’ll probably get lots of contacts from promoters on your Discord, Twitter, Instagram, and more. But most of them are scams, so don’t pay right away. If you have a promoter that looks attractive to you, be sure to check the promoter’s social media accounts or website to see who he/she is. They basically charge in dollars. The amount they charge isn’t cheap, but promoters with lots of followers may have some temporary effect on your project. Some promoters accept 50% prepaid and 50% postpaid. If you can afford it, it might be worth a try. I never ask them, though.
When Should the Promotion Activities Start?
You may be worried that if you promote your project before it starts, someone will copy your project (artworks). It is true that some projects have actually suffered such damage. I don’t have a clear answer to this question right now, but:
- Do not publish all the information about your project too early
- The information should be released little by little
- Creating artworks that no one can easily copy
I think these are important.
If anyone has a good idea, please share it!
About Giveaways
When hosting giveaways, you’ll probably use multiple social media platforms. You may want to grow your Discord server faster. But if joining the Discord server is included in the giveaway requirements, some people hate it. I recommend holding giveaways for each platform. On Twitter and Reddit, you should just add the words “Discord members-only giveaway is being held now! Please join us if you like!”.
If you want to easily pick a giveaway winner in your browser, I recommend Twitter Picker.
Precautions for Distributing Free NFTs
If you want to increase your Twitter followers and Discord members, you can actually get a lot of people by holding events such as giveaways and invite contests. However, distributing many free NFTs at once can be dangerous. Some people who want free NFTs, as soon as they get a free one, sell it at a very low price on marketplaces such as OpenSea. They don’t care about your project and are only thinking about replacing their own “free” NFTs with Ethereum. The lower the floor price of your NFTs, the lower the value of your NFTs (project). Try to think of ways to get people to “buy” your NFTs as much as possible.
Ethereum vs. Polygon
Even though Ethereum has high gas fees, NFT projects on the Ethereum network are still mainstream and popular. On the other hand, Polygon has very low gas fees and fast transaction processing, but NFT projects on the Polygon network are not very popular.
Why? There are several reasons, but the biggest one is that it’s a lot of work to get MATIC (on Polygon blockchain, use MATIC instead of ETH) ready to use. Simply put, you need to bridge your tokens to the Polygon chain. So people need to do this first before minting your NFTs on your website. It may not be a big deal for those who are familiar with crypto and blockchain, but it may be complicated for those who are not. I hope that the tedious work will be simplified in the near future.
If you are confident that your NFTs will be purchased even if they are expensive, or if the total supply of your NFTs is low, you may choose Ethereum. If you just want to save money, you should choose Polygon. Keep in mind that gas fees are incurred not only when minting, but also when performing some of your smart contract functions and when transferring your NFTs.
If I were to launch a new NFT project, I would probably choose Ethereum or Solana.
Conclusion
Some people may want to start an NFT project to make money, but don’t forget to enjoy your own project. Several months ago, I was playing with creating generative art by imitating the CryptoPunks. I found out that auto-generated artworks would be more interesting than I had imagined, and since then I’ve been completely absorbed in generative art.
This is one of the Skulls In Love artworks:
This character wears a cowboy hat, black slim sunglasses, and a kimono. If anyone looks like this, I can’t help laughing!
The Skulls In Love NFTs can be minted for a small amount of MATIC on the official website. Please give it a try to see what kind of unique characters will appear 💀💖
Thank you for reading to the end. I hope this article will be helpful to those who want to launch an NFT project in the future ✨

Vitalik
3 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3
