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Jim Siwek

Jim Siwek

3 years ago

In 2022, can a lone developer be able to successfully establish a SaaS product?

More on Entrepreneurship/Creators

Jerry Keszka

Jerry Keszka

3 years ago

10 Crazy Useful Free Websites No One Told You About But You Needed

The internet is a massive information resource. With so much stuff, it's easy to forget about useful websites. Here are five essential websites you may not have known about.

Image from Canva selected by the author. The author assumes responsibility for the copyright

1. Companies.tools

Companies.tools are what successful startups employ. This website offers a curated selection of design, research, coding, support, and feedback resources. Ct has the latest app development platform and greatest client feedback method.

2. Excel Formula Bot

Excel Formula Bot can help if you forget a formula. Formula Bot uses AI to convert text instructions into Excel formulas, so you don't have to remember them.

Just tell the Bot what to do, and it will do it. Excel Formula Bot can calculate sales tax and vacation days. When you're stuck, let the Bot help.

3.TypeLit

TypeLit helps you improve your typing abilities while reading great literature.

TypeLit.io lets you type any book or dozens of preset classics. TypeLit provides real-time feedback on accuracy and speed.

Goals and progress can be tracked. Why not improve your typing and learn great literature with TypeLit?

4. Calm Schedule

Finding a meeting time that works for everyone is difficult. Personal and business calendars might be difficult to coordinate.

Synchronize your two calendars to save time and avoid problems. You may avoid searching through many calendars for conflicts and keep your personal information secret. Having one source of truth for personal and work occasions will help you never miss another appointment.
https://calmcalendar.com/

5. myNoise

myNoise makes the outside world quieter. myNoise is the right noise for a noisy office or busy street.

If you can't locate the right noise, make it. MyNoise unlocks the world. Shut out distractions. Thank your ears.

6. Synthesia

Professional videos require directors, filmmakers, editors, and animators. Now, thanks to AI, you can generate high-quality videos without video editing experience.

AI avatars are crucial. You can design a personalized avatar using a web-based software like synthesia.io. Our avatars can lip-sync in over 60 languages, so you can make worldwide videos. There's an AI avatar for every video goal.

Not free. Amazing service, though.

7. Cleaning-up-images

Have you shot a wonderful photo just to notice something in the background? You may have a beautiful headshot but wish to erase an imperfection.

Cleanup.pictures removes undesirable objects from photos. Our algorithms will eliminate the selected object.

Cleanup.pictures can help you obtain the ideal shot every time. Next time you take images, let Cleanup.pictures fix any flaws.

8. PDF24 Tools

Editing a PDF can be a pain. Most of us don't know Adobe Acrobat's functionalities. Why buy something you'll rarely use? Better options exist.

PDF24 is an online PDF editor that's free and subscription-free. Rotate, merge, split, compress, and convert PDFs in your browser. PDF24 makes document signing easy.

Upload your document, sign it (or generate a digital signature), and download it. It's easy and free. PDF24 is a free alternative to pricey PDF editing software.

9. Class Central

Finding online classes is much easier. Class Central has classes from Harvard, Stanford, Coursera, Udemy, and Google, Amazon, etc. in one spot.

Whether you want to acquire a new skill or increase your knowledge, you'll find something. New courses bring variety.

10. Rome2rio

Foreign travel offers countless transport alternatives. How do you get from A to B? It’s easy!

Rome2rio will show you the best method to get there, including which mode of transport is ideal.

  • Plane

  • Car

  • Train

  • Bus

  • Ferry

  • Driving

  • Shared bikes

  • Walking

Do you know any free, useful websites?

Navdeep Yadav

Navdeep Yadav

2 years ago

31 startup company models (with examples)

Many people find the internet's various business models bewildering.

This article summarizes 31 startup e-books.

Types of Startup

1. Using the freemium business model (free plus premium),

The freemium business model offers basic software, games, or services for free and charges for enhancements.

Examples include Slack, iCloud, and Google Drive

Provide a rudimentary, free version of your product or service to users.

Graphic Credit: Business Model toolbox

Google Drive and Dropbox offer 15GB and 2GB of free space but charge for more.

Freemium business model details (Click here)

2. The Business Model of Subscription

Subscription business models sell a product or service for recurring monthly or yearly revenue.

Graphic Credit: Business Model toolbox

Examples: Tinder, Netflix, Shopify, etc

It's the next step to Freemium if a customer wants to pay monthly for premium features.

Types of Subscription Business Models

Subscription Business Model (Click here)

3. A market-based business strategy

It's an e-commerce site or app where third-party sellers sell products or services.

Examples are Amazon and Fiverr.

Marketplace Business Model
  • On Amazon's marketplace, a third-party vendor sells a product.

  • Freelancers on Fiverr offer specialized skills like graphic design.

Marketplace's business concept is explained.

4. Business plans using aggregates

In the aggregator business model, the service is branded.

Uber, Airbnb, and other examples

Airbnb Aggregator Business Model

Marketplace and Aggregator business models differ.

Aggregators Vs Market Place

Amazon and Fiverr link merchants and customers and take a 10-20% revenue split.

Uber and Airbnb-style aggregator Join these businesses and provide their products.

5. The pay-as-you-go concept of business

This is a consumption-based pricing system. Cloud companies use it.

Example: Amazon Web Service and Google Cloud Platform (GCP) (AWS)

Pay-as-you-go pricing in AWS

AWS, an Amazon subsidiary, offers over 200 pay-as-you-go cloud services.

“In short, the more you use the more you pay”

Types of Pay-as-you-plan

When it's difficult to divide clients into pricing levels, pay-as-you is employed.

6. The business model known as fee-for-service (FFS)

FFS charges fixed and variable fees for each successful payment.

For instance, PayU, Paypal, and Stripe

Stripe charges 2.9% + 30 per payment.

Fee-for-service (FFS) business model

These firms offer a payment gateway to take consumer payments and deposit them to a business account.

Fintech business model

7. EdTech business strategy

In edtech, you generate money by selling material or teaching as a service.

Most popular revenue model in EdTech

edtech business models

Freemium When course content is free but certification isn't, e.g. Coursera

FREE TRIAL SkillShare offers free trials followed by monthly or annual subscriptions.

Self-serving marketplace approach where you pick what to learn.

Ad-revenue model The company makes money by showing adverts to its huge user base.

Lock-in business strategy

Lock in prevents customers from switching to a competitor's brand or offering.

It uses switching costs or effort to transmit (soft lock-in), improved brand experience, or incentives.

Apple, SAP, and other examples

Graphic Credit: Business Model toolbox

Apple offers an iPhone and then locks you in with extra hardware (Watch, Airpod) and platform services (Apple Store, Apple Music, cloud, etc.).

9. Business Model for API Licensing

APIs let third-party apps communicate with your service.

How do APIs work?

Uber and Airbnb use Google Maps APIs for app navigation.

Examples are Google Map APIs (Map), Sendgrid (Email), and Twilio (SMS).

Types of APIs business model

Business models for APIs

  1. Free: The simplest API-driven business model that enables unrestricted API access for app developers. Google Translate and Facebook are two examples.

  2. Developer Pays: Under this arrangement, service providers such as AWS, Twilio, Github, Stripe, and others must be paid by application developers.

  3. The developer receives payment: These are the compensated content producers or developers who distribute the APIs utilizing their work. For example, Amazon affiliate programs

10. Open-source enterprise

Open-source software can be inspected, modified, and improved by anybody.

For instance, use Firefox, Java, or Android.

Product with Open source business model

Google paid Mozilla $435,702 million to be their primary search engine in 2018.

Open-source software profits in six ways.

  1. Paid assistance The Project Manager can charge for customization because he is quite knowledgeable about the codebase.

  2. A full database solution is available as a Software as a Service (MongoDB Atlas), but there is a fee for the monitoring tool.

  3. Open-core design R studio is a better GUI substitute for open-source applications.

  4. sponsors of GitHub Sponsorships benefit the developers in full.

  5. demands for paid features Earn Money By Developing Open Source Add-Ons for Current Products

Open-source business model

11. The business model for data

If the software or algorithm collects client data to improve or monetize the system.

Open AI GPT3 gets smarter with use.

Graphic Credit: Business Model toolbox

Foursquare allows users to exchange check-in locations.

Later, they compiled large datasets to enable retailers like Starbucks launch new outlets.

12. Business Model Using Blockchain

Blockchain is a distributed ledger technology that allows firms to deploy smart contracts without a central authority.

Examples include Alchemy, Solana, and Ethereum.

blockchain business model

Business models using blockchain

  1. Economy of tokens or utility When a business uses a token business model, it issues some kind of token as one of the ways to compensate token holders or miners. For instance, Solana and Ethereum

  2. Bitcoin Cash P2P Business Model Peer-to-peer (P2P) blockchain technology permits direct communication between end users. as in IPFS

  3. Enterprise Blockchain as a Service (Baas) BaaS focuses on offering ecosystem services similar to those offered by Amazon (AWS) and Microsoft (Azure) in the web 3 sector. Example: Ethereum Blockchain as a Service with Bitcoin (EBaaS).

  4. Blockchain-Based Aggregators With AWS for blockchain, you can use that service by making an API call to your preferred blockchain. As an illustration, Alchemy offers nodes for many blockchains.

13. The free-enterprise model

In the freeterprise business model, free professional accounts are led into the funnel by the free product and later become B2B/enterprise accounts.

For instance, Slack and Zoom

Freeterprise business model

Freeterprise companies flourish through collaboration.

Loom wants you to join your workspace for an enterprise account.

Start with a free professional account to build an enterprise.

14. Business plan for razor blades

It's employed in hardware where one piece is sold at a loss and profits are made through refills or add-ons.

Gillet razor & blades, coffee machine & beans, HP printer & cartridge, etc.

Razor blade/Bait and hook business model

Sony sells the Playstation console at a loss but makes up for it by selling games and charging for online services.

Advantages of the Razor-Razorblade Method

  1. lowers the risk a customer will try a product. enables buyers to test the goods and services without having to pay a high initial investment.

  2. The product's ongoing revenue stream has the potential to generate sales that much outweigh the original investments.

Razor blade business model

15. The business model of direct-to-consumer (D2C)

In D2C, the company sells directly to the end consumer through its website using a third-party logistic partner.

Examples include GymShark and Kylie Cosmetics.

Direct-to-consumer business Model

D2C brands can only expand via websites, marketplaces (Amazon, eBay), etc.

Traditional Retailer vs D2C business model

D2C benefits

  • Lower reliance on middlemen = greater profitability

  • You now have access to more precise demographic and geographic customer data.

  • Additional space for product testing

  • Increased customisation throughout your entire product line-Inventory Less

16. Business model: White Label vs. Private Label

Private label/White label products are made by a contract or third-party manufacturer.

Most amazon electronics are made in china and white-labeled.

Amazon supplements and electronics.

White-label business model

Contract manufacturers handle everything after brands select product quantities on design labels.

17. The franchise model

The franchisee uses the franchisor's trademark, branding, and business strategy (company).

For instance, KFC, Domino's, etc.

Master Franchise business model

Subway, Domino, Burger King, etc. use this business strategy.

Opening your restaurant vs Frenchies

Many people pick a franchise because opening a restaurant is risky.

18. Ad-based business model

Social media and search engine giants exploit search and interest data to deliver adverts.

Google, Meta, TikTok, and Snapchat are some examples.

Ad-based business model

Users don't pay for the service or product given, e.g. Google users don't pay for searches.

In exchange, they collected data and hyper-personalized adverts to maximize revenue.

19. Business plan for octopuses

Each business unit functions separately but is connected to the main body.

Instance: Oyo

OYO’s Octopus business model

OYO is Asia's Airbnb, operating hotels, co-working, co-living, and vacation houses.

20, Transactional business model, number

Sales to customers produce revenue.

E-commerce sites and online purchases employ SSL.

Goli is an ex-GymShark.

Transactional business model

21. The peer-to-peer (P2P) business model

In P2P, two people buy and sell goods and services without a third party or platform.

Consider OLX.

OLX Business Model

22. P2P lending as a manner of operation

In P2P lending, one private individual (P2P Lender) lends/invests or borrows money from another (P2P Borrower).

Instance: Kabbage

P2P Lending as a business model

Social lending lets people lend and borrow money directly from each other without an intermediary financial institution.

23. A business model for brokers

Brokerages charge a commission or fee for their services.

Examples include eBay, Coinbase, and Robinhood.

Brokerage business model

Brokerage businesses are common in Real estate, finance, and online and operate on this model.

Types of brokerage business model
  1. Buy/sell similar models Examples include financial brokers, insurance brokers, and others who match purchase and sell transactions and charge a commission.

  2. These brokers charge an advertiser a fee based on the date, place, size, or type of an advertisement. This is known as the classified-advertiser model. For instance, Craiglist

24. Drop shipping as an industry

Dropshipping allows stores to sell things without holding physical inventories.

Drop shipping Business model

When a customer orders, use a third-party supplier and logistic partners.

Retailer product portfolio and customer experience Fulfiller The consumer places the order.

Dropshipping advantages

  • Less money is needed (Low overhead-No Inventory or warehousing)

  • Simple to start (costs under $100)

  • flexible work environment

  • New product testing is simpler

25. Business Model for Space as a Service

It's centered on a shared economy that lets millennials live or work in communal areas without ownership or lease.

Consider WeWork and Airbnb.

WeWork business model

WeWork helps businesses with real estate, legal compliance, maintenance, and repair.

Space as a Service Business Model

26. The business model for third-party logistics (3PL)

In 3PL, a business outsources product delivery, warehousing, and fulfillment to an external logistics company.

Examples include Ship Bob, Amazon Fulfillment, and more.

Third-Party Logistics (3PL)

3PL partners warehouse, fulfill, and return inbound and outbound items for a charge.

Inbound logistics involves bringing products from suppliers to your warehouse.

Outbound logistics refers to a company's production line, warehouse, and customer.

Inbound and outbound in 3PL

27. The last-mile delivery paradigm as a commercial strategy

Last-mile delivery is the collection of supply chain actions that reach the end client.

Examples include Rappi, Gojek, and Postmates.

gojek business model

Last-mile is tied to on-demand and has a nighttime peak.

28. The use of affiliate marketing

Affiliate marketing involves promoting other companies' products and charging commissions.

Examples include Hubspot, Amazon, and Skillshare.

Affiliate business model

Your favorite youtube channel probably uses these short amazon links to get 5% of sales.

affiliate link from a youtube video.

Affiliate marketing's benefits

  • In exchange for a success fee or commission, it enables numerous independent marketers to promote on its behalf.

  • Ensure system transparency by giving the influencers a specific tracking link and an online dashboard to view their profits.

  • Learn about the newest bargains and have access to promotional materials.

29. The business model for virtual goods

This is an in-app purchase for an intangible product.

Examples include PubG, Roblox, Candy Crush, etc.

virtual goods business model

Consumables are like gaming cash that runs out. Non-consumable products provide a permanent advantage without repeated purchases.

30. Business Models for Cloud Kitchens

Ghost, Dark, Black Box, etc.

Delivery-only restaurant.

These restaurants don't provide dine-in, only delivery.

For instance, NextBite and Faasos

Cloud kitchen business model

31. Crowdsourcing as a Business Model

Crowdsourcing = Using the crowd as a platform's source.

In crowdsourcing, you get support from people around the world without hiring them.

Crowdsourcing Business model

Crowdsourcing sites

  1. Open-Source Software gives access to the software's source code so that developers can edit or enhance it. Examples include Firefox browsers and Linux operating systems.

  2. Crowdfunding The oculus headgear would be an example of crowdfunding in essence, with no expectations.

ANTHONY P.

ANTHONY P.

2 years ago

Startups are difficult. Streamlining the procedure for creating the following unicorn.

New ventures are exciting. It's fun to imagine yourself rich, successful, and famous (if that's your thing). How you'll help others and make your family proud. This excitement can pull you forward for years, even when you intuitively realize that the path you're on may not lead to your desired success.

Know when to change course. Switching course can mean pivoting or changing direction.

In this not-so-short blog, I'll describe the journey of building your dream. And how the journey might look when you think you're building your dream, but fall short of that vision. Both can feel similar in the beginning, but there are subtle differences.

Let’s dive in.

How an exciting journey to a dead end looks and feels.

You want to help many people. You're business-minded, creative, and ambitious. You jump into entrepreneurship. You're excited, free, and in control.

I'll use tech as an example because that's what I know best, but this applies to any entrepreneurial endeavor.

So you start learning the basics of your field, say coding/software development. You read books, take courses, and may even join a bootcamp. You start practicing, and the journey begins. Once you reach a certain level of skill (which can take months, usually 12-24), you gain the confidence to speak with others in the field and find common ground. You might attract a co-founder this way with time. You and this person embark on a journey (Tip: the idea you start with is rarely the idea you end with).

Amateur mistake #1: You spend months building a product before speaking to customers.

Building something pulls you forward blindly. You make mistakes, avoid customers, and build with your co-founder or small team in the dark for months, usually 6-12 months.

You're excited when the product launches. We'll be billionaires! The market won't believe it. This excites you and the team. Launch.

….

Nothing happens.

Some people may sign up out of pity, only to never use the product or service again.

You and the team are confused, discouraged and in denial. They don't get what we've built yet. We need to market it better, we need to talk to more investors, someone will understand our vision.

This is a hopeless path, and your denial could last another 6 months. If you're lucky, while talking to consumers and investors (which you should have done from the start), someone who has been there before would pity you and give you an idea to pivot into that can create income.

Suppose you get this idea and pivot your business. Again, you've just pivoted into something limited by what you've already built. It may be a revenue-generating idea, but it's rarely new. Now you're playing catch-up, doing something others are doing but you can do better. (Tip #2: Don't be late.) Your chances of winning are slim, and you'll likely never catch up.

You're finally seeing revenue and feel successful. You can compete, but if you're not a first mover, you won't earn enough over time. You'll get by or work harder than ever to earn what a skilled trade could provide. You didn't go into business to stress out and make $100,000 or $200,000 a year. When you can make the same amount by becoming a great software developer, electrician, etc.

You become stuck. Either your firm continues this way for years until you realize there isn't enough growth to recruit a strong team and remove yourself from day-to-day operations due to competition. Or a catastrophic economic event forces you to admit that what you were building wasn't new and unique and wouldn't get you where you wanted to be.

This realization could take 6-10 years. No kidding.

The good news is, you’ve learned a lot along the way and this information can be used towards your next venture (if you have the energy).

Key Lesson: Don’t build something if you aren’t one of the first in the space building it just for the sake of building something.

-

Let's discuss what it's like to build something that can make your dream come true.

Case 2: Building something the market loves is difficult but rewarding.

It starts with a problem that hasn't been adequately solved for a long time but is now solvable due to technology. Or a new problem due to a change in how things are done.

Let's examine each example.

Example #1: Mass communication. The problem is now solvable due to some technological breakthrough.

Twitter — One of the first web 2 companies that became successful with the rise of smart mobile computing.

People can share their real-time activities via mobile device with friends, family, and strangers. Web 2 and smartphones made it easy and fun.

Example #2: A new problem has emerged due to some change in the way things are conducted.

Zoom- A web-conferencing company that reached massive success due to the movement towards “work from home”, remote/hybrid work forces.

Online web conferencing allows for face-to-face communication.

-

These two examples show how to build a unicorn-type company. It's a mix of solving the right problem at the right time, either through a technological breakthrough that opens up new opportunities or by fundamentally changing how people do things.

Let's find these opportunities.

Start by examining problems, such as how the world has changed and how we can help it adapt. It can also be both. Start team brainstorming. Research technologies, current world-trends, use common sense, and make a list. Then, choose the top 3 that you're most excited about and seem most workable based on your skillsets, values, and passion.

Once you have this list, create the simplest MVP you can and test it with customers. The prototype can be as simple as a picture or diagram of user flow and end-user value. No coding required. Market-test. Twitter's version 1 was simple. It was a web form that asked, "What are you doing?" Then publish it from your phone. A global status update, wherever you are. Currently, this company has a $50 billion market cap.

Here's their MVP screenshot.

Small things grow. Tiny. Simplify.

Remember Frequency and Value when brainstorming. Your product is high frequency (Twitter, Instagram, Snapchat, TikTok) or high value (Airbnb for renting travel accommodations), or both (Gmail).

Once you've identified product ideas that meet the above criteria, they're simple, have a high frequency of use, or provide deep value. You then bring it to market in the simplest, most cost-effective way. You can sell a half-working prototype with imagination and sales skills. You need just enough of a prototype to convey your vision to a user or customer.

With this, you can approach real people. This will do one of three things: give you a green light to continue on your vision as is, show you that there is no opportunity and people won't use it, or point you in a direction that is a blend of what you've come up with and what the customer / user really wants, and you update the prototype and go back to the maze. Repeat until you have enough yeses and conviction to build an MVP.

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Mark Shpuntov

Mark Shpuntov

3 years ago

How to Produce a Month's Worth of Content for Social Media in a Day

New social media producers' biggest error

Photo by Libby Penner on Unsplash

The Treadmill of Social Media Content

New creators focus on the wrong platforms.

They post to Instagram, Twitter, TikTok, etc.

They create daily material, but it's never enough for social media algorithms.

Creators recognize they're on a content creation treadmill.

They have to keep publishing content daily just to stay on the algorithm’s good side and avoid losing the audience they’ve built on the platform.

This is exhausting and unsustainable, causing creator burnout.

They focus on short-lived platforms, which is an issue.

Comparing low- and high-return social media platforms

Social media networks are great for reaching new audiences.

Their algorithm is meant to viralize material.

Social media can use you for their aims if you're not careful.

To master social media, focus on the right platforms.

To do this, we must differentiate low-ROI and high-ROI platforms:

Low ROI platforms are ones where content has a short lifespan. High ROI platforms are ones where content has a longer lifespan.

A tweet may be shown for 12 days. If you write an article or blog post, it could get visitors for 23 years.

ROI is drastically different.

New creators have limited time and high learning curves.

Nothing is possible.

First create content for high-return platforms.

ROI for social media platforms

Here are high-return platforms:

  1. Your Blog - A single blog article can rank and attract a ton of targeted traffic for a very long time thanks to the power of SEO.

  2. YouTube - YouTube has a reputation for showing search results or sidebar recommendations for videos uploaded 23 years ago. A superb video you make may receive views for a number of years.

  3. Medium - A platform dedicated to excellent writing is called Medium. When you write an article about a subject that never goes out of style, you're building a digital asset that can drive visitors indefinitely.

These high ROI platforms let you generate content once and get visitors for years.

This contrasts with low ROI platforms:

  1. Twitter

  2. Instagram

  3. TikTok

  4. LinkedIn

  5. Facebook

The posts you publish on these networks have a 23-day lifetime. Instagram Reels and TikToks are exceptions since viral content can last months.

If you want to make content creation sustainable and enjoyable, you must focus the majority of your efforts on creating high ROI content first. You can then use the magic of repurposing content to publish content to the lower ROI platforms to increase your reach and exposure.

How To Use Your Content Again

So, you’ve decided to focus on the high ROI platforms.

Great!

You've published an article or a YouTube video.

You worked hard on it.

Now you have fresh stuff.

What now?

If you are not repurposing each piece of content for multiple platforms, you are throwing away your time and efforts.

You've created fantastic material, so why not distribute it across platforms?

Repurposing Content Step-by-Step

For me, it's writing a blog article, but you might start with a video or podcast.

The premise is the same regardless of the medium.

Start by creating content for a high ROI platform (YouTube, Blog Post, Medium). Then, repurpose, edit, and repost it to the lower ROI platforms.

Here's how to repurpose pillar material for other platforms:

  1. Post the article on your blog.

  2. Put your piece on Medium (use the canonical link to point to your blog as the source for SEO)

  3. Create a video and upload it to YouTube using the talking points from the article.

  4. Rewrite the piece a little, then post it to LinkedIn.

  5. Change the article's format to a Thread and share it on Twitter.

  6. Find a few quick quotes throughout the article, then use them in tweets or Instagram quote posts.

  7. Create a carousel for Instagram and LinkedIn using screenshots from the Twitter Thread.

  8. Go through your film and select a few valuable 30-second segments. Share them on LinkedIn, Facebook, Twitter, TikTok, YouTube Shorts, and Instagram Reels.

  9. Your video's audio can be taken out and uploaded as a podcast episode.

If you (or your team) achieve all this, you'll have 20-30 pieces of social media content.

If you're just starting, I wouldn't advocate doing all of this at once.

Instead, focus on a few platforms with this method.

You can outsource this as your company expands. (If you'd want to learn more about content repurposing, contact me.)

You may focus on relevant work while someone else grows your social media on autopilot.

You develop high-ROI pillar content, and it's automatically chopped up and posted on social media.

This lets you use social media algorithms without getting sucked in.

Thanks for reading!

Vitalik

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

Trevor Stark

Trevor Stark

2 years ago

Peter Thiels's Multi-Billion Dollar Net Worth's Unknown Philosopher

Peter Thiel studied philosophy as an undergraduate.

Peter Thiel and Elon Musk, Co-Founders of PayPal

Peter Thiel has $7.36 billion.

Peter is a world-ranked chess player, has a legal degree, and has written profitable novels.

In 1999, he co-founded PayPal with Max Levchin, which merged with X.com.

Peter Thiel made $55 million after selling the company to eBay for $1.5 billion in 2002.

You may be wondering…

How did Peter turn $55 million into his now multi-billion dollar net worth?

One amazing investment?

Facebook.

Thiel was Facebook's first external investor. He bought 10% of the company for $500,000 in 2004.

This investment returned 159% annually, 200x in 8 years.

By 2012, Thiel sold almost all his Facebook shares, becoming a billionaire.

What was the investment thesis of Peter?

This investment appeared ridiculous. Facebook was an innovative startup.

Thiel's $500,000 contribution transformed Facebook.

Screenshot of Facebook in 2004 (Source)

Harvard students have access to Facebook's 8 features and 1 photo per profile.

How did Peter determine that this would be a wise investment, then?

Facebook is a mimetic desire machine.

Social media's popularity is odd. Why peek at strangers' images on a computer?

Peter Thiel studied under French thinker Rene Girard at Stanford.

Mimetic Desire explains social media's success.

Mimetic Desire is the idea that humans desire things simply because other people do.

If nobody wanted it, would you?

Would you desire a family, a luxury car, or expensive clothes if no one else did? Girard says no.

People we admire affect our aspirations because we're social animals. Every person has a role model.

Our nonreligious culture implies role models are increasingly other humans, not God.

The idea explains why social media influencers are so powerful.

Why would Andrew Tate or Kim Kardashian matter if people weren't mimetic?

Humanity is fundamentally motivated by social comparison.

Facebook takes advantage of this need for social comparison, and puts it on a global scale.

It aggregates photographs and updates from millions of individuals.

Facebook mobile allows 24/7 social comparison.

Thiel studied mimetic desire with Girard and realized Facebook exploits the urge for social comparison to gain money.

Social media is more significant and influential than ever, despite Facebook's decline.

Thiel and Girard show that applied philosophy (particularly in business) can be immensely profitable.