An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3
(Edited)
Hackernoon
4 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
4 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Elnaz Sarraf
3 years ago
Why Bitcoin's Crash Could Be Good for Investors
The crypto market crashed in June 2022. Bitcoin and other cryptocurrencies hit their lowest prices in over a year, causing market panic. Some believe this crash will benefit future investors.
Before I discuss how this crash might help investors, let's examine why it happened. Inflation in the U.S. reached a 30-year high in 2022 after Russia invaded Ukraine. In response, the U.S. Federal Reserve raised interest rates by 0.5%, the most in almost 20 years. This hurts cryptocurrencies like Bitcoin. Higher interest rates make people less likely to invest in volatile assets like crypto, so many investors sold quickly.
The crypto market collapsed. Bitcoin, Ethereum, and Binance dropped 40%. Other cryptos crashed so hard they were delisted from almost every exchange. Bitcoin peaked in April 2022 at $41,000, but after the May interest rate hike, it crashed to $28,000. Bitcoin investors were worried. Even in bad times, this crash is unprecedented.
Bitcoin wasn't "doomed." Before the crash, LUNA was one of the top 5 cryptos by market cap. LUNA was trading around $80 at the start of May 2022, but after the rate hike?
Less than 1 cent. LUNA lost 99.99% of its value in days and was removed from every crypto exchange. Bitcoin's "crash" isn't as devastating when compared to LUNA.
Many people said Bitcoin is "due" for a LUNA-like crash and that the only reason it hasn't crashed is because it's bigger. Still false. If so, Bitcoin should be worth zero by now. We didn't. Instead, Bitcoin reached 28,000, then 29k, 30k, and 31k before falling to 18k. That's not the world's greatest recovery, but it shows Bitcoin's safety.
Bitcoin isn't falling constantly. It fell because of the initial shock of interest rates, but not further. Now, Bitcoin's value is more likely to rise than fall. Bitcoin's low price also attracts investors. They know what prices Bitcoin can reach with enough hype, and they want to capitalize on low prices before it's too late.
Bitcoin's crash was bad, but in a way it wasn't. To understand, consider 2021. In March 2021, Bitcoin surpassed $60k for the first time. Elon Musk's announcement in May that he would no longer support Bitcoin caused a massive crash in the crypto market. In May 2017, Bitcoin's price hit $29,000. Elon Musk's statement isn't worth more than the Fed raising rates. Many expected this big announcement to kill Bitcoin.
Not so. Bitcoin crashed from $58k to $31k in 2021. Bitcoin fell from $41k to $28k in 2022. This crash is smaller. Bitcoin's price held up despite tensions and stress, proving investors still believe in it. What happened after the initial crash in the past?
Bitcoin fell until mid-July. This is also something we’re not seeing today. After a week, Bitcoin began to improve daily. Bitcoin's price rose after mid-July. Bitcoin's price fluctuated throughout the rest of 2021, but it topped $67k in November. Despite no major changes, the peak occurred after the crash. Elon Musk seemed uninterested in crypto and wasn't likely to change his mind soon. What triggered this peak? Nothing, really. What really happened is that people got over the initial statement. They forgot.
Internet users have goldfish-like attention spans. People quickly forgot the crash's cause and were back investing in crypto months later. Despite the market's setbacks, more crypto investors emerged by the end of 2017. Who gained from these peaks? Bitcoin investors who bought low. Bitcoin not only recovered but also doubled its ROI. It was like a movie, and it shows us what to expect from Bitcoin in the coming months.
The current Bitcoin crash isn't as bad as the last one. LUNA is causing market panic. LUNA and Bitcoin are different cryptocurrencies. LUNA crashed because Terra wasn’t able to keep its peg with the USD. Bitcoin is unanchored. It's one of the most decentralized investments available. LUNA's distrust affected crypto prices, including Bitcoin, but it won't last forever.
This is why Bitcoin will likely rebound in the coming months. In 2022, people will get over the rise in interest rates and the crash of LUNA, just as they did with Elon Musk's crypto stance in 2021. When the world moves on to the next big controversy, Bitcoin's price will soar.
Bitcoin may recover for another reason. Like controversy, interest rates fluctuate. The Russian invasion caused this inflation. World markets will stabilize, prices will fall, and interest rates will drop.
Next, lower interest rates could boost Bitcoin's price. Eventually, it will happen. The U.S. economy can't sustain such high interest rates. Investors will put every last dollar into Bitcoin if interest rates fall again.
Bitcoin has proven to be a stable investment. This boosts its investment reputation. Even if Ethereum dethrones Bitcoin as crypto king one day (or any other crypto, for that matter). Bitcoin may stay on top of the crypto ladder for a while. We'll have to wait a few months to see if any of this is true.
This post is a summary. Read the full article here.
Coinbase
4 years ago
10 Predictions for Web3 and the Cryptoeconomy for 2022
By Surojit Chatterjee, Chief Product Officer
2021 proved to be a breakout year for crypto with BTC price gaining almost 70% yoy, Defi hitting $150B in value locked, and NFTs emerging as a new category. Here’s my view through the crystal ball into 2022 and what it holds for our industry:
1. Eth scalability will improve, but newer L1 chains will see substantial growth — As we welcome the next hundred million users to crypto and Web3, scalability challenges for Eth are likely to grow. I am optimistic about improvements in Eth scalability with the emergence of Eth2 and many L2 rollups. Traction of Solana, Avalanche and other L1 chains shows that we’ll live in a multi-chain world in the future. We’re also going to see newer L1 chains emerge that focus on specific use cases such as gaming or social media.
2. There will be significant usability improvements in L1-L2 bridges — As more L1 networks gain traction and L2s become bigger, our industry will desperately seek improvements in speed and usability of cross-L1 and L1-L2 bridges. We’re likely to see interesting developments in usability of bridges in the coming year.
3. Zero knowledge proof technology will get increased traction — 2021 saw protocols like ZkSync and Starknet beginning to get traction. As L1 chains get clogged with increased usage, ZK-rollup technology will attract both investor and user attention. We’ll see new privacy-centric use cases emerge, including privacy-safe applications, and gaming models that have privacy built into the core. This may also bring in more regulator attention to crypto as KYC/AML could be a real challenge in privacy centric networks.
4. Regulated Defi and emergence of on-chain KYC attestation — Many Defi protocols will embrace regulation and will create separate KYC user pools. Decentralized identity and on-chain KYC attestation services will play key roles in connecting users’ real identity with Defi wallet endpoints. We’ll see more acceptance of ENS type addresses, and new systems from cross chain name resolution will emerge.
5. Institutions will play a much bigger role in Defi participation — Institutions are increasingly interested in participating in Defi. For starters, institutions are attracted to higher than average interest-based returns compared to traditional financial products. Also, cost reduction in providing financial services using Defi opens up interesting opportunities for institutions. However, they are still hesitant to participate in Defi. Institutions want to confirm that they are only transacting with known counterparties that have completed a KYC process. Growth of regulated Defi and on-chain KYC attestation will help institutions gain confidence in Defi.
6. Defi insurance will emerge — As Defi proliferates, it also becomes the target of security hacks. According to London-based firm Elliptic, total value lost by Defi exploits in 2021 totaled over $10B. To protect users from hacks, viable insurance protocols guaranteeing users’ funds against security breaches will emerge in 2022.
7. NFT Based Communities will give material competition to Web 2.0 social networks — NFTs will continue to expand in how they are perceived. We’ll see creator tokens or fan tokens take more of a first class seat. NFTs will become the next evolution of users’ digital identity and passport to the metaverse. Users will come together in small and diverse communities based on types of NFTs they own. User created metaverses will be the future of social networks and will start threatening the advertising driven centralized versions of social networks of today.
8. Brands will start actively participating in the metaverse and NFTs — Many brands are realizing that NFTs are great vehicles for brand marketing and establishing brand loyalty. Coca-Cola, Campbell’s, Dolce & Gabbana and Charmin released NFT collectibles in 2021. Adidas recently launched a new metaverse project with Bored Ape Yacht Club. We’re likely to see more interesting brand marketing initiatives using NFTs. NFTs and the metaverse will become the new Instagram for brands. And just like on Instagram, many brands may start as NFT native. We’ll also see many more celebrities jumping in the bandwagon and using NFTs to enhance their personal brand.
9. Web2 companies will wake up and will try to get into Web3 — We’re already seeing this with Facebook trying to recast itself as a Web3 company. We’re likely to see other big Web2 companies dipping their toes into Web3 and metaverse in 2022. However, many of them are likely to create centralized and closed network versions of the metaverse.
10. Time for DAO 2.0 — We’ll see DAOs become more mature and mainstream. More people will join DAOs, prompting a change in definition of employment — never receiving a formal offer letter, accepting tokens instead of or along with fixed salaries, and working in multiple DAO projects at the same time. DAOs will also confront new challenges in terms of figuring out how to do M&A, run payroll and benefits, and coordinate activities in larger and larger organizations. We’ll see a plethora of tools emerge to help DAOs execute with efficiency. Many DAOs will also figure out how to interact with traditional Web2 companies. We’re likely to see regulators taking more interest in DAOs and make an attempt to educate themselves on how DAOs work.
Thanks to our customers and the ecosystem for an incredible 2021. Looking forward to another year of building the foundations for Web3. Wagmi.
Chris
2 years ago
What the World's Most Intelligent Investor Recently Said About Crypto
Cryptoshit. This thing is crazy to buy.
Charlie Munger is revered and powerful in finance.
Munger, vice chairman of Berkshire Hathaway, is noted for his wit, no-nonsense attitude to investment, and ability to spot promising firms and markets.
Munger's crypto views have upset some despite his reputation as a straight shooter.
“There’s only one correct answer for intelligent people, just totally avoid all the people that are promoting it.” — Charlie Munger
The Munger Interview on CNBC (4:48 secs)
This Monday, CNBC co-anchor Rebecca Quick interviewed Munger and brought up his 2007 statement, "I'm not allowed to have an opinion on this subject until I can present the arguments against my viewpoint better than the folks who are supporting it."
Great investing and life advice!
If you can't explain the opposing reasons, you're not informed enough to have an opinion.
In today's world, it's important to grasp both sides of a debate before supporting one.
Rebecca inquired:
Does your Wall Street Journal article on banning cryptocurrency apply? If so, would you like to present the counterarguments?
Mungers reply:
I don't see any viable counterarguments. I think my opponents are idiots, hence there is no sensible argument against my position.
Consider his words.
Do you believe Munger has studied both sides?
He said, "I assume my opponents are idiots, thus there is no sensible argument against my position."
This is worrisome, especially from a guy who once encouraged studying both sides before forming an opinion.
Munger said:
National currencies have benefitted humanity more than almost anything else.
Hang on, I think we located the perpetrator.
Munger thinks crypto will replace currencies.
False.
I doubt he studied cryptocurrencies because the name is deceptive.
He misread a headline as a Dollar destroyer.
Cryptocurrencies are speculations.
Like Tesla, Amazon, Apple, Google, Microsoft, etc.
Crypto won't replace dollars.
In the interview with CNBC, Munger continued:
“I’m not proud of my country for allowing this crap, what I call the cryptoshit. It’s worthless, it’s no good, it’s crazy, it’ll do nothing but harm, it’s anti-social to allow it.” — Charlie Munger
Not entirely inaccurate.
Daily cryptos are established solely to pump and dump regular investors.
Let's get into Munger's crypto aversion.
Rat poison is bitcoin.
Munger famously dubbed Bitcoin rat poison and a speculative bubble that would implode.
Partially.
But the bubble broke. Since 2021, the market has fallen.
Scam currencies and NFTs are being eliminated, which I like.
Whoa.
Why does Munger doubt crypto?
Mungers thinks cryptocurrencies has no intrinsic value.
He worries about crypto fraud and money laundering.
Both are valid issues.
Yet grouping crypto is intellectually dishonest.
Ethereum, Bitcoin, Solana, Chainlink, Flow, and Dogecoin have different purposes and values (not saying they’re all good investments).
Fraudsters who hurt innocents will be punished.
Therefore, complaining is useless.
Why not stop it? Repair rather than complain.
Regrettably, individuals today don't offer solutions.
Blind Areas for Mungers
As with everyone, Mungers' bitcoin views may be impacted by his biases and experiences.
OK.
But Munger has always advocated classic value investing and may be wary of investing in an asset outside his expertise.
Mungers' banking and insurance investments may influence his bitcoin views.
Could a coworker or acquaintance have told him crypto is bad and goes against traditional finance?
Right?
Takeaways
Do you respect Charlie Mungers?
Yes and no, like any investor or individual.
To understand Mungers' bitcoin beliefs, you must be critical.
Mungers is a successful investor, but his views about bitcoin should be considered alongside other viewpoints.
Munger’s success as an investor has made him an influencer in the space.
Influence gives power.
He controls people's thoughts.
Munger's ok. He will always be heard.
I'll do so cautiously.
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Jano le Roux
3 years ago
My Top 11 Tools For Building A Modern Startup, With A Free Plan
The best free tools are probably unknown to you.
Modern startups are easy to build.
Start with free tools.
Let’s go.
Web development — Webflow
Code-free HTML, CSS, and JS.
Webflow isn't like Squarespace, Wix, or Shopify.
It's a super-fast no-code tool for professionals to construct complex, highly-responsive websites and landing pages.
Webflow can help you add animations like those on Apple's website to your own site.
I made the jump from WordPress a few years ago and it changed my life.
No damn plugins. No damn errors. No damn updates.
The best, you can get started on Webflow for free.
Data tracking — Airtable
Spreadsheet wings.
Airtable combines spreadsheet flexibility with database power without code.
Airtable is modern.
Airtable has modularity.
Scaling Airtable is simple.
Airtable, one of the most adaptable solutions on this list, is perfect for client data management.
Clients choose customized service packages. Airtable consolidates data so you can automate procedures like invoice management and focus on your strengths.
Airtable connects with so many tools that rarely creates headaches. Airtable scales when you do.
Airtable's flexibility makes it a potential backend database.
Design — Figma
Better, faster, easier user interface design.
Figma rocks!
It’s fast.
It's free.
It's adaptable
First, design in Figma.
Iterate.
Export development assets.
Figma lets you add more team members as your company grows to work on each iteration simultaneously.
Figma is web-based, so you don't need a powerful PC or Mac to start.
Task management — Trello
Unclock jobs.
Tacky and terrifying task management products abound. Trello isn’t.
Those that follow Marie Kondo will appreciate Trello.
Everything is clean.
Nothing is complicated.
Everything has a place.
Compared to other task management solutions, Trello is limited. And that’s good. Too many buttons lead to too many decisions lead to too many hours wasted.
Trello is a must for teamwork.
Domain email — Zoho
Free domain email hosting.
Professional email is essential for startups. People relied on monthly payments for too long. Nope.
Zoho offers 5 free professional emails.
It doesn't have Google's UI, but it works.
VPN — Proton VPN
Fast Swiss VPN protects your data and privacy.
Proton VPN is secure.
Proton doesn't record any data.
Proton is based in Switzerland.
Swiss privacy regulation is among the most strict in the world, therefore user data are protected. Switzerland isn't a 14 eye country.
Journalists and activists trust Proton to secure their identities while accessing and sharing information authoritarian governments don't want them to access.
Web host — Netlify
Free fast web hosting.
Netlify is a scalable platform that combines your favorite tools and APIs to develop high-performance sites, stores, and apps through GitHub.
Serverless functions and environment variables preserve API keys.
Netlify's free tier is unmissable.
100GB of free monthly bandwidth.
Free 125k serverless operations per website each month.
Database — MongoDB
Create a fast, scalable database.
MongoDB is for small and large databases. It's a fast and inexpensive database.
Free for the first million reads.
Then, for each million reads, you must pay $0.10.
MongoDB's free plan has:
Encryption from end to end
Continual authentication
field-level client-side encryption
If you have a large database, you can easily connect MongoDB to Webflow to bypass CMS limits.
Automation — Zapier
Time-saving tip: automate repetitive chores.
Zapier simplifies life.
Zapier syncs and connects your favorite apps to do impossibly awesome things.
If your online store is connected to Zapier, a customer's purchase can trigger a number of automated actions, such as:
The customer is being added to an email chain.
Put the information in your Airtable.
Send a pre-programmed postcard to the customer.
Alexa, set the color of your smart lights to purple.
Zapier scales when you do.
Email & SMS marketing — Omnisend
Email and SMS marketing campaigns.
This is an excellent Mailchimp option for magical emails. Omnisend's processes simplify email automation.
I love the interface's cleanliness.
Omnisend's free tier includes web push notifications.
Send up to:
500 emails per month
60 maximum SMSs
500 Web Push Maximum
Forms and surveys — Tally
Create flexible forms that people enjoy.
Typeform is clean but restricting. Sometimes you need to add many questions. Tally's needed sometimes.
Tally is flexible and cheaper than Typeform.
99% of Tally's features are free and unrestricted, including:
Unlimited forms
Countless submissions
Collect payments
File upload
Tally lets you examine what individuals contributed to forms before submitting them to see where they get stuck.
Airtable and Zapier connectors automate things further. If you pay, you can apply custom CSS to fit your brand.
See.
Free tools are the greatest.
Let's use them to launch a startup.
Jano le Roux
3 years ago
Here's What I Learned After 30 Days Analyzing Apple's Microcopy
Move people with tiny words.
Apple fanboy here.
Macs are awesome.
Their iPhones rock.
$19 cloths are great.
$999 stands are amazing.
I love Apple's microcopy even more.
It's like the marketing goddess bit into the Apple logo and blessed the world with microcopy.
I took on a 30-day micro-stalking mission.
Every time I caught myself wasting time on YouTube, I had to visit Apple’s website to learn the secrets of the marketing goddess herself.
We've learned. Golden apples are calling.
Cut the friction
Benefit-first, not commitment-first.
Brands lose customers through friction.
Most brands don't think like customers.
Brands want sales.
Brands want newsletter signups.
Here's their microcopy:
“Buy it now.”
“Sign up for our newsletter.”
Both are difficult. They ask for big commitments.
People are simple creatures. Want pleasure without commitment.
Apple nails this.
So, instead of highlighting the commitment, they highlight the benefit of the commitment.
Saving on the latest iPhone sounds easier than buying it. Everyone saves, but not everyone buys.
A subtle change in framing reduces friction.
Apple eliminates customer objections to reduce friction.
Less customer friction means simpler processes.
Apple's copy expertly reassures customers about shipping fees and not being home. Apple assures customers that returning faulty products is easy.
Apple knows that talking to a real person is the best way to reduce friction and improve their copy.
Always rhyme
Learn about fine rhyme.
Poets make things beautiful with rhyme.
Copywriters use rhyme to stand out.
Apple’s copywriters have mastered the art of corporate rhyme.
Two techniques are used.
1. Perfect rhyme
Here, rhymes are identical.
2. Imperfect rhyme
Here, rhyming sounds vary.
Apple prioritizes meaning over rhyme.
Apple never forces rhymes that don't fit.
It fits so well that the copy seems accidental.
Add alliteration
Alliteration always entertains.
Alliteration repeats initial sounds in nearby words.
Apple's copy uses alliteration like no other brand I've seen to create a rhyming effect or make the text more fun to read.
For example, in the sentence "Sam saw seven swans swimming," the initial "s" sound is repeated five times. This creates a pleasing rhythm.
Microcopy overuse is like pouring ketchup on a Michelin-star meal.
Alliteration creates a memorable phrase in copywriting. It's subtler than rhyme, and most people wouldn't notice; it simply resonates.
I love how Apple uses alliteration and contrast between "wonders" and "ease".
Assonance, or repeating vowels, isn't Apple's thing.
You ≠ Hero, Customer = Hero
Your brand shouldn't be the hero.
Because they'll be using your product or service, your customer should be the hero of your copywriting. With your help, they should feel like they can achieve their goals.
I love how Apple emphasizes what you can do with the machine in this microcopy.
It's divine how they position their tools as sidekicks to help below.
This one takes the cake:
Dialogue-style writing
Conversational copy engages.
Excellent copy Like sharing gum with a friend.
This helps build audience trust.
Apple does this by using natural connecting words like "so" and phrases like "But that's not all."
Snowclone-proof
The mother of all microcopy techniques.
A snowclone uses an existing phrase or sentence to create a new one. The new phrase or sentence uses the same structure but different words.
It’s usually a well know saying like:
To be or not to be.
This becomes a formula:
To _ or not to _.
Copywriters fill in the blanks with cause-related words. Example:
To click or not to click.
Apple turns "survival of the fittest" into "arrival of the fittest."
It's unexpected and surprises the reader.
So this was fun.
But my fun has just begun.
Microcopy is 21st-century poetry.
I came as an Apple fanboy.
I leave as an Apple fanatic.
Now I’m off to find an apple tree.
Cause you know how it goes.
(Apples, trees, etc.)
This post is a summary. Original post available here.
Mangu Solutions
3 years ago
Growing a New App to $15K/mo in 6 Months [SaaS Case Study]
Discover How We Used Facebook Ads to Grow a New Mobile App from $0 to $15K MRR in Just 6 Months and Our Strategy to Hit $100K a Month.
Our client introduced a mobile app for Poshmark resellers in December and wanted as many to experience it and subscribe to the monthly plan.
An Error We Committed
We initiated a Facebook ad campaign with a "awareness" goal, not "installs." This sent them to a landing page that linked to the iPhone App Store and Android Play Store. Smart, right?
We got some installs, but we couldn't tell how many came from the ad versus organic/other channels because the objective we chose only reported landing page clicks, not app installs.
We didn't know which interest groups/audiences had the best cost per install (CPI) to optimize and scale our budget.
After spending $700 without adequate data (installs and trials report), we stopped the campaign and worked with our client's app developer to set up app events tracking.
This allowed us to create an installs campaign and track installs, trials, and purchases (in some cases).
Finding a Successful Audience
Once we knew what ad sets brought in what installs at what cost, we began optimizing and testing other interest groups and audiences, growing the profitable low CPI ones and eliminating the high CPI ones.
We did all our audience testing using an ABO campaign (Ad Set Budget Optimization), spending $10 to $30 on each ad set for three days and optimizing afterward. All ad sets under $30 were moved to a CBO campaign (Campaign Budget Optimization).
We let Facebook's AI decide how much to spend on each ad set, usually the one most likely to convert at the lowest cost.
If the CBO campaign maintains a nice CPI, we keep increasing the budget by $50 every few days or duplicating it sometimes in order to double the budget. This is how we've scaled to $400/day profitably.
Finding Successful Creatives
Per campaign, we tested 2-6 images/videos. Same ad copy and CTA. There was no clear winner because some images did better with some interest groups.
The image above with mail packages, for example, got us a cheap CPI of $9.71 from our Goodwill Stores interest group but, a high $48 CPI from our lookalike audience. Once we had statistically significant data, we turned off the high-cost ad.
New marketers who are just discovering A/B testing may assume it's black and white — winner and loser. However, Facebook ads' machine learning and reporting has gotten so sophisticated that it's hard to call a creative a flat-out loser, but rather a 'bad fit' for some audiences, and perfect for others.
You can see how each creative performs across age groups and optimize.
How Many Installs Did It Take Us to Earn $15K Per Month?
Six months after paying $25K, we got 1,940 app installs, 681 free trials, and 522 $30 monthly subscriptions. 522 * $30 gives us $15,660 in monthly recurring revenue (MRR).
Next, what? $100K per month
The conversation above is with the app's owner. We got on a 30-minute call where I shared how I plan to get the app to be making $100K a month like I’ve done for other businesses.
Reverse Engineering $100K
Formula:
For $100K/month, we need 3,334 people to pay $30/month. 522 people pay that. We need 2,812 more paid users.
522 paid users from 1,940 installs is a 27% conversion rate. To hit $100K/month, we need 10,415 more installs. Assuming...
With a $400 daily ad spend, we average 40 installs per day. This means that if everything stays the same, it would take us 260 days (around 9 months) to get to $100K a month (MRR).
Conclusion
You must market your goods to reach your income objective (without waiting forever). Paid ads is the way to go if you hate knocking on doors or irritating friends and family (who aren’t scalable anyways).
You must also test and optimize different angles, audiences, interest groups, and creatives.