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
3 years ago
👏 Awesome post! When is part 3 coming?

Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto

Percy Bolmér
3 years ago
Ethereum No Longer Consumes A Medium-Sized Country's Electricity To Run
The Merge cut Ethereum's energy use by 99.5%.
The Crypto community celebrated on September 15, 2022. This day, Ethereum Merged. The entire blockchain successfully merged with the Beacon chain, and it was so smooth you barely noticed.
Many have waited, dreaded, and longed for this day.
Some investors feared the network would break down, while others envisioned a seamless merging.
Speculators predict a successful Merge will lead investors to Ethereum. This could boost Ethereum's popularity.
What Has Changed Since The Merge
The merging transitions Ethereum mainnet from PoW to PoS.
PoW sends a mathematical riddle to computers worldwide (miners). First miner to solve puzzle updates blockchain and is rewarded.
The puzzles sent are power-intensive to solve, so mining requires a lot of electricity. It's sent to every miner competing to solve it, requiring duplicate computation.
PoS allows investors to stake their coins to validate a new transaction. Instead of validating a whole block, you validate a transaction and get the fees.
You can validate instead of mine. A validator stakes 32 Ethereum. After staking, the validator can validate future blocks.
Once a validator validates a block, it's sent to a randomly selected group of other validators. This group verifies that a validator is not malicious and doesn't validate fake blocks.
This way, only one computer needs to solve or validate the transaction, instead of all miners. The validated block must be approved by a small group of validators, causing duplicate computation.
PoS is more secure because validating fake blocks results in slashing. You lose your bet tokens. If a validator signs a bad block or double-signs conflicting blocks, their ETH is burned.
Theoretically, Ethereum has one block every 12 seconds, so a validator forging a block risks burning 1 Ethereum for 12 seconds of transactions. This makes mistakes expensive and risky.
What Impact Does This Have On Energy Use?
Cryptocurrency is a natural calamity, sucking electricity and eating away at the earth one transaction at a time.
Many don't know the environmental impact of cryptocurrencies, yet it's tremendous.
A single Ethereum transaction used to use 200 kWh and leave a large carbon imprint. This update reduces global energy use by 0.2%.
Ethereum will submit a challenge to one validator, and that validator will forward it to randomly selected other validators who accept it.
This reduces the needed computing power.
They expect a 99.5% reduction, therefore a single transaction should cost 1 kWh.
Carbon footprint is 0.58 kgCO2, or 1,235 VISA transactions.
This is a big Ethereum blockchain update.
I love cryptocurrency and Mother Earth.

Jayden Levitt
3 years ago
The country of El Salvador's Bitcoin-obsessed president lost $61.6 million.
It’s only a loss if you sell, right?
Nayib Bukele proclaimed himself “the world’s coolest dictator”.
His jokes aren't clear.
El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.
His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.
He's popular, though.
Bukele won 53% of the vote by fighting violent crime and opposition party corruption.
El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.
They were powerless.
Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.
He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.
Transfer and deposit costs.
This makes logical sense when the president’s theatrics don’t blind you.
El Salvador's Bukele revealed plans to make bitcoin legal tender.
Remittances total $5.9 billion (23%) of the country's expenses.
Anything that reduces costs could boost the economy.
The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).
According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.
In 2021, 36% of El Salvadorans had bank accounts.
Large rural countries like Kenya seem to have resolved their unbanked dilemma.
An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.
Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.
96% of Kenyans utilize M-PESA, which doesn't require a bank account.
The software involves human agents who hang out with cash and a phone.
These people are like ATMs.
You offer them cash to deposit money in your mobile money account or withdraw cash.
In a country with a faulty banking system, cash availability and a safe place to deposit it are important.
William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.
Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.
The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.
Currently, El Salvador has 2,301 Bitcoin.
At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.
Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.
It's still falling.
This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.
The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.
You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.

Ryan Weeks
3 years ago
Terra fiasco raises TRON's stablecoin backstop
After Terra's algorithmic stablecoin collapsed in May, TRON announced a plan to increase the capital backing its own stablecoin.
USDD, a near-carbon copy of Terra's UST, arrived on the TRON blockchain on May 5. TRON founder Justin Sun says USDD will be overcollateralized after initially being pegged algorithmically to the US dollar.
A reserve of cryptocurrencies and stablecoins will be kept at 130 percent of total USDD issuance, he said. TRON described the collateral ratio as "guaranteed" and said it would begin publishing real-time updates on June 5.
Currently, the reserve contains 14,040 bitcoin (around $418 million), 140 million USDT, 1.9 billion TRX, and 8.29 billion TRX in a burning contract.
Sun: "We want to hybridize USDD." We have an algorithmic stablecoin and TRON DAO Reserve.
algorithmic failure
USDD was designed to incentivize arbitrageurs to keep its price pegged to the US dollar by trading TRX, TRON's token, and USDD. Like Terra, TRON signaled its intent to establish a bitcoin and cryptocurrency reserve to support USDD in extreme market conditions.
Still, Terra's UST failed despite these safeguards. The stablecoin veered sharply away from its dollar peg in mid-May, bringing down Terra's LUNA and wiping out $40 billion in value in days. In a frantic attempt to restore the peg, billions of dollars in bitcoin were sold and unprecedented volumes of LUNA were issued.
Sun believes USDD, which has a total circulating supply of $667 million, can be backed up.
"Our reserve backing is diversified." Bitcoin and stablecoins are included. USDC will be a small part of Circle's reserve, he said.
TRON's news release lists the reserve's assets as bitcoin, TRX, USDC, USDT, TUSD, and USDJ.
All Bitcoin addresses will be signed so everyone knows they belong to us, Sun said.
Not giving in
Sun told that the crypto industry needs "decentralized" stablecoins that regulators can't touch.
Sun said the Luna Foundation Guard, a Singapore-based non-profit that raised billions in cryptocurrency to buttress UST, mismanaged the situation by trying to sell to panicked investors.
He said, "We must be ahead of the market." We want to stabilize the market and reduce volatility.
Currently, TRON finances most of its reserve directly, but Sun says the company hopes to add external capital soon.
Before its demise, UST holders could park the stablecoin in Terra's lending platform Anchor Protocol to earn 20% interest, which many deemed unsustainable. TRON's JustLend is similar. Sun hopes to raise annual interest rates from 17.67% to "around 30%."
This post is a summary. Read full article here
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ANTHONY P.
3 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.

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.
Matthew Royse
3 years ago
5 Tips for Concise Writing
Here's how to be clear.
“I have only made this letter longer because I have not had the time to make it shorter.” — French mathematician, physicist, inventor, philosopher, and writer Blaise Pascal
Concise.
People want this. We tend to repeat ourselves and use unnecessary words.
Being vague frustrates readers. It focuses their limited attention span on figuring out what you're saying rather than your message.
Edit carefully.
“Examine every word you put on paper. You’ll find a surprising number that don’t serve any purpose.” — American writer, editor, literary critic, and teacher William Zinsser
How do you write succinctly?
Here are three ways to polish your writing.
1. Delete
Your readers will appreciate it if you delete unnecessary words. If a word or phrase is essential, keep it. Don't force it.
Many readers dislike bloated sentences. Ask yourself if cutting a word or phrase will change the meaning or dilute your message.
For example, you could say, “It’s absolutely essential that I attend this meeting today, so I know the final outcome.” It’s better to say, “It’s critical I attend the meeting today, so I know the results.”
Key takeaway
Delete actually, completely, just, full, kind of, really, and totally. Keep the necessary words, cut the rest.
2. Just Do It
Don't tell readers your plans. Your readers don't need to know your plans. Who are you?
Don't say, "I want to highlight our marketing's problems." Our marketing issues are A, B, and C. This cuts 5–7 words per sentence.
Keep your reader's attention on the essentials, not the fluff. What are you doing? You won't lose readers because you get to the point quickly and don't build up.
Key takeaway
Delete words that don't add to your message. Do something, don't tell readers you will.
3. Cut Overlap
You probably repeat yourself unintentionally. You may add redundant sentences when brainstorming. Read aloud to detect overlap.
Remove repetition from your writing. It's important to edit our writing and thinking to avoid repetition.
Key Takeaway
If you're repeating yourself, combine sentences to avoid overlap.
4. Simplify
Write as you would to family or friends. Communicate clearly. Don't use jargon. These words confuse readers.
Readers want specifics, not jargon. Write simply. Done.
Most adults read at 8th-grade level. Jargon and buzzwords make speech fluffy. This confuses readers who want simple language.
Key takeaway
Ensure all audiences can understand you. USA Today's 5th-grade reading level is intentional. They want everyone to understand.
5. Active voice
Subjects perform actions in active voice. When you write in passive voice, the subject receives the action.
For example, “the board of directors decided to vote on the topic” is an active voice, while “a decision to vote on the topic was made by the board of directors” is a passive voice.
Key takeaway
Active voice clarifies sentences. Active voice is simple and concise.
Bringing It All Together
Five tips help you write clearly. Delete, just do it, cut overlap, use simple language, and write in an active voice.
Clear writing is effective. It's okay to occasionally use unnecessary words or phrases. Realizing it is key. Check your writing.
Adding words costs.
Write more concisely. People will appreciate it and read your future articles, emails, and messages. Spending extra time will increase trust and influence.
“Not that the story need be long, but it will take a long while to make it short.” — Naturalist, essayist, poet, and philosopher Henry David Thoreau
