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
JEFF JOHN ROBERTS
3 years ago
What just happened in cryptocurrency? A plain-English Q&A about Binance's FTX takedown.
Crypto people have witnessed things. They've seen big hacks, mind-boggling swindles, and amazing successes. They've never seen a day like Tuesday, when the world's largest crypto exchange murdered its closest competition.
Here's a primer on Binance and FTX's lunacy and why it matters if you're new to crypto.
What happened?
CZ, a shrewd Chinese-Canadian billionaire, runs Binance. FTX, a newcomer, has challenged Binance in recent years. SBF (Sam Bankman-Fried)—a young American with wild hair—founded FTX (initials are a thing in crypto).
Last weekend, CZ complained about SBF's lobbying and then exploited Binance's market power to attack his competition.
How did CZ do that?
CZ invested in SBF's new cryptocurrency exchange when they were friends. CZ sold his investment in FTX for FTT when he no longer wanted it. FTX clients utilize those tokens to get trade discounts, although they are less liquid than Bitcoin.
SBF made a mistake by providing CZ just too many FTT tokens, giving him control over FTX. It's like Pepsi handing Coca-Cola a lot of stock it could sell at any time. CZ got upset with SBF and flooded the market with FTT tokens.
SBF owns a trading fund with many FTT tokens, therefore this was catastrophic. SBF sought to defend FTT's worth by selling other assets to buy up the FTT tokens flooding the market, but it didn't succeed, and as FTT's value plummeted, his liabilities exceeded his assets. By Tuesday, his companies were insolvent, so he sold them to his competition.
Crazy. How could CZ do that?
CZ likely did this to crush a rising competition. It was also personal. In recent months, regulators have been tough toward the crypto business, and Binance and FTX have been trying to stay on their good side. CZ believed SBF was poisoning U.S. authorities by saying CZ was linked to China, so CZ took retribution.
“We supported previously, but we won't pretend to make love after divorce. We're neutral. But we won't assist people that push against other industry players behind their backs," CZ stated in a tragic tweet on Sunday. He crushed his rival's company two days later.
So does Binance now own FTX?
No. Not yet. CZ has only stated that Binance signed a "letter of intent" to acquire FTX. CZ and SBF say Binance will protect FTX consumers' funds.
Who’s to blame?
You could blame CZ for using his control over FTX to destroy it. SBF is also being criticized for not disclosing the full overlap between FTX and his trading company, which controlled plenty of FTT. If he had been upfront, someone might have warned FTX about this vulnerability earlier, preventing this mess.
Others have alleged that SBF utilized customer monies to patch flaws in his enterprises' balance accounts. That happened to multiple crypto startups that collapsed this spring, which is unfortunate. These are allegations, not proof.
Why does this matter? Isn't this common in crypto?
Crypto is notorious for shady executives and pranks. FTX is the second-largest crypto business, and SBF was largely considered as the industry's golden boy who would help it get on authorities' good side. Thus far.
Does this affect cryptocurrency prices?
Short-term, it's bad. Prices fell on suspicions that FTX was in peril, then rallied when Binance rescued it, only to fall again later on Tuesday.
These occurrences have hurt FTT and SBF's Solana token. It appears like a huge token selloff is affecting the rest of the market. Bitcoin fell 10% and Ethereum 15%, which is bad but not catastrophic for the two largest coins by market cap.
joyce shen
3 years ago
Framework to Evaluate Metaverse and Web3
Everywhere we turn, there's a new metaverse or Web3 debut. Microsoft recently announced a $68.7 BILLION cash purchase of Activision.
Like AI in 2013 and blockchain in 2014, NFT growth in 2021 feels like this year's metaverse and Web3 growth. We are all bombarded with information, conflicting signals, and a sensation of FOMO.
How can we evaluate the metaverse and Web3 in a noisy, new world? My framework for evaluating upcoming technologies and themes is shown below. I hope you will also find them helpful.
Understand the “pipes” in a new space.
Whatever people say, Metaverse and Web3 will have to coexist with the current Internet. Companies who host, move, and store data over the Internet have a lot of intriguing use cases in Metaverse and Web3, whether in infrastructure, data analytics, or compliance. Hence the following point.
## Understand the apps layer and their infrastructure.
Gaming, crypto exchanges, and NFT marketplaces would not exist today if not for technology that enables rapid app creation. Yes, according to Chainalysis and other research, 30–40% of Ethereum is self-hosted, with the rest hosted by large cloud providers. For Microsoft to acquire Activision makes strategic sense. It's not only about the games, but also the infrastructure that supports them.
Follow the money
Understanding how money and wealth flow in a complex and dynamic environment helps build clarity. Unless you are exceedingly wealthy, you have limited ability to significantly engage in the Web3 economy today. Few can just buy 10 ETH and spend it in one day. You must comprehend who benefits from the process, and how that 10 ETH circulates now and possibly tomorrow. Major holders and players control supply and liquidity in any market. Today, most Web3 apps are designed to increase capital inflow so existing significant holders can utilize it to create a nascent Web3 economy. When you see a new Metaverse or Web3 application, remember how money flows.
What is the use case?
What does the app do? If there is no clear use case with clear makers and consumers solving a real problem, then the euphoria soon fades, and the only stakeholders who remain enthused are those who have too much to lose.
Time is a major competition that is often overlooked.
We're only busier, but each day is still 24 hours. Using new apps may mean that time is lost doing other things. The user must be eager to learn. Metaverse and Web3 vs. our time? I don't think we know the answer yet (at least for working adults whose cost of time is higher).
I don't think we know the answer yet (at least for working adults whose cost of time is higher).
People and organizations need security and transparency.
For new technologies or apps to be widely used, they must be safe, transparent, and trustworthy. What does secure Metaverse and Web3 mean? This is an intriguing subject for both the business and public sectors. Cloud adoption grew in part due to improved security and data protection regulations.
The following frameworks can help analyze and understand new technologies and emerging technological topics, unless you are a significant investment fund with the financial ability to gamble on numerous initiatives and essentially form your own “index fund”.
I write on VC, startups, and leadership.
More on https://www.linkedin.com/in/joycejshen/ and https://joyceshen.substack.com/
This writing is my own opinion and does not represent investment advice.
ANDREW SINGER
3 years ago
Crypto seen as the ‘future of money’ in inflation-mired countries
Crypto as the ‘future of money' in inflation-stricken nations
Citizens of devalued currencies “need” crypto. “Nice to have” in the developed world.
According to Gemini's 2022 Global State of Crypto report, cryptocurrencies “evolved from what many considered a niche investment into an established asset class” last year.
More than half of crypto owners in Brazil (51%), Hong Kong (51%), and India (54%), according to the report, bought cryptocurrency for the first time in 2021.
The study found that inflation and currency devaluation are powerful drivers of crypto adoption, especially in emerging market (EM) countries:
“Respondents in countries that have seen a 50% or greater devaluation of their currency against the USD over the last decade were more than 5 times as likely to plan to purchase crypto in the coming year.”
Between 2011 and 2021, the real lost 218 percent of its value against the dollar, and 45 percent of Brazilians surveyed by Gemini said they planned to buy crypto in 2019.
The rand (South Africa's currency) has fallen 103 percent in value over the last decade, second only to the Brazilian real, and 32 percent of South Africans expect to own crypto in the coming year. Mexico and India, the third and fourth highest devaluation countries, followed suit.
Compared to the US dollar, Hong Kong and the UK currencies have not devalued in the last decade. Meanwhile, only 5% and 8% of those surveyed in those countries expressed interest in buying crypto.
What can be concluded? Noah Perlman, COO of Gemini, sees various crypto use cases depending on one's location.
‘Need to have' investment in countries where the local currency has devalued against the dollar, whereas in the developed world it is still seen as a ‘nice to have'.
Crypto as money substitute
As an adjunct professor at New York University School of Law, Winston Ma distinguishes between an asset used as an inflation hedge and one used as a currency replacement.
Unlike gold, he believes Bitcoin (BTC) is not a “inflation hedge”. They acted more like growth stocks in 2022. “Bitcoin correlated more closely with the S&P 500 index — and Ether with the NASDAQ — than gold,” he told Cointelegraph. But in the developing world, things are different:
“Inflation may be a primary driver of cryptocurrency adoption in emerging markets like Brazil, India, and Mexico.”
According to Justin d'Anethan, institutional sales director at the Amber Group, a Singapore-based digital asset firm, early adoption was driven by countries where currency stability and/or access to proper banking services were issues. Simply put, he said, developing countries want alternatives to easily debased fiat currencies.
“The larger flows may still come from institutions and developed countries, but the actual users may come from places like Lebanon, Turkey, Venezuela, and Indonesia.”
“Inflation is one of the factors that has and continues to drive adoption of Bitcoin and other crypto assets globally,” said Sean Stein Smith, assistant professor of economics and business at Lehman College.
But it's only one factor, and different regions have different factors, says Stein Smith. As a “instantaneously accessible, traceable, and cost-effective transaction option,” investors and entrepreneurs increasingly recognize the benefits of crypto assets. Other places promote crypto adoption due to “potential capital gains and returns”.
According to the report, “legal uncertainty around cryptocurrency,” tax questions, and a general education deficit could hinder adoption in Asia Pacific and Latin America. In Africa, 56% of respondents said more educational resources were needed to explain cryptocurrencies.
Not only inflation, but empowering our youth to live better than their parents without fear of failure or allegiance to legacy financial markets or products, said Monica Singer, ConsenSys South Africa lead. Also, “the issue of cash and remittances is huge in Africa, as is the issue of social grants.”
Money's future?
The survey found that Brazil and Indonesia had the most cryptocurrency ownership. In each country, 41% of those polled said they owned crypto. Only 20% of Americans surveyed said they owned cryptocurrency.
These markets are more likely to see cryptocurrencies as the future of money. The survey found:
“The majority of respondents in Latin America (59%) and Africa (58%) say crypto is the future of money.”
Brazil (66%), Nigeria (63%), Indonesia (61%), and South Africa (57%). Europe and Australia had the fewest believers, with Denmark at 12%, Norway at 15%, and Australia at 17%.
Will the Ukraine conflict impact adoption?
The poll was taken before the war. Will the devastating conflict slow global crypto adoption growth?
With over $100 million in crypto donations directly requested by the Ukrainian government since the war began, Stein Smith says the war has certainly brought crypto into the mainstream conversation.
“This real-world demonstration of decentralized money's power could spur wider adoption, policy debate, and increased use of crypto as a medium of exchange.”
But the war may not affect all developing nations. “The Ukraine war has no impact on African demand for crypto,” Others loom larger. “Yes, inflation, but also a lack of trust in government in many African countries, and a young demographic very familiar with mobile phones and the internet.”
A major success story like Mpesa in Kenya has influenced the continent and may help accelerate crypto adoption. Creating a plan when everyone you trust fails you is directly related to the African spirit, she said.
On the other hand, Ma views the Ukraine conflict as a sort of crisis check for cryptocurrencies. For those in emerging markets, the Ukraine-Russia war has served as a “stress test” for the cryptocurrency payment rail, he told Cointelegraph.
“These emerging markets may see the greatest future gains in crypto adoption.”
Inflation and currency devaluation are persistent global concerns. In such places, Bitcoin and other cryptocurrencies are now seen as the “future of money.” Not in the developed world, but that could change with better regulation and education. Inflation and its impact on cash holdings are waking up even Western nations.
Read original post here.
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Jake Prins
3 years ago
What are NFTs 2.0 and what issues are they meant to address?
New standards help NFTs reach their full potential.
NFTs lack interoperability and functionality. They have great potential but are mostly speculative. To maximize NFTs, we need flexible smart contracts.
Current requirements are too restrictive.
Most NFTs are based on ERC-721, which makes exchanging them easy. CryptoKitties, a popular online game, used the 2017 standard to demonstrate NFTs' potential.
This simple standard includes a base URI and incremental IDs for tokens. Add the tokenID to the base URI to get the token's metadata.
This let creators collect NFTs. Many NFT projects store metadata on IPFS, a distributed storage network, but others use Google Drive. NFT buyers often don't realize that if the creators delete or move the files, their NFT is just a pointer.
This isn't the standard's biggest issue. There's no way to validate NFT projects.
Creators are one of the most important aspects of art, but nothing is stored on-chain.
ERC-721 contracts only have a name and symbol.
Most of the data on OpenSea's collection pages isn't from the NFT's smart contract. It was added through a platform input field, so it's in the marketplace's database. Other websites may have different NFT information.
In five years, your NFT will be just a name, symbol, and ID.
Your NFT doesn't mention its creators. Although the smart contract has a public key, it doesn't reveal who created it.
The NFT's creators and their reputation are crucial to its value. Think digital fashion and big brands working with well-known designers when more professionals use NFTs. Don't you want them in your NFT?
Would paintings be as valuable if their artists were unknown? Would you believe it's real?
Buying directly from an on-chain artist would reduce scams. Current standards don't allow this data.
Most creator profiles live on centralized marketplaces and could disappear. Current platforms have outpaced underlying standards. The industry's standards are lagging.
For NFTs to grow beyond pointers to a monkey picture file, we may need to use new Web3-based standards.
Introducing NFTs 2.0
Fabian Vogelsteller, creator of ERC-20, developed new web3 standards. He proposed LSP7 Digital Asset and LSP8 Identifiable Digital Asset, also called NFT 2.0.
NFT and token metadata inputs are extendable. Changes to on-chain metadata inputs allow NFTs to evolve. Instead of public keys, the contract can have Universal Profile addresses attached. These profiles show creators' faces and reputations. NFTs can notify asset receivers, automating smart contracts.
LSP7 and LSP8 use ERC725Y. Using a generic data key-value store gives contracts much-needed features:
The asset can be customized and made to stand out more by allowing for unlimited data attachment.
Recognizing changes to the metadata
using a hash reference for metadata rather than a URL reference
This base will allow more metadata customization and upgradeability. These guidelines are:
Genuine and Verifiable Now, the creation of an NFT by a specific Universal Profile can be confirmed by smart contracts.
Dynamic NFTs can update Flexible & Updatable Metadata, allowing certain things to evolve over time.
Protected metadata Now, secure metadata that is readable by smart contracts can be added indefinitely.
Better NFTS prevent the locking of NFTs by only being sent to Universal Profiles or a smart contract that can interact with them.
Summary
NFTS standards lack standardization and powering features, limiting the industry.
ERC-721 is the most popular NFT standard, but it only represents incremental tokenIDs without metadata or asset representation. No standard sender-receiver interaction or security measures ensure safe asset transfers.
NFT 2.0 refers to the new LSP7-DigitalAsset and LSP8-IdentifiableDigitalAsset standards.
They have new standards for flexible metadata, secure transfers, asset representation, and interactive transfer.
With NFTs 2.0 and Universal Profiles, creators could build on-chain reputations.
NFTs 2.0 could bring the industry's needed innovation if it wants to move beyond trading profile pictures for speculation.
Al Anany
2 years ago
Notion AI Might Destroy Grammarly and Jasper
The trick Notion could use is simply Facebook-ing the hell out of them.
*Time travel to fifteen years ago.* Future-Me: “Hey! What are you up to?” Old-Me: “I am proofreading an article. It’s taking a few hours, but I will be done soon.” Future-Me: “You know, in the future, you will be using a google chrome plugin called Grammarly that will help you easily proofread articles in half that time.” Old-Me: “What is… Google Chrome?” Future-Me: “Gosh…”
I love Grammarly. It’s one of those products that I personally feel the effects of. I mean, Space X is a great company. But I am not a rocket writing this article in space (or am I?)…
No, I’m not. So I don’t personally feel a connection to Space X. So, if a company collapse occurs in the morning, I might write about it. But I will have zero emotions regarding it.
Yet, if Grammarly fails tomorrow, I will feel 1% emotionally distressed. So looking at the title of this article, you’d realize that I am betting against them. This is how much I believe in the critical business model that’s taking over the world, the one of Notion.
Notion How frequently do you go through your notes?
Grammarly is everywhere, which helps its success. Grammarly is available when you update LinkedIn on Chrome. Grammarly prevents errors in Google Docs.
My internal concentration isn't apparent in the previous paragraph. Not Grammarly. I should have used Chrome to make a Google doc and LinkedIn update. Without this base, Grammarly will be useless.
So, welcome to this business essay.
Grammarly provides a solution.
Another issue is resolved by Jasper.
Your entire existence is supposed to be contained within Notion.
New Google Chrome is offline. It's an all-purpose notepad (in the near future.)
How should I start my blog? Enter it in Note.
an update on LinkedIn? If you mention it, it might be automatically uploaded there (with little help from another app.)
An advanced thesis? You can brainstorm it with your coworkers.
This ad sounds great! I won't cry if Notion dies tomorrow.
I'll reread the following passages to illustrate why I think Notion could kill Grammarly and Jasper.
Notion is a fantastic app that incubates your work.
Smartly, they began with note-taking.
Hopefully, your work will be on Notion. Grammarly and Jasper are still must-haves.
Grammarly will proofread your typing while Jasper helps with copywriting and AI picture development.
They're the best, therefore you'll need them. Correct? Nah.
Notion might bombard them with Facebook posts.
Notion: “Hi Grammarly, do you want to sell your product to us?” Grammarly: “Dude, we are more valuable than you are. We’ve even raised $400m, while you raised $342m. Our last valuation round put us at $13 billion, while yours put you at $10 billion. Go to hell.” Notion: “Okay, we’ll speak again in five years.”
Notion: “Jasper, wanna sell?” Jasper: “Nah, we’re deep into AI and the field. You can’t compete with our people.” Notion: “How about you either sell or you turn into a Snapchat case?” Jasper: “…”
Notion is your home. Grammarly is your neighbor. Your track is Jasper.
What if you grew enough vegetables in your backyard to avoid the supermarket? No more visits.
What if your home had a beautiful treadmill? You won't rush outside as much (I disagree with my own metaphor). (You get it.)
It's Facebooking. Instagram Stories reduced your Snapchat usage. Notion will reduce your need to use Grammarly.
The Final Piece of the AI Puzzle
Let's talk about Notion first, since you've probably read about it everywhere.
They raised $343 million, as I previously reported, and bought four businesses
According to Forbes, Notion will have more than 20 million users by 2022. The number of users is up from 4 million in 2020.
If raising $1.8 billion was impressive, FTX wouldn't have fallen.
This article compares the basic product to two others. Notion is a day-long app.
Notion has released Notion AI to support writers. It's early, so it's not as good as Jasper. Then-Jasper isn't now-Jasper. In five years, Notion AI will be different.
With hard work, they may construct a Jasper-like writing assistant. They have resources and users.
At this point, it's all speculation. Jasper's copywriting is top-notch. Grammarly's proofreading is top-notch. Businesses are constrained by user activities.
If Notion's future business movements are strategic, they might become a blue ocean shark (or get acquired by an unbelievable amount.)
I love business mental teasers, so tell me:
How do you feel? Are you a frequent Notion user?
Do you dispute my position? I enjoy hearing opposing viewpoints.
Ironically, I proofread this with Grammarly.
Nikhil Vemu
3 years ago
7 Mac Tips You Never Knew You Needed
Unleash the power of the Option key ⌥
#1 Open a link in the Private tab first.
Previously, if I needed to open a Safari link in a private window, I would:
copied the URL with the right click command,
choose File > New Private Window to open a private window, and
clicked return after pasting the URL.
I've found a more straightforward way.
Right-clicking a link shows this, right?
Hold option (⌥) for:
Click Open Link in New Private Window while holding.
Finished!
#2. Instead of searching for specific characters, try this
You may use unicode for business or school. Most people Google them when they need them.
That is lengthy!
You can type some special characters just by pressing ⌥ and a key.
For instance
• ⌥+2 -> ™ (Trademark)
• ⌥+0 -> ° (Degree)
• ⌥+G -> © (Copyright)
• ⌥+= -> ≠ (Not equal to)
• ⌥+< -> ≤ (Less than or equal to)
• ⌥+> -> ≥ (Greater then or equal to)
• ⌥+/ -> ÷ (Different symbol for division)#3 Activate Do Not Disturb silently.
Do Not Disturb when sharing my screen is awkward for me (because people may think Im trying to hide some secret notifications).
Here's another method.
Hold ⌥ and click on Time (at the extreme right on the menu-bar).
Now, DND is activated (secretly!). To turn it off, do it again.
Note: This works only for DND focus.#4. Resize a window starting from its center
Although this is rarely useful, it is still a hidden trick.
When you resize a window, the opposite edge or corner is used as the pivot, right?
However, if you want to resize it with its center as the pivot, hold while doing so.
#5. Yes, Cut-Paste is available on Macs as well (though it is slightly different).
I call it copy-move rather than cut-paste. This is how it works.
Carry it out.
Choose a file (by clicking on it), then copy it (⌘+C).
Go to a new location on your Mac. Do you use ⌘+V to paste it? However, to move it, press ⌘+⌥+V.
This removes the file from its original location and copies it here. And it works exactly like cut-and-paste on Windows.
#6. Instantly expand all folders
Set your Mac's folders to List view.
Assume you have one folder with multiple subfolders, each of which contains multiple files. And you wanted to look at every single file that was over there.
How would you do?
You're used to clicking the ⌄ glyph near the folder and each subfolder to expand them all, right? Instead, hold down ⌥ while clicking ⌄ on the parent folder.
This is what happens next.
Everything expands.
View/Copy a file's path as an added bonus
If you want to see the path of a file in Finder, select it and hold ⌥, and you'll see it at the bottom for a moment.
To copy its path, right-click on the folder and hold down ⌥ to see this
Click on Copy <"folder name"> as Pathname to do it.
#7 "Save As"
I was irritated by the lack of "Save As" in Pages when I first got a Mac (after 15 years of being a Windows guy).
It was necessary for me to save the file as a new file, in a different location, with a different name, or both.
Unfortunately, I couldn't do it on a Mac.
However, I recently discovered that it appears when you hold ⌥ when in the File menu.
Yay!