An approximate introduction to how zk-SNARKs are possible (part 1)
You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.
In the context of blockchains, this has 2 very powerful applications: Perhaps the most powerful cryptographic technology to come out of the last decade is general-purpose succinct zero knowledge proofs, usually called zk-SNARKs ("zero knowledge succinct arguments of knowledge"). A zk-SNARK allows you to generate a proof that some computation has some particular output, in such a way that the proof can be verified extremely quickly even if the underlying computation takes a very long time to run. The "ZK" part adds an additional feature: the proof can keep some of the inputs to the computation hidden.
You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.
In the context of blockchains, this has two very powerful applications:
- Scalability: if a block takes a long time to verify, one person can verify it and generate a proof, and everyone else can just quickly verify the proof instead
- Privacy: you can prove that you have the right to transfer some asset (you received it, and you didn't already transfer it) without revealing the link to which asset you received. This ensures security without unduly leaking information about who is transacting with whom to the public.
But zk-SNARKs are quite complex; indeed, as recently as in 2014-17 they were still frequently called "moon math". The good news is that since then, the protocols have become simpler and our understanding of them has become much better. This post will try to explain how ZK-SNARKs work, in a way that should be understandable to someone with a medium level of understanding of mathematics.
Why ZK-SNARKs "should" be hard
Let us take the example that we started with: we have a number (we can encode "cow" followed by the secret input as an integer), we take the SHA256 hash of that number, then we do that again another 99,999,999 times, we get the output, and we check what its starting digits are. This is a huge computation.
A "succinct" proof is one where both the size of the proof and the time required to verify it grow much more slowly than the computation to be verified. If we want a "succinct" proof, we cannot require the verifier to do some work per round of hashing (because then the verification time would be proportional to the computation). Instead, the verifier must somehow check the whole computation without peeking into each individual piece of the computation.
One natural technique is random sampling: how about we just have the verifier peek into the computation in 500 different places, check that those parts are correct, and if all 500 checks pass then assume that the rest of the computation must with high probability be fine, too?
Such a procedure could even be turned into a non-interactive proof using the Fiat-Shamir heuristic: the prover computes a Merkle root of the computation, uses the Merkle root to pseudorandomly choose 500 indices, and provides the 500 corresponding Merkle branches of the data. The key idea is that the prover does not know which branches they will need to reveal until they have already "committed to" the data. If a malicious prover tries to fudge the data after learning which indices are going to be checked, that would change the Merkle root, which would result in a new set of random indices, which would require fudging the data again... trapping the malicious prover in an endless cycle.
But unfortunately there is a fatal flaw in naively applying random sampling to spot-check a computation in this way: computation is inherently fragile. If a malicious prover flips one bit somewhere in the middle of a computation, they can make it give a completely different result, and a random sampling verifier would almost never find out.
It only takes one deliberately inserted error, that a random check would almost never catch, to make a computation give a completely incorrect result.
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? There is a clever solution.
see part 2
(Edited)
More on Web3 & Crypto
CyberPunkMetalHead
3 years ago
195 countries want Terra Luna founder Do Kwon
Interpol has issued a red alert on Terraform Labs' CEO, South Korean prosecutors said.
After the May crash of Terra Luna revealed tax evasion issues, South Korean officials filed an arrest warrant for Do Kwon, but he is missing.
Do Kwon is now a fugitive in 195 countries after Seoul prosecutors placed him to Interpol's red list. Do Kwon hasn't commented since then. The red list allows any country's local authorities to apprehend Do Kwon.
Do Dwon and Terraform Labs were believed to have moved to Singapore days before the $40 billion wipeout, but Singapore authorities said he fled the country on September 17. Do Kwon tweeted that he wasn't on the run and cited privacy concerns.
Do Kwon was not on the red list at the time and said he wasn't "running," only to reply to his own tweet saying he hasn't jogged in a while and needed to trim calories.
Whether or not it makes sense to read too much into this, the reality is that Do Kwon is now on Interpol red list, despite the firmly asserts on twitter that he does absolutely nothing to hide.
UPDATE:
South Korean authorities are investigating alleged withdrawals of over $60 million U.S. and seeking to freeze these assets. Korean authorities believe a new wallet exchanged over 3000 BTC through OKX and Kucoin.
Do Kwon and the Luna Foundation Guard (of whom Do Kwon is a key member of) have declined all charges and dubbed this disinformation.
Singapore's Luna Foundation Guard (LFG) manages the Terra Ecosystem.
The Legal Situation
Multiple governments are searching for Do Kwon and five other Terraform Labs employees for financial markets legislation crimes.
South Korean authorities arrested a man suspected of tax fraud and Ponzi scheme.
The U.S. SEC is also examining Terraform Labs on how UST was advertised as a stablecoin. No legal precedent exists, so it's unclear what's illegal.
The future of Terraform Labs, Terra, and Terra 2 is unknown, and despite what Twitter shills say about LUNC, the company remains in limbo awaiting a decision that will determine its fate. This project isn't a wise investment.
Robert Kim
4 years ago
Crypto Legislation Might Progress Beyond Talk in 2022
Financial regulators have for years attempted to apply existing laws to the multitude of issues created by digital assets. In 2021, leading federal regulators and members of Congress have begun to call for legislation to address these issues. As a result, 2022 may be the year when federal legislation finally addresses digital asset issues that have been growing since the mining of the first Bitcoin block in 2009.
Digital Asset Regulation in the Absence of Legislation
So far, Congress has left the task of addressing issues created by digital assets to regulatory agencies. Although a Congressional Blockchain Caucus formed in 2016, House and Senate members introduced few bills addressing digital assets until 2018. As of October 2021, Congress has not amended federal laws on financial regulation, which were last significantly revised by the Dodd-Frank Act in 2010, to address digital asset issues.
In the absence of legislation, issues that do not fit well into existing statutes have created problems. An example is the legal status of digital assets, which can be considered to be either securities or commodities, and can even shift from one to the other over time. Years after the SEC’s 2017 report applying the definition of a security to digital tokens, the SEC and the CFTC have yet to clarify the distinction between securities and commodities for the thousands of digital assets in existence.
SEC Chair Gary Gensler has called for Congress to act, stating in August, “We need additional Congressional authorities to prevent transactions, products, and platforms from falling between regulatory cracks.” Gensler has reached out to Sen. Elizabeth Warren (D-Ma.), who has expressed her own concerns about the need for legislation.
Legislation on Digital Assets in 2021
While regulators and members of Congress talked about the need for legislation, and the debate over cryptocurrency tax reporting in the 2021 infrastructure bill generated headlines, House and Senate bills proposing specific solutions to various issues quietly started to emerge.
Digital Token Sales
Several House bills attempt to address securities law barriers to digital token sales—some of them by building on ideas proposed by regulators in past years.
Exclusion from the definition of a security. Congressional Blockchain Caucus members have been introducing bills to exclude digital tokens from the definition of a security since 2018, and they have revived those bills in 2021. They include the Token Taxonomy Act of 2021 (H.R. 1628), successor to identically named bills in 2018 and 2019, and the Securities Clarity Act (H.R. 4451), successor to a 2020 namesake.
Safe harbor. SEC Commissioner Hester Peirce proposed a regulatory safe harbor for token sales in 2020, and two 2021 bills have proposed statutory safe harbors. Rep. Patrick McHenry (R-N.C.), Republican leader of the House Financial Services Committee, introduced a Clarity for Digital Tokens Act of 2021 (H.R. 5496) that would amend the Securities Act to create a safe harbor providing a grace period of exemption from Securities Act registration requirements. The Digital Asset Market Structure and Investor Protection Act (H.R. 4741) from Rep. Don Beyer (D-Va.) would amend the Securities Exchange Act to define a new type of security—a “digital asset security”—and add issuers of digital asset securities to an existing provision for delayed registration of securities.
Stablecoins
Stablecoins—digital currencies linked to the value of the U.S. dollar or other fiat currencies—have not yet been the subject of regulatory action, although Treasury Secretary Janet Yellen and Federal Reserve Chair Jerome Powell have each underscored the need to create a regulatory framework for them. The Beyer bill proposes to create a regulatory regime for stablecoins by amending Title 31 of the U.S. Code. Treasury Department approval would be required for any “digital asset fiat-based stablecoin” to be issued or used, under an application process to be established by Treasury in consultation with the Federal Reserve, the SEC, and the CFTC.
Serious consideration for any of these proposals in the current session of Congress may be unlikely. A spate of autumn bills on crypto ransom payments (S. 2666, S. 2923, S. 2926, H.R. 5501) shows that Congress is more inclined to pay attention first to issues that are more spectacular and less arcane. Moreover, the arcaneness of digital asset regulatory issues is likely only to increase further, now that major industry players such as Coinbase and Andreessen Horowitz are starting to roll out their own regulatory proposals.
Digital Dollar vs. Digital Yuan
Impetus to pass legislation on another type of digital asset, a central bank digital currency (CBDC), may come from a different source: rivalry with China.
China established itself as a world leader in developing a CBDC with a pilot project launched in 2020, and in 2021, the People’s Bank of China announced that its CBDC will be used at the Beijing Winter Olympics in February 2022. Republican Senators responded by calling for the U.S. Olympic Committee to forbid use of China’s CBDC by U.S. athletes in Beijing and introducing a bill (S. 2543) to require a study of its national security implications.
The Beijing Olympics could motivate a legislative mandate to accelerate implementation of a U.S. digital dollar, which the Federal Reserve has been in the process of considering in 2021. Antecedents to such legislation already exist. A House bill sponsored by 46 Republicans (H.R. 4792) has a provision that would require the Treasury Department to assess China’s CBDC project and report on the status of Federal Reserve work on a CBDC, and the Beyer bill includes a provision amending the Federal Reserve Act to authorize issuing a digital dollar.
Both parties are likely to support creating a digital dollar. The Covid-19 pandemic made a digital dollar for delivery of relief payments a popular idea in 2020, and House Democrats introduced bills with provisions for creating one in 2020 and 2021. Bipartisan support for a bill on a digital dollar, based on concerns both foreign and domestic in nature, could result.
International rivalry and bipartisan support may make the digital dollar a gateway issue for digital asset legislation in 2022. Legislative work on a digital dollar may open the door for considering further digital asset issues—including the regulatory issues that have been emerging for years—in 2022 and beyond.
OnChain Wizard
3 years ago
How to make a >800 million dollars in crypto attacking the once 3rd largest stablecoin, Soros style
Everyone is talking about the $UST attack right now, including Janet Yellen. But no one is talking about how much money the attacker made (or how brilliant it was). Lets dig in.
Our story starts in late March, when the Luna Foundation Guard (or LFG) starts buying BTC to help back $UST. LFG started accumulating BTC on 3/22, and by March 26th had a $1bn+ BTC position. This is leg #1 that made this trade (or attack) brilliant.
The second leg comes in the form of the 4pool Frax announcement for $UST on April 1st. This added the second leg needed to help execute the strategy in a capital efficient way (liquidity will be lower and then the attack is on).
We don't know when the attacker borrowed 100k BTC to start the position, other than that it was sold into Kwon's buying (still speculation). LFG bought 15k BTC between March 27th and April 11th, so lets just take the average price between these dates ($42k).
So you have a ~$4.2bn short position built. Over the same time, the attacker builds a $1bn OTC position in $UST. The stage is now set to create a run on the bank and get paid on your BTC short. In anticipation of the 4pool, LFG initially removes $150mm from 3pool liquidity.
The liquidity was pulled on 5/8 and then the attacker uses $350mm of UST to drain curve liquidity (and LFG pulls another $100mm of liquidity).
But this only starts the de-pegging (down to 0.972 at the lows). LFG begins selling $BTC to defend the peg, causing downward pressure on BTC while the run on $UST was just getting started.
With the Curve liquidity drained, the attacker used the remainder of their $1b OTC $UST position ($650mm or so) to start offloading on Binance. As withdrawals from Anchor turned from concern into panic, this caused a real de-peg as people fled for the exits
So LFG is selling $BTC to restore the peg while the attacker is selling $UST on Binance. Eventually the chain gets congested and the CEXs suspend withdrawals of $UST, fueling the bank run panic. $UST de-pegs to 60c at the bottom, while $BTC bleeds out.
The crypto community panics as they wonder how much $BTC will be sold to keep the peg. There are liquidations across the board and LUNA pukes because of its redemption mechanism (the attacker very well could have shorted LUNA as well). BTC fell 25% from $42k on 4/11 to $31.3k
So how much did our attacker make? There aren't details on where they covered obviously, but if they are able to cover (or buy back) the entire position at ~$32k, that means they made $952mm on the short.
On the $350mm of $UST curve dumps I don't think they took much of a loss, lets assume 3% or just $11m. And lets assume that all the Binance dumps were done at 80c, thats another $125mm cost of doing business. For a grand total profit of $815mm (bf borrow cost).
BTC was the perfect playground for the trade, as the liquidity was there to pull it off. While having LFG involved in BTC, and foreseeing they would sell to keep the peg (and prevent LUNA from dying) was the kicker.
Lastly, the liquidity being low on 3pool in advance of 4pool allowed the attacker to drain it with only $350mm, causing the broader panic in both BTC and $UST. Any shorts on LUNA would've added a lot of P&L here as well, with it falling -65% since 5/7.
And for the reply guys, yes I know a lot of this involves some speculation & assumptions. But a lot of money was made here either way, and I thought it would be cool to dive into how they did it.
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Quant Galore
3 years ago
I created BAW-IV Trading because I was short on money.
More retail traders means faster, more sophisticated, and more successful methods.
Tech specifications
Only requires a laptop and an internet connection.
We'll use OpenBB's research platform for data/analysis.
Pricing and execution on Options-Quant
Background
You don't need to know the arithmetic details to use this method.
Black-Scholes is a popular option pricing model. It's best for pricing European options. European options are only exercisable at expiration, unlike American options. American options are always exercisable.
American options carry a premium to cover for the risk of early exercise. The Black-Scholes model doesn't account for this premium, hence it can't price genuine, traded American options.
Barone-Adesi-Whaley (BAW) model. BAW modifies Black-Scholes. It accounts for exercise risk premium and stock dividends. It adds the option's early exercise value to the Black-Scholes value.
The trader need not know the formulaic derivations of this model.
https://ir.nctu.edu.tw/bitstream/11536/14182/1/000264318900005.pdf
Strategy
This strategy targets implied volatility. First, we'll locate liquid options that expire within 30 days and have minimal implied volatility.
After selecting the option that meets the requirements, we price it to get the BAW implied volatility (we choose BAW because it's a more accurate Black-Scholes model). If estimated implied volatility is larger than market volatility, we'll capture the spread.
(Calculated IV — Market IV) = (Profit)
Some approaches to target implied volatility are pricey and inaccessible to individual investors. The best and most cost-effective alternative is to acquire a straddle and delta hedge. This may sound terrifying and pricey, but as shown below, it's much less so.
The Trade
First, we want to find our ideal option, so we use OpenBB terminal to screen for options that:
Have an IV at least 5% lower than the 20-day historical IV
Are no more than 5% out-of-the-money
Expire in less than 30 days
We query:
stocks/options/screen/set low_IV/scr --export Output.csv
This uses the screener function to screen for options that satisfy the above criteria, which we specify in the low IV preset (more on custom presets here). It then saves the matching results to a csv(Excel) file for viewing and analysis.
Stick to liquid names like SPY, AAPL, and QQQ since getting out of a position is just as crucial as getting in. Smaller, illiquid names have higher inefficiencies, which could restrict total profits.
We calculate IV using the BAWbisection model (the bisection is a method of calculating IV, more can be found here.) We price the IV first.
According to the BAW model, implied volatility at this level should be priced at 26.90%. When re-pricing the put, IV is 24.34%, up 3%.
Now it's evident. We must purchase the straddle (long the call and long the put) assuming the computed implied volatility is more appropriate and efficient than the market's. We just want to speculate on volatility, not price fluctuations, thus we delta hedge.
The Fun Starts
We buy both options for $7.65. (x100 multiplier). Initial delta is 2. For every dollar the stock price swings up or down, our position value moves $2.
We want delta to be 0 to avoid price vulnerability. A delta of 0 suggests our position's value won't change from underlying price changes. Being delta-hedged allows us to profit/lose from implied volatility. Shorting 2 shares makes us delta-neutral.
That's delta hedging. (Share price * shares traded) = $330.7 to become delta-neutral. You may have noted that delta is not truly 0.00. This is common since delta-hedging means getting as near to 0 as feasible, since it is rare for deltas to align at 0.00.
Now we're vulnerable to changes in Vega (and Gamma, but given we're dynamically hedging, it's not a big risk), or implied volatility. We wanted to gamble that the position's IV would climb by at least 2%, so we'll maintain it delta-hedged and watch IV.
Because the underlying moves continually, the option's delta moves continuously. A trader can short/long 5 AAPL shares at most. Paper trading lets you practice delta-hedging. Being quick-footed will help with this tactic.
Profit-Closing
As expected, implied volatility rose. By 10 minutes before market closure, the call's implied vol rose to 27% and the put's to 24%. This allowed us to sell the call for $4.95 and the put for $4.35, creating a profit of $165.
You may pull historical data to see how this trade performed. Note the implied volatility and pricing in the final options chain for August 5, 2022 (the position date).
Final Thoughts
Congratulations, that was a doozy. To reiterate, we identified tickers prone to increased implied volatility by screening OpenBB's low IV setting. We double-checked the IV by plugging the price into Options-BAW Quant's model. When volatility was off, we bought a straddle and delta-hedged it. Finally, implied volatility returned to a normal level, and we profited on the spread.
The retail trading space is very quickly catching up to that of institutions. Commissions and fees used to kill this method, but now they cost less than $5. Watching momentum, technical analysis, and now quantitative strategies evolve is intriguing.
I'm not linked with these sites and receive no financial benefit from my writing.
Tell me how your experience goes and how I helped; I love success tales.
Jon Brosio
3 years ago
This Landing Page is a (Legal) Money-Printing Machine
and it’s easy to build.
A landing page with good copy is a money-maker.
Let's be honest, page-builder templates are garbage.
They can help you create a nice-looking landing page, but not persuasive writing.
Over the previous 90 days, I've examined 200+ landing pages.
What's crazy?
Top digital entrepreneurs use a 7-part strategy to bring in email subscribers, generate prospects, and (passively) sell their digital courses.
Steal this 7-part landing page architecture to maximize digital product sales.
The offer
Landing pages require offers.
Newsletter, cohort, or course offer.
Your reader should see this offer first. Includind:
Headline
Imagery
Call-to-action
Clear, persuasive, and simplicity are key. Example: the Linkedin OS course home page of digital entrepreneur Justin Welsh offers:
A distinctly defined problem
Everyone needs an enemy.
You need an opponent on your landing page. Problematic.
Next, employ psychology to create a struggle in your visitor's thoughts.
Don't be clever here; label your customer's problem. The more particular you are, the bigger the situation will seem.
When you build a clear monster, you invite defeat. I appreciate Theo Ohene's Growth Roadmaps landing page.
Exacerbation of the effects
Problem identification doesn't motivate action.
What would an unresolved problem mean?
This is landing page copy. When you describe the unsolved problem's repercussions, you accomplish several things:
You write a narrative (and stories are remembered better than stats)
You cause the reader to feel something.
You help the reader relate to the issue
Important!
My favorite script is:
"Sure, you can let [problem] go untreated. But what will happen if you do? Soon, you'll begin to notice [new problem 1] will start to arise. That might bring up [problem 2], etc."
Take the copywriting course, digital writer and entrepreneur Dickie Bush illustrates below when he labels the problem (see: "poor habit") and then illustrates the repercussions.
The tale of transformation
Every landing page needs that "ah-ha!" moment.
Transformation stories do this.
Did you find a solution? Someone else made the discovery? Have you tested your theory?
Next, describe your (or your subject's) metamorphosis.
Kieran Drew nails his narrative (and revelation) here. Right before the disclosure, he introduces his "ah-ha!" moment:
Testimonials
Social proof completes any landing page.
Social proof tells the reader, "If others do it, it must be worthwhile."
This is your argument.
Positive social proof helps (obviously).
Offer "free" training in exchange for a testimonial if you need social evidence. This builds social proof.
Most social proof is testimonies (recommended). Kurtis Hanni's creative take on social proof (using a screenshot of his colleague) is entertaining.
Bravo.
Reveal your offer
Now's the moment to act.
Describe the "bundle" that provides the transformation.
Here's:
Course
Cohort
Ebook
Whatever you're selling.
Include a product or service image, what the consumer is getting ("how it works"), the price, any "free" bonuses (preferred), and a CTA ("buy now").
Clarity is key. Don't make a cunning offer. Make sure your presentation emphasizes customer change (benefits). Dan Koe's Modern Mastery landing page makes an offer. Consider:
An ultimatum
Offering isn't enough.
You must give your prospect an ultimatum.
They can buy your merchandise from you.
They may exit the webpage.
That’s it.
It's crucial to show what happens if the reader does either. Stress the consequences of not buying (again, a little consequence amplification). Remind them of the benefits of buying.
I appreciate Charles Miller's product offer ending:
The top online creators use a 7-part landing page structure:
Offer the service
Describe the problem
Amplify the consequences
Tell the transformational story
Include testimonials and social proof.
Reveal the offer (with any bonuses if applicable)
Finally, give the reader a deadline to encourage them to take action.
Sequence these sections to develop a landing page that (essentially) prints money.
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.
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?