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
Marco Manoppo
3 years ago
Failures of DCG and Genesis
Don't sleep with your own sister.
70% of lottery winners go broke within five years. You've heard the last one. People who got rich quickly without setbacks and hard work often lose it all. My father said, "Easy money is easily lost," and a wealthy friend who owns a family office said, "The first generation makes it, the second generation spends it, and the third generation blows it."
This is evident. Corrupt politicians in developing countries live lavishly, buying their third wives' fifth Hermès bag and celebrating New Year's at The Brando Resort. A successful businessperson from humble beginnings is more conservative with money. More so if they're atom-based, not bit-based. They value money.
Crypto can "feel" easy. I have nothing against capital market investing. The global financial system is shady, but that's another topic. The problem started when those who took advantage of easy money started affecting other businesses. VCs did minimal due diligence on FTX because they needed deal flow and returns for their LPs. Lenders did minimum diligence and underwrote ludicrous loans to 3AC because they needed revenue.
Alameda (hence FTX) and 3AC made "easy money" Genesis and DCG aren't. Their businesses are more conventional, but they underestimated how "easy money" can hurt them.
Genesis has been the victim of easy money hubris and insolvency, losing $1 billion+ to 3AC and $200M to FTX. We discuss the implications for the broader crypto market.
Here are the quick takeaways:
Genesis is one of the largest and most notable crypto lenders and prime brokerage firms.
DCG and Genesis have done related party transactions, which can be done right but is a bad practice.
Genesis owes DCG $1.5 billion+.
If DCG unwinds Grayscale's GBTC, $9-10 billion in BTC will hit the market.
DCG will survive Genesis.
What happened?
Let's recap the FTX shenanigan from two weeks ago. Shenanigans! Delphi's tweet sums up the craziness. Genesis has $175M in FTX.
Cred's timeline: I hate bad crisis management. Yes, admitting their balance sheet hole right away might've sparked more panic, and there's no easy way to convey your trouble, but no one ever learns.
By November 23, rumors circulated online that the problem could affect Genesis' parent company, DCG. To address this, Barry Silbert, Founder, and CEO of DCG released a statement to shareholders.
A few things are confirmed thanks to this statement.
DCG owes $1.5 billion+ to Genesis.
$500M is due in 6 months, and the rest is due in 2032 (yes, that’s not a typo).
Unless Barry raises new cash, his last-ditch efforts to repay the money will likely push the crypto market lower.
Half a year of GBTC fees is approximately $100M.
They can pay $500M with GBTC.
With profits, sell another port.
Genesis has hired a restructuring adviser, indicating it is in trouble.
Rehypothecation
Every crypto problem in the past year seems to be rehypothecation between related parties, excessive leverage, hubris, and the removal of the money printer. The Bankless guys provided a chart showing 2021 crypto yield.
In June 2022, @DataFinnovation published a great investigation about 3AC and DCG. Here's a summary.
3AC borrowed BTC from Genesis and pledged it to create Grayscale's GBTC shares.
3AC uses GBTC to borrow more money from Genesis.
This lets 3AC leverage their capital.
3AC's strategy made sense because GBTC had a premium, creating "free money."
GBTC's discount and LUNA's implosion caused problems.
3AC lost its loan money in LUNA.
Margin called on 3ACs' GBTC collateral.
DCG bought GBTC to avoid a systemic collapse and a larger discount.
Genesis lost too much money because 3AC can't pay back its loan. DCG "saved" Genesis, but the FTX collapse hurt Genesis further, forcing DCG and Genesis to seek external funding.
bruh…
Learning Experience
Co-borrowing. Unnecessary rehypothecation. Extra space. Governance disaster. Greed, hubris. Crypto has repeatedly shown it can recreate traditional financial system disasters quickly. Working in crypto is one of the best ways to learn crazy financial tricks people will do for a quick buck much faster than if you dabble in traditional finance.
Moving Forward
I think the crypto industry needs to consider its future. This is especially true for professionals. I'm not trying to scare you. In 2018 and 2020, I had doubts. No doubts now. Detailing the crypto industry's potential outcomes helped me gain certainty and confidence in its future. This includes VCs' benefits and talking points during the bull market, as well as what would happen if government regulations became hostile, etc. Even if that happens, I'm certain. This is permanent. I may write a post about that soon.
Sincerely,
M.
Vitalik
4 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3
Ben
3 years ago
The Real Value of Carbon Credit (Climate Coin Investment)
Disclaimer : This is not financial advice for any investment.
TL;DR
You might not have realized it, but as we move toward net zero carbon emissions, the globe is already at war.
According to the Paris Agreement of COP26, 64% of nations have already declared net zero, and the issue of carbon reduction has already become so important for businesses that it affects their ability to survive. Furthermore, the time when carbon emission standards will be defined and controlled on an individual basis is becoming closer.
Since 2017, the market for carbon credits has experienced extraordinary expansion as a result of widespread talks about carbon credits. The carbon credit market is predicted to expand much more once net zero is implemented and carbon emission rules inevitably tighten.
Hello! Ben here from Nonce Classic. Nonce Classic has recently confirmed the tremendous growth potential of the carbon credit market in the midst of a major trend towards the global goal of net zero (carbon emissions caused by humans — carbon reduction by humans = 0 ). Moreover, we too believed that the questions and issues the carbon credit market suffered from the last 30–40yrs could be perfectly answered through crypto technology and that is why we have added a carbon credit crypto project to the Nonce Classic portfolio. There have been many teams out there that have tried to solve environmental problems through crypto but very few that have measurable experience working in the carbon credit scene. Thus we have put in our efforts to find projects that are not crypto projects created for the sake of issuing tokens but projects that pragmatically use crypto technology to combat climate change by solving problems of the current carbon credit market. In that process, we came to hear of Climate Coin, a veritable carbon credit crypto project, and us Nonce Classic as an accelerator, have begun contributing to its growth and invested in its tokens. Starting with this article, we plan to publish a series of articles explaining why the carbon credit market is bullish, why we invested in Climate Coin, and what kind of project Climate Coin is specifically. In this first article let us understand the carbon credit market and look into its growth potential! Let’s begin :)
The Unavoidable Entry of the Net Zero Era
Net zero means... Human carbon emissions are balanced by carbon reduction efforts. A non-environmentalist may find it hard to accept that net zero is attainable by 2050. Global cooperation to save the earth is happening faster than we imagine.
In the Paris Agreement of COP26, concluded in Glasgow, UK on Oct. 31, 2021, nations pledged to reduce worldwide yearly greenhouse gas emissions by more than 50% by 2030 and attain net zero by 2050. Governments throughout the world have pledged net zero at the national level and are holding each other accountable by submitting Nationally Determined Contributions (NDC) every five years to assess implementation. 127 of 198 nations have declared net zero.
Each country's 1.5-degree reduction plans have led to carbon reduction obligations for companies. In places with the strictest environmental regulations, like the EU, companies often face bankruptcy because the cost of buying carbon credits to meet their carbon allowances exceeds their operating profits. In this day and age, minimizing carbon emissions and securing carbon credits are crucial.
Recent SEC actions on climate change may increase companies' concerns about reducing emissions. The SEC required all U.S. stock market companies to disclose their annual greenhouse gas emissions and climate change impact on March 21, 2022. The SEC prepared the proposed regulation through in-depth analysis and stakeholder input since last year. Three out of four SEC members agreed that it should pass without major changes. If the regulation passes, it will affect not only US companies, but also countless companies around the world, directly or indirectly.
Even companies not listed on the U.S. stock market will be affected and, in most cases, required to disclose emissions. Companies listed on the U.S. stock market with significant greenhouse gas emissions or specific targets are subject to stricter emission standards (Scope 3) and disclosure obligations, which will magnify investigations into all related companies. Greenhouse gas emissions can be calculated three ways. Scope 1 measures carbon emissions from a company's facilities and transportation. Scope 2 measures carbon emissions from energy purchases. Scope 3 covers all indirect emissions from a company's value chains.
The SEC's proposed carbon emission disclosure mandate and regulations are one example of how carbon credit policies can cross borders and affect all parties. As such incidents will continue throughout the implementation of net zero, even companies that are not immediately obligated to disclose their carbon emissions must be prepared to respond to changes in carbon emission laws and policies.
Carbon reduction obligations will soon become individual. Individual consumption has increased dramatically with improved quality of life and convenience, despite national and corporate efforts to reduce carbon emissions. Since consumption is directly related to carbon emissions, increasing consumption increases carbon emissions. Countries around the world have agreed that to achieve net zero, carbon emissions must be reduced on an individual level. Solutions to individual carbon reduction are being actively discussed and studied under the term Personal Carbon Trading (PCT).
PCT is a system that allows individuals to trade carbon emission quotas in the form of carbon credits. Individuals who emit more carbon than their allotment can buy carbon credits from those who emit less. European cities with well-established carbon credit markets are preparing for net zero by conducting early carbon reduction prototype projects. The era of checking product labels for carbon footprints, choosing low-emissions transportation, and worrying about hot shower emissions is closer than we think.
The Market for Carbon Credits Is Expanding Fearfully
Compliance and voluntary carbon markets make up the carbon credit market.
A Compliance Market enforces carbon emission allowances for actors. Companies in industries that previously emitted a lot of carbon are included in the mandatory carbon market, and each government receives carbon credits each year. If a company's emissions are less than the assigned cap and it has extra carbon credits, it can sell them to other companies that have larger emissions and require them (Cap and Trade). The annual number of free emission permits provided to companies is designed to decline, therefore companies' desire for carbon credits will increase. The compliance market's yearly trading volume will exceed $261B in 2020, five times its 2017 level.
In the Voluntary Market, carbon reduction is voluntary and carbon credits are sold for personal reasons or to build market participants' eco-friendly reputations. Even if not in the compliance market, it is typical for a corporation to be obliged to offset its carbon emissions by acquiring voluntary carbon credits. When a company seeks government or company investment, it may be denied because it is not net zero. If a significant shareholder declares net zero, the companies below it must execute it. As the world moves toward ESG management, becoming an eco-friendly company is no longer a strategic choice to gain a competitive edge, but an important precaution to not fall behind. Due to this eco-friendly trend, the annual market volume of voluntary emission credits will approach $1B by November 2021. The voluntary credit market is anticipated to reach $5B to $50B by 2030. (TSCVM 2021 Report)
In conclusion
This article analyzed how net zero, a target promised by countries around the world to combat climate change, has brought governmental, corporate, and human changes. We discussed how these shifts will become more obvious as we approach net zero, and how the carbon credit market would increase exponentially in response. In the following piece, let's analyze the hurdles impeding the carbon credit market's growth, how the project we invested in tries to tackle these issues, and why we chose Climate Coin. Wait! Jim Skea, co-chair of the IPCC working group, said,
“It’s now or never, if we want to limit global warming to 1.5°C” — Jim Skea
Join nonceClassic’s community:
Telegram: https://t.me/non_stock
Youtube: https://www.youtube.com/channel/UCqeaLwkZbEfsX35xhnLU2VA
Twitter: @nonceclassic
Mail us : general@nonceclassic.org
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Sony and LEGO raise $2 billion for Epic Games' metaverse
‘Kid-friendly’ project holds $32 billion valuation
Epic Games announced today that it has raised $2 billion USD from Sony Group Corporation and KIRKBI (holding company of The LEGO Group). Both companies contributed $1 billion to Epic Games' upcoming ‘metaverse' project.
“We need partners who share our vision as we reimagine entertainment and play. Our partnership with Sony and KIRKBI has found this,” said Epic Games CEO Tim Sweeney. A new metaverse will be built where players can have fun with friends and brands create creative and immersive experiences, as well as creators thrive.
Last week, LEGO and Epic Games announced their plans to create a family-friendly metaverse where kids can play, interact, and create in digital environments. The service's users' safety and security will be prioritized.
With this new round of funding, Epic Games' project is now valued at $32 billion.
“Epic Games is known for empowering creators large and small,” said KIRKBI CEO Sren Thorup Srensen. “We invest in trends that we believe will impact the world we and our children will live in. We are pleased to invest in Epic Games to support their continued growth journey, with a long-term focus on the future metaverse.”
Epic Games is expected to unveil its metaverse plans later this year, including its name, details, services, and release date.
Alex Mathers
3 years ago
8 guidelines to help you achieve your objectives 5x fast
If you waste time every day, even though you're ambitious, you're not alone.
Many of us could use some new time-management strategies, like these:
Focus on the following three.
You're thinking about everything at once.
You're overpowered.
It's mental. We just have what's in front of us. So savor the moment's beauty.
Prioritize 1-3 things.
To be one of the most productive people you and I know, follow these steps.
Get along with boredom.
Many of us grow bored, sweat, and turn on Netflix.
We shout, "I'm rarely bored!" Look at me! I'm happy.
Shut it, Sally.
You're not making wonderful things for the world. Boredom matters.
If you can sit with it for a second, you'll get insight. Boredom? Breathe.
Go blank.
Then watch your creativity grow.
Check your MacroVision once more.
We don't know what to do with our time, which contributes to time-wasting.
Nobody does, either. Jeff Bezos won't hand-deliver that crap to you.
Daily vision checks are required.
Also:
What are 5 things you'd love to create in the next 5 years?
You're soul-searching. It's food.
Return here regularly, and you'll adore the high you get from doing valuable work.
Improve your thinking.
What's Alex's latest nonsense?
I'm talking about overcoming our own thoughts. Worrying wastes so much time.
Too many of us are assaulted by lies, myths, and insecurity.
Stop letting your worries massage you into a worried coma like a Thai woman.
Optimizing your thoughts requires accepting what you can't control.
It means letting go of unhelpful thoughts and returning to the moment.
Keep your blood sugar level.
I gave up gluten, donuts, and sweets.
This has really boosted my energy.
Blood-sugar-spiking carbs make us irritable and tired.
These day-to-day ups and downs aren't productive. It's crucial.
Know how your diet affects insulin levels. Now I have more energy and can do more without clenching my teeth.
Reduce harmful carbs to boost energy.
Create a focused setting for yourself.
When we optimize the mind, we have more energy and use our time better because we're not tense.
Changing our environment can also help us focus. Disabling alerts is one example.
Too hot makes me procrastinate and irritable.
List five items that hinder your productivity.
You may be amazed at how much you may improve by removing distractions.
Be responsible.
Accountability is a time-saver.
Creating an emotional pull to finish things.
Writing down our goals makes us accountable.
We can engage a coach or work with an accountability partner to feel horrible if we don't show up and finish on time.
‘Hey Jake, I’m going to write 1000 words every day for 30 days — you need to make sure I do.’ ‘Sure thing, Nathan, I’ll be making sure you check in daily with me.’
Tick.
You might also blog about your ambitions to show your dedication.
Now you can't hide when you promised to appear.
Acquire a liking for bravery.
Boldness changes everything.
I sometimes feel lazy and wonder why. If my food and sleep are in order, I should assess my footing.
Most of us live backward. Doubtful. Uncertain. Feelings govern us.
Backfooting isn't living. It's lame, and you'll soon melt. Live boldly now.
Be assertive.
Get disgustingly into everything. Expand.
Even if it's hard, stop being a b*tch.
Those that make Mr. Bold Bear their spirit animal benefit. Save time to maximize your effect.
Jerry Keszka
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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/
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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.
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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.
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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.
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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.
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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.
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Whether you want to acquire a new skill or increase your knowledge, you'll find something. New courses bring variety.
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Rome2rio will show you the best method to get there, including which mode of transport is ideal.
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Do you know any free, useful websites?