Use popular Ethereum dapps with Coinbase’s new dapp wallet and browser

Decentralized autonomous organizations (DAOs) and decentralized finance (DeFi) have gained popularity in the last year (DAOs). The total value locked (TVL) of DeFi investments on the Ethereum blockchain has grown to over $110B USD, while NFTs sales have grown to over $30B USD in the last 12 months (LTM). New innovative real-world applications are emerging every day.

Today, a small group of Coinbase app users can access Ethereum-based dapps. Buying NFTs on Coinbase NFT and OpenSea, trading on Uniswap and Sushiswap, and borrowing and lending on Curve and Compound are examples.

Our new dapp wallet and dapp browser enable you to access and explore web3 directly from your Coinbase app.

Web3 in the Coinbase app

Users can now access dapps without a recovery phrase. This innovative dapp wallet experience uses Multi-Party Computation (MPC) technology to secure your on-chain wallet. This wallet's design allows you and Coinbase to share the 'key.' If you lose access to your device, the key to your dapp wallet is still safe and Coinbase can help recover it.

Set up your new dapp wallet by clicking the "Browser" tab in the Android app's navigation bar. Once set up, the Coinbase app's new dapp browser lets you search, discover, and use Ethereum-based dapps.

Looking forward

We want to enable everyone to seamlessly and safely participate in web3, and today’s launch is another step on that journey. We're rolling out the new dapp wallet and browser in the US on Android first to a small subset of users and plan to expand soon. Stay tuned!

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.

Many believe Putin is simply sabre rattling and intimidating us. They see no threat of nuclear war. We can send NATO troops into Ukraine without risking a nuclear war.

I keep reading that Putin is just using nuclear blackmail and that a strong leader will call the bluff. That, in my opinion, misunderstands the danger of sending NATO into Ukraine.
It assumes that once NATO moves in, Putin can either push the red nuclear button or not.
Sure, Putin won't go nuclear if NATO invades Ukraine. So we're safe? Can't we just move NATO?

No, because history has taught us that wars often escalate far beyond our initial expectations. One domino falls, knocking down another. That's why having clear boundaries is vital. Crossing a seemingly harmless line can set off a chain of events that are unstoppable once started.
One example is WWI. The assassin of Archduke Franz Ferdinand could not have known that his actions would kill millions. They couldn't have known that invading Serbia to punish them for not handing over the accomplices would start a world war. Every action triggered a counter-action, plunging Europe into a brutal and bloody war. Each leader saw their actions as limited, not realizing how they kept the dominos falling.

Nobody can predict the future, but it's easy to imagine how NATO intervention could trigger a chain of events leading to a total war. Let me suggest some outcomes.
NATO creates a no-fly-zone. In retaliation, Russia bombs NATO airfields. Russia may see this as a limited counter-move that shouldn't cause further NATO escalation. They think it's a reasonable response to force NATO out of Ukraine. Nobody has yet thought to use the nuke.
Will NATO act? Polish airfields bombed, will they be stuck? Is this an article 5 event? If so, what should be done?

It could happen. Maybe NATO sends troops into Ukraine to punish Russia. Maybe NATO will bomb Russian airfields.

Putin's response Is bombing Russian airfields an invasion or an attack? Remember that Russia has always used nuclear weapons for defense, not offense. But let's not panic, let's assume Russia doesn't go nuclear.

Maybe Russia retaliates by attacking NATO military bases with planes. Maybe they use ships to attack military targets. How does NATO respond? Will they fight Russia in Ukraine or escalate? Will they invade Russia or attack more military installations there?
Seen the pattern? As each nation responds, smaller limited military operations can grow in scope.

So far, the Russian military has shown that they begin with less brutal methods. As losses and failures increase, brutal means are used. Syria had the same. Assad used chemical weapons and attacked hospitals, schools, residential areas, etc.
A NATO invasion of Ukraine would cost Russia dearly. “Oh, this isn't looking so good, better pull out and finish this war,” do you think? No way. Desperate, they will resort to more brutal tactics. If desperate, Russia has a huge arsenal of ugly weapons. They have nerve agents, chemical weapons, and other nasty stuff.

What happens if Russia uses chemical weapons? What if Russian nerve agents kill NATO soldiers horribly? West calls for retaliation will grow. Will we invade Russia? Will we bomb them?

We are angry and determined to punish war criminal Putin, so NATO tanks may be heading to Moscow. We want vengeance for his chemical attacks and bombing of our cities.
Do you think the distance between that red nuclear button and Putin's finger will be that far once NATO tanks are on their way to Moscow?

We might avoid a nuclear apocalypse. A NATO invasion force or even Western cities may be used by Putin. Not as destructive as ICBMs. Putin may think we won't respond to tactical nukes with a full nuclear counterattack. Why would we risk a nuclear Holocaust by launching ICBMs on Russia?

Maybe. My point is that at every stage of the escalation, one party may underestimate the other's response. This war is spiraling out of control and the chances of a nuclear exchange are increasing. Nobody really wants it.

Fear, anger, and resentment cause it. If Putin and his inner circle decide their time is up, they may no longer care about the rest of the world. We saw it with Hitler. Hitler, seeing the end of his empire, ordered the destruction of Germany. Nobody should win if he couldn't. He wanted to destroy everything, including Paris.

In other words, the danger isn't what happens after NATO intervenes The danger is the potential chain reaction. Gambling has a psychological equivalent. It's best to exit when you've lost less. We humans are willing to take small risks for big rewards. To avoid losses, we are willing to take high risks. Daniel Kahneman describes this behavior in his book Thinking, Fast and Slow.

And so bettors who have lost a lot begin taking bigger risks to make up for it. We get a snowball effect. NATO involvement in the Ukraine conflict is akin to entering a casino and placing a bet. We'll start taking bigger risks as we start losing to Russian retaliation. That's the game's psychology.

It's impossible to stop. So will politicians and citizens from both Russia and the West, until we risk the end of human civilization.

You can avoid spiraling into ever larger bets in the Casino by drawing a hard line and declaring “I will not enter that Casino.” We're doing it now. We supply Ukraine. We send money and intelligence but don't cross that crucial line.

It's difficult to watch what happened in Bucha without demanding NATO involvement. What should we do? Of course, I'm not in charge. I'm a writer. My hope is that people will think about the consequences of the actions we demand. My hope is that you think ahead not just one step but multiple dominos.

More and more, we are driven by our emotions. We cannot act solely on emotion in matters of life and death. If we make the wrong choice, more people will die.

Read the original post here.