Quantum Cryptography Explained CryptoCoins Info Club

Technical: Confidential Transactions and Their Implementation Tradeoffs

As requested by estradata here: https://old.reddit.com/Bitcoin/comments/iylou9/what_are_some_of_the_latest_innovations_in_the/g6heez1/
It is a general issue that crops up at the extremes of cryptography, with quantum breaks being just one of the extremes of (classical) cryptography.

Computational vs Information-Theoretic

The dichotomy is between computationally infeasible vs informationally-theoretic infeasible. Basically:
Quantum breaks represent a possible reduction in computational infeasibility of certain things, but not information-theoretic infeasibility.
For example, suppose you want to know what 256-bit preimages map to 256-bit hashes. In theory, you just need to build a table with 2256 entries and start from 0x0000000000000000000000000000000000000000000000000000000000000000 and so on. This is computationally infeasible, but not information-theoretic infeasible.
However, suppose you want to know what preimages, of any size, map to 256-bit hashes. Since the preimages can be of any size, after finishing with 256-bit preimages, you have to proceed to 257-bit preimages. And so on. And there is no size limit, so you will literally never finish. Even if you lived forever, you would not complete it. This is information-theoretic infeasible.

Commitments

How does this relate to confidential transactions? Basically, every confidential transaction simply hides the value behind a homomorphic commitment. What is a homomorphic commitment? Okay, let's start with commitments. A commitment is something which lets you hide something, and later reveal what you hid. Until you reveal it, even if somebody has access to the commitment, they cannot reverse it to find out what you hid. This is called the "hiding property" of commitments. However, when you do reveal it (or "open the commitment"), then you cannot replace what you hid with some other thing. This is called the "binding property" of commitments.
For example, a hash of a preimage is a commitment. Suppose I want to commit to something. For example, I want to show that I can predict the future using the energy of a spare galaxy I have in my pocket. I can hide that something by hashing a description of the future. Then I can give the hash to you. You still cannot learn the future, because it's just a hash, and you can't reverse the hash ("hiding"). But suppose the future event occurs. I can reveal that I did, in fact, know the future. So I give you the description, and you hash it and compare it to the hash I gave earlier. Because of preimage resistance, I cannot retroactively change what I hid in the hash, so what I gave must have been known to me at the time that I gave you the commitment i..e. hash ("binding").

Homomorphic Commitments

A homomorphic commitment simply means that if I can do certain operations on preimages of the commitment scheme, there are certain operations on the commitments that would create similar ("homo") changes ("morphic") to the commitments. For example, suppose I have a magical function h() which is a homomorphic commitment scheme. It can hide very large (near 256-bit) numbers. Then if h() is homomorphic, there may be certain operations on numbers behind the h() that have homomorphisms after the h(). For example, I might have an operation <+> that is homomorphic in h() on +, or in other words, if I have two large numbers a and b, then h(a + b) = h(a) <+> h(b). + and <+> are different operations, but they are homomorphic to each other.
For example, elliptic curve scalars and points have homomorphic operations. Scalars (private keys) are "just" very large near-256-bit numbers, while points are a scalar times a standard generator point G. Elliptic curve operations exist where there is a <+> between points that is homomorphic on standard + on scalars, and a <*> between a scalar and a point that is homomorphic on standard * multiplication on scalars.
For example, suppose I have two large scalars a and b. I can use elliptic curve points as a commitment scheme: I can take a <*> G to generate a point A. It is hiding since nobody can learn what a is unless I reveal it (a and A can be used in standard ECDSA private-public key cryptography, with the scalar a as the private key and the point A as the public key, and the a cannot be derived even if somebody else knows A). Thus, it is hiding. At the same time, for a particular point A and standard generator point G, there is only one possible scalar a which when "multiplied" with G yields A. So scalars and elliptic curve points are a commitment scheme, with both hiding and binding properties.
Now, as mentioned there is a <+> operation on points that is homomorphic to the + operation on corresponding scalars. For example, suppose there are two scalars a and b. I can compute (a + b) <*> G to generate a particular point. But even if I don't know scalars a and b, but I do know points A = a <*> G and B = b <*> G, then I can use A <+> B to derive (a + b) <*> G (or equivalently, (a <*> G) <+> (b <*> G) == (a + b) <*> G). This makes points a homomorphic commitment scheme on scalars.

Confidential Transactions: A Sketch

This is useful since we can easily use the near-256-bit scalars in SECP256K1 elliptic curves to easily represent values in a monetary system, and hide those values by using a homomorphic commitment scheme. We can use the hiding property to prevent people from learning the values of the money we are sending and receiving.
Now, in a proper cryptocurrency, a normal, non-coinbase transaction does not create or destroy coins: the values of the input coins are equal to the value of the output coins. We can use a homomorphic commitment scheme. Suppose I have a transaction that consumes an input value a and creates two output values b and c. That is, a = b + c, i.e. the sum of all inputs a equals the sum of all outputs b and c. But remember, with a homomorphic commitment scheme like elliptic curve points, there exists a <+> operation on points that is homomorphic to the ordinary school-arithmetic + addition on large numbers. So, confidential transactions can use points a <*> G as input, and points b <*> G and c <*> G as output, and we can easily prove that a <*> G = (b <*> G) <+> (c <*> G) if a = b + c, without revealing a, b, or c to anyone.

Pedersen Commitments

Actually, we cannot just use a <*> G as a commitment scheme in practice. Remember, Bitcoin has a cap on the number of satoshis ever to be created, and it's less than 253 satoshis, which is fairly trivial. I can easily compute all values of a <*> G for all values of a from 0 to 253 and know which a <*> G corresponds to which actual amount a. So in confidential transactions, we cannot naively use a <*> G commitments, we need Pedersen commitments.
If you know what a "salt" is, then Pedersen commitments are fairly obvious. A "salt" is something you add to e.g. a password so that the hash of the password is much harder to attack. Humans are idiots and when asked to generate passwords, will output a password that takes less than 230 possibilities, which is fairly easy to grind. So what you do is that you "salt" a password by prepending a random string to it. You then hash the random string + password, and store the random string --- the salt --- together with the hash in your database. Then when somebody logs in, you take the password, prepend the salt, hash, and check if the hash matches with the in-database hash, and you let them log in. Now, with a hash, even if somebody copies your password database, the can't get the password. They're hashed. But with a salt, even techniques like rainbow tables make a hacker's life even harder. They can't hash a possible password and check every hash in your db for something that matches. Instead, if they get a possible password, they have to prepend each salt, hash, then compare. That greatly increases the computational needs of a hacker, which is why salts are good.
What a Pedersen commitment is, is a point a <*> H, where a is the actual value you commit to, plus <+> another point r <*> G. H here is a second standard generator point, different from G. The r is the salt in the Pedersen commitment. It makes it so that even if you show (a <*> H) <+> (r <*> G) to somebody, they can't grind all possible values of a and try to match it with your point --- they also have to grind r (just as with the password-salt example above). And r is much larger, it can be a true near-256-bit number that is the range of scalars in SECP256K1, whereas a is constrained to "reasonable" numbers of satoshi, which cannot exceed 21 million Bitcoins.
Now, in order to validate a transaction with input a and outputs b and c, you only have to prove a = b + c. Suppose we are hiding those amounts using Pedersen commitments. You have an input of amount a, and you know a and r. The blockchain has an amount (a <*> H) <+> (r <*> G). In order to create the two outputs b and c, you just have to create two new r scalars such that r = r[0] + r[1]. This is trivial, you just select a new random r[0] and then compute r[1] = r - r[0], it's just basic algebra.
Then you create a transaction consuming the input (a <*> H) <+> (r <*> G) and outputs (b <*> H) <+> (r[0] <*> G) and (c <*> H) <+> (r[1] <*> G). You know that a = b + c, and r = r[0] + r[1], while fullnodes around the world, who don't know any of the amounts or scalars involved, can just take the points (a <*> H) <+> (r <*> G) and see if it equals (b <*> H) <+> (r[0] <*> G) <+> (c <*> H) <+> (r[1] <*> G). That is all that fullnodes have to validate, they just need to perform <+> operations on points and comparison on points, and from there they validate transactions, all without knowing the actual values involved.

Computational Binding, Information-Theoretic Hiding

Like all commitments, Pedersen Commitments are binding and hiding.
However, there are really two kinds of commitments:
What does this mean? It's just a measure of how "impossible" binding vs hiding is. Pedersen commitments are computationally binding, meaning that in theory, a user of this commitment with arbitrary time and space and energy can, in theory, replace the amount with something else. However, it is information-theoretic hiding, meaning an attacker with arbitrary time and space and energy cannot figure out exactly what got hidden behind the commitment.
But why?
Now, we have been using a and a <*> G as private keys and public keys in ECDSA and Schnorr. There is an operation <*> on a scalar and a point that generates another point, but we cannot "revrese" this operation. For example, even if I know A, and know that A = a <*> G, but do not know a, I cannot derive a --- there is no operation between A G that lets me know a.
Actually there is: I "just" need to have so much time, space, and energy that I just start counting a from 0 to 2256 and find which a results in A = a <*> G. This is a computational limit: I don't have a spare universe in my back pocket I can use to do all those computations.
Now, replace a with h and A with H. Remember that Pedersen commitments use a "second" standard generator point. The generator points G and H are "not really special" --- they are just random points on the curve that we selected and standardized. There is no operation H G such that I can learn h where H = h <*> G, though if I happen to have a spare universe in my back pocket I can "just" brute force it.
Suppose I do have a spare universe in my back pocket, and learn h = H G such that H = h <*> G. What can I do in Pedersen commitments?
Well, I have an amount a that is committed to by (a <*> H) <+> (r <*> G). But I happen to know h! Suppose I want to double my money a without involving Elon Musk. Then:
That is what we mean by computationally binding: if I can compute h such that H = h <*> G, then I can find another number which opens the same commitment. And of course I'd make sure that number is much larger than what I originally had in that address!
Now, the reason why it is "only" computationally binding is that it is information-theoretically hiding. Suppose somebody knows h, but has no money in the cryptocurrency. All they see are points. They can try to find what the original amounts are, but because any amount can be mapped to "the same" point with knowledge of h (e.g. in the above, a and 2 * a got mapped to the same point by "just" replacing the salt r with r - a * h; this can be done for 3 * a, 4 * a etc.), they cannot learn historical amounts --- the a in historical amounts could be anything.
The drawback, though, is that --- as seen above --- arbitrary inflation is now introduced once somebody knows h. They can multiply their money by any arbitrary factor with knowledge of h.
It is impossible to have both perfect hiding (i.e. historical amounts remain hidden even after a computational break) and perfect binding (i.e. you can't later open the commitment to a different, much larger, amount).
Pedersen commitments just happen to have perfect hiding, but only computationally-infeasible binding. This means they allow hiding historical values, but in case of anything that allows better computational power --- including but not limited to quantum breaks --- they allow arbitrary inflation.

Changing The Tradeoffs with ElGamal Commitments

An ElGamal commitment is just a Pedersen commitment, but with the point r <*> G also stored in a separate section of the transaction.
This commits the r, and fixes it to a specific value. This prevents me from opening my (a <*> H) <+> (r <*> G) as ((2 * a) <*> H) <+> ((r - a * h) <*> G), because the (r - a * h) would not match the r <*> G sitting in a separate section of the transaction. This forces me to be bound to that specific value, and no amount of computation power will let me escape --- it is information-theoretically binding i.e. perfectly binding.
But that is now computationally hiding. An evil surveillor with arbitrary time and space can focus on the r <*> G sitting in a separate section of the transaction, and grind r from 0 to 2256 to determine what r matches that point. Then from there, they can negate r to get (-r) <*> G and add it to the (a <*> H) <+> (r <*> G) to get a <*> H, and then grind that to determine the value a. With massive increases in computational ability --- including but not limited to quantum breaks --- an evil surveillor can see all the historical amounts of confidential transactions.

Conclusion

This is the source of the tradeoff: either you design confidential transactions so in case of a quantum break, historical transactions continue to hide their amounts, but inflation of the money is now unavoidable, OR you make the money supply sacrosanct, but you potentially sacrifice amount hiding in case of some break, including but not limited to quantum breaks.
submitted by almkglor to Bitcoin [link] [comments]

Technical: Taproot: Why Activate?

This is a follow-up on https://old.reddit.com/Bitcoin/comments/hqzp14/technical_the_path_to_taproot_activation/
Taproot! Everybody wants it!! But... you might ask yourself: sure, everybody else wants it, but why would I, sovereign Bitcoin HODLer, want it? Surely I can be better than everybody else because I swapped XXX fiat for Bitcoin unlike all those nocoiners?
And it is important for you to know the reasons why you, o sovereign Bitcoiner, would want Taproot activated. After all, your nodes (or the nodes your wallets use, which if you are SPV, you hopefully can pester to your wallet vendoimplementor about) need to be upgraded in order for Taproot activation to actually succeed instead of becoming a hot sticky mess.
First, let's consider some principles of Bitcoin.
I'm sure most of us here would agree that the above are very important principles of Bitcoin and that these are principles we would not be willing to remove. If anything, we would want those principles strengthened (especially the last one, financial privacy, which current Bitcoin is only sporadically strong with: you can get privacy, it just requires effort to do so).
So, how does Taproot affect those principles?

Taproot and Your /Coins

Most HODLers probably HODL their coins in singlesig addresses. Sadly, switching to Taproot would do very little for you (it gives a mild discount at spend time, at the cost of a mild increase in fee at receive time (paid by whoever sends to you, so if it's a self-send from a P2PKH or bech32 address, you pay for this); mostly a wash).
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash, so the Taproot output spends 12 bytes more; spending from a P2WPKH requires revealing a 32-byte public key later, which is not needed with Taproot, and Taproot signatures are about 9 bytes smaller than P2WPKH signatures, but the 32 bytes plus 9 bytes is divided by 4 because of the witness discount, so it saves about 11 bytes; mostly a wash, it increases blockweight by about 1 virtual byte, 4 weight for each Taproot-output-input, compared to P2WPKH-output-input).
However, as your HODLings grow in value, you might start wondering if multisignature k-of-n setups might be better for the security of your savings. And it is in multisignature that Taproot starts to give benefits!
Taproot switches to using Schnorr signing scheme. Schnorr makes key aggregation -- constructing a single public key from multiple public keys -- almost as trivial as adding numbers together. "Almost" because it involves some fairly advanced math instead of simple boring number adding, but hey when was the last time you added up your grocery list prices by hand huh?
With current P2SH and P2WSH multisignature schemes, if you have a 2-of-3 setup, then to spend, you need to provide two different signatures from two different public keys. With Taproot, you can create, using special moon math, a single public key that represents your 2-of-3 setup. Then you just put two of your devices together, have them communicate to each other (this can be done airgapped, in theory, by sending QR codes: the software to do this is not even being built yet, but that's because Taproot hasn't activated yet!), and they will make a single signature to authorize any spend from your 2-of-3 address. That's 73 witness bytes -- 18.25 virtual bytes -- of signatures you save!
And if you decide that your current setup with 1-of-1 P2PKH / P2WPKH addresses is just fine as-is: well, that's the whole point of a softfork: backwards-compatibility; you can receive from Taproot users just fine, and once your wallet is updated for Taproot-sending support, you can send to Taproot users just fine as well!
(P2WPKH and P2WSH -- SegWit v0 -- addresses start with bc1q; Taproot -- SegWit v1 --- addresses start with bc1p, in case you wanted to know the difference; in bech32 q is 0, p is 1)
Now how about HODLers who keep all, or some, of their coins on custodial services? Well, any custodial service worth its salt would be doing at least 2-of-3, or probably something even bigger, like 11-of-15. So your custodial service, if it switched to using Taproot internally, could save a lot more (imagine an 11-of-15 getting reduced from 11 signatures to just 1!), which --- we can only hope! --- should translate to lower fees and better customer service from your custodial service!
So I think we can say, very accurately, that the Bitcoin principle --- that YOU are in control of your money --- can only be helped by Taproot (if you are doing multisignature), and, because P2PKH and P2WPKH remain validly-usable addresses in a Taproot future, will not be harmed by Taproot. Its benefit to this principle might be small (it mostly only benefits multisignature users) but since it has no drawbacks with this (i.e. singlesig users can continue to use P2WPKH and P2PKH still) this is still a nice, tidy win!
(even singlesig users get a minor benefit, in that multisig users will now reduce their blockchain space footprint, so that fees can be kept low for everybody; so for example even if you have your single set of private keys engraved on titanium plates sealed in an airtight box stored in a safe buried in a desert protected by angry nomads riding giant sandworms because you're the frickin' Kwisatz Haderach, you still gain some benefit from Taproot)
And here's the important part: if P2PKH/P2WPKH is working perfectly fine with you and you decide to never use Taproot yourself, Taproot will not affect you detrimentally. First do no harm!

Taproot and Your Contracts

No one is an island, no one lives alone. Give and you shall receive. You know: by trading with other people, you can gain expertise in some obscure little necessity of the world (and greatly increase your productivity in that little field), and then trade the products of your expertise for necessities other people have created, all of you thereby gaining gains from trade.
So, contracts, which are basically enforceable agreements that facilitate trading with people who you do not personally know and therefore might not trust.
Let's start with a simple example. You want to buy some gewgaws from somebody. But you don't know them personally. The seller wants the money, you want their gewgaws, but because of the lack of trust (you don't know them!! what if they're scammers??) neither of you can benefit from gains from trade.
However, suppose both of you know of some entity that both of you trust. That entity can act as a trusted escrow. The entity provides you security: this enables the trade, allowing both of you to get gains from trade.
In Bitcoin-land, this can be implemented as a 2-of-3 multisignature. The three signatories in the multisgnature would be you, the gewgaw seller, and the escrow. You put the payment for the gewgaws into this 2-of-3 multisignature address.
Now, suppose it turns out neither of you are scammers (whaaaat!). You receive the gewgaws just fine and you're willing to pay up for them. Then you and the gewgaw seller just sign a transaction --- you and the gewgaw seller are 2, sufficient to trigger the 2-of-3 --- that spends from the 2-of-3 address to a singlesig the gewgaw seller wants (or whatever address the gewgaw seller wants).
But suppose some problem arises. The seller gave you gawgews instead of gewgaws. Or you decided to keep the gewgaws but not sign the transaction to release the funds to the seller. In either case, the escrow is notified, and if it can sign with you to refund the funds back to you (if the seller was a scammer) or it can sign with the seller to forward the funds to the seller (if you were a scammer).
Taproot helps with this: like mentioned above, it allows multisignature setups to produce only one signature, reducing blockchain space usage, and thus making contracts --- which require multiple people, by definition, you don't make contracts with yourself --- is made cheaper (which we hope enables more of these setups to happen for more gains from trade for everyone, also, moon and lambos).
(technology-wise, it's easier to make an n-of-n than a k-of-n, making a k-of-n would require a complex setup involving a long ritual with many communication rounds between the n participants, but an n-of-n can be done trivially with some moon math. You can, however, make what is effectively a 2-of-3 by using a three-branch SCRIPT: either 2-of-2 of you and seller, OR 2-of-2 of you and escrow, OR 2-of-2 of escrow and seller. Fortunately, Taproot adds a facility to embed a SCRIPT inside a public key, so you can have a 2-of-2 Taprooted address (between you and seller) with a SCRIPT branch that can instead be spent with 2-of-2 (you + escrow) OR 2-of-2 (seller + escrow), which implements the three-branched SCRIPT above. If neither of you are scammers (hopefully the common case) then you both sign using your keys and never have to contact the escrow, since you are just using the escrow public key without coordinating with them (because n-of-n is trivial but k-of-n requires setup with communication rounds), so in the "best case" where both of you are honest traders, you also get a privacy boost, in that the escrow never learns you have been trading on gewgaws, I mean ewww, gawgews are much better than gewgaws and therefore I now judge you for being a gewgaw enthusiast, you filthy gewgawer).

Taproot and Your Contracts, Part 2: Cryptographic Boogaloo

Now suppose you want to buy some data instead of things. For example, maybe you have some closed-source software in trial mode installed, and want to pay the developer for the full version. You want to pay for an activation code.
This can be done, today, by using an HTLC. The developer tells you the hash of the activation code. You pay to an HTLC, paying out to the developer if it reveals the preimage (the activation code), or refunding the money back to you after a pre-agreed timeout. If the developer claims the funds, it has to reveal the preimage, which is the activation code, and you can now activate your software. If the developer does not claim the funds by the timeout, you get refunded.
And you can do that, with HTLCs, today.
Of course, HTLCs do have problems:
Fortunately, with Schnorr (which is enabled by Taproot), we can now use the Scriptless Script constuction by Andrew Poelstra. This Scriptless Script allows a new construction, the PTLC or Pointlocked Timelocked Contract. Instead of hashes and preimages, just replace "hash" with "point" and "preimage" with "scalar".
Or as you might know them: "point" is really "public key" and "scalar" is really a "private key". What a PTLC does is that, given a particular public key, the pointlocked branch can be spent only if the spender reveals the private key of the given public key to you.
Another nice thing with PTLCs is that they are deniable. What appears onchain is just a single 2-of-2 signature between you and the developemanufacturer. It's like a magic trick. This signature has no special watermarks, it's a perfectly normal signature (the pledge). However, from this signature, plus some datta given to you by the developemanufacturer (known as the adaptor signature) you can derive the private key of a particular public key you both agree on (the turn). Anyone scraping the blockchain will just see signatures that look just like every other signature, and as long as nobody manages to hack you and get a copy of the adaptor signature or the private key, they cannot get the private key behind the public key (point) that the pointlocked branch needs (the prestige).
(Just to be clear, the public key you are getting the private key from, is distinct from the public key that the developemanufacturer will use for its funds. The activation key is different from the developer's onchain Bitcoin key, and it is the activation key whose private key you will be learning, not the developer's/manufacturer's onchain Bitcoin key).
So:
Taproot lets PTLCs exist onchain because they enable Schnorr, which is a requirement of PTLCs / Scriptless Script.
(technology-wise, take note that Scriptless Script works only for the "pointlocked" branch of the contract; you need normal Script, or a pre-signed nLockTimed transaction, for the "timelocked" branch. Since Taproot can embed a script, you can have the Taproot pubkey be a 2-of-2 to implement the Scriptless Script "pointlocked" branch, then have a hidden script that lets you recover the funds with an OP_CHECKLOCKTIMEVERIFY after the timeout if the seller does not claim the funds.)

Quantum Quibbles!

Now if you were really paying attention, you might have noticed this parenthetical:
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash...)
So wait, Taproot uses raw 32-byte public keys, and not public key hashes? Isn't that more quantum-vulnerable??
Well, in theory yes. In practice, they probably are not.
It's not that hashes can be broken by quantum computes --- they're still not. Instead, you have to look at how you spend from a P2WPKH/P2PKH pay-to-public-key-hash.
When you spend from a P2PKH / P2WPKH, you have to reveal the public key. Then Bitcoin hashes it and checks if this matches with the public-key-hash, and only then actually validates the signature for that public key.
So an unconfirmed transaction, floating in the mempools of nodes globally, will show, in plain sight for everyone to see, your public key.
(public keys should be public, that's why they're called public keys, LOL)
And if quantum computers are fast enough to be of concern, then they are probably fast enough that, in the several minutes to several hours from broadcast to confirmation, they have already cracked the public key that is openly broadcast with your transaction. The owner of the quantum computer can now replace your unconfirmed transaction with one that pays the funds to itself. Even if you did not opt-in RBF, miners are still incentivized to support RBF on RBF-disabled transactions.
So the extra hash is not as significant a protection against quantum computers as you might think. Instead, the extra hash-and-compare needed is just extra validation effort.
Further, if you have ever, in the past, spent from the address, then there exists already a transaction indelibly stored on the blockchain, openly displaying the public key from which quantum computers can derive the private key. So those are still vulnerable to quantum computers.
For the most part, the cryptographers behind Taproot (and Bitcoin Core) are of the opinion that quantum computers capable of cracking Bitcoin pubkeys are unlikely to appear within a decade or two.
So:
For now, the homomorphic and linear properties of elliptic curve cryptography provide a lot of benefits --- particularly the linearity property is what enables Scriptless Script and simple multisignature (i.e. multisignatures that are just 1 signature onchain). So it might be a good idea to take advantage of them now while we are still fairly safe against quantum computers. It seems likely that quantum-safe signature schemes are nonlinear (thus losing these advantages).

Summary

I Wanna Be The Taprooter!

So, do you want to help activate Taproot? Here's what you, mister sovereign Bitcoin HODLer, can do!

But I Hate Taproot!!

That's fine!

Discussions About Taproot Activation

submitted by almkglor to Bitcoin [link] [comments]

Quantum Resistance

Before jumping to conclusions about this post, know that I am not looking to spread any FUD but rather am trying to understand a forthcoming risk and potential solutions from an unbiased standpoint. My research has not yielded any definitive answer so I am turning here to seek direction from those more knowledgable than me.
--
When it comes to predicting quantum computing's ability to break Bitcoin cryptographically, I've seen estimates as small as two years and as large as 25 years. Either way, it is easily conceivable that quantum processors will improve to the point of threatening Bitcoin as a reliable form of currency and store of value.
One way to prevent vulnerability to quantum threats is by storing Bitcoin in an address that has only ever received Bitcoin and never sent it. Although, this is an unrealistic mitigant for an asset/currency that is intended to be bought and sold, for all trust will be lost in the network once quantum computing becomes powerful enough to hack Bitcoin. Nobody will place any value in a currency that can be hacked by sending a transaction.
Another argument I've seen is that once quantum computing is strong enough to hack Bitcoin's cryptography, Bitcoin will be a non-factor compared to the other digital security breakdowns that will have transpired. For example, nuclear codes, bank accounts, digital privacy, etc. However, those centralized networks will have the ability to preemptively update their internal security to the standard required in a quantum computing world. In a similar manner, cryptocurrency and blockchain as a whole will survive such transition via improved cryptography.
But when it comes to Bitcoin specifically, will it be possible to generate consensus among the miners to switch to a quantum resistant protocol? My research has found conflicting perspectives - one side being that in order to upgrade Bitcoin's security, it would require manual movement of coins to a new address by all users, and a burning of the coins that did not move after a "sufficient" amount of time. Burning one's assets would undoubtedly not hold in a court of law. Even if we are still several years away, an unsolvable existential threat on the horizon would be priced into the value of Bitcoin and drive it down to zero.
With that being said, are there any feasible solutions to bring Bitcoin to quantum resistance? How can Bitcoin survive this threat in the long run? What is being done currently to resolve such problem?
submitted by fuegoblue to Bitcoin [link] [comments]

Scaling Reddit Community Points with Arbitrum Rollup: a piece of cake

Scaling Reddit Community Points with Arbitrum Rollup: a piece of cake
https://preview.redd.it/b80c05tnb9e51.jpg?width=2550&format=pjpg&auto=webp&s=850282c1a3962466ed44f73886dae1c8872d0f31
Submitted for consideration to The Great Reddit Scaling Bake-Off
Baked by the pastry chefs at Offchain Labs
Please send questions or comments to [[email protected] ](mailto:[email protected])
1. Overview
We're excited to submit Arbitrum Rollup for consideration to The Great Reddit Scaling Bake-Off. Arbitrum Rollup is the only Ethereum scaling solution that supports arbitrary smart contracts without compromising on Ethereum's security or adding points of centralization. For Reddit, this means that Arbitrum can not only scale the minting and transfer of Community Points, but it can foster a creative ecosystem built around Reddit Community Points enabling points to be used in a wide variety of third party applications. That's right -- you can have your cake and eat it too!
Arbitrum Rollup isn't just Ethereum-style. Its Layer 2 transactions are byte-for-byte identical to Ethereum, which means Ethereum users can continue to use their existing addresses and wallets, and Ethereum developers can continue to use their favorite toolchains and development environments out-of-the-box with Arbitrum. Coupling Arbitrum’s tooling-compatibility with its trustless asset interoperability, Reddit not only can scale but can onboard the entire Ethereum community at no cost by giving them the same experience they already know and love (well, certainly know).
To benchmark how Arbitrum can scale Reddit Community Points, we launched the Reddit contracts on an Arbitrum Rollup chain. Since Arbitrum provides full Solidity support, we didn't have to rewrite the Reddit contracts or try to mimic their functionality using an unfamiliar paradigm. Nope, none of that. We launched the Reddit contracts unmodified on Arbitrum Rollup complete with support for minting and distributing points. Like every Arbitrum Rollup chain, the chain included a bridge interface in which users can transfer Community Points or any other asset between the L1 and L2 chains. Arbitrum Rollup chains also support dynamic contract loading, which would allow third-party developers to launch custom ecosystem apps that integrate with Community Points on the very same chain that runs the Reddit contracts.
1.1 Why Ethereum
Perhaps the most exciting benefit of distributing Community Points using a blockchain is the ability to seamlessly port points to other applications and use them in a wide variety of contexts. Applications may include simple transfers such as a restaurant that allows Redditors to spend points on drinks. Or it may include complex smart contracts -- such as placing Community Points as a wager for a multiparty game or as collateral in a financial contract.
The common denominator between all of the fun uses of Reddit points is that it needs a thriving ecosystem of both users and developers, and the Ethereum blockchain is perhaps the only smart contract platform with significant adoption today. While many Layer 1 blockchains boast lower cost or higher throughput than the Ethereum blockchain, more often than not, these attributes mask the reality of little usage, weaker security, or both.
Perhaps another platform with significant usage will rise in the future. But today, Ethereum captures the mindshare of the blockchain community, and for Community Points to provide the most utility, the Ethereum blockchain is the natural choice.
1.2 Why Arbitrum
While Ethereum's ecosystem is unmatched, the reality is that fees are high and capacity is too low to support the scale of Reddit Community Points. Enter Arbitrum. Arbitrum Rollup provides all of the ecosystem benefits of Ethereum, but with orders of magnitude more capacity and at a fraction of the cost of native Ethereum smart contracts. And most of all, we don't change the experience from users. They continue to use the same wallets, addresses, languages, and tools.
Arbitrum Rollup is not the only solution that can scale payments, but it is the only developed solution that can scale both payments and arbitrary smart contracts trustlessly, which means that third party users can build highly scalable add-on apps that can be used without withdrawing money from the Rollup chain. If you believe that Reddit users will want to use their Community Points in smart contracts--and we believe they will--then it makes the most sense to choose a single scaling solution that can support the entire ecosystem, eliminating friction for users.
We view being able to run smart contracts in the same scaling solution as fundamentally critical since if there's significant demand in running smart contracts from Reddit's ecosystem, this would be a load on Ethereum and would itself require a scaling solution. Moreover, having different scaling solutions for the minting/distribution/spending of points and for third party apps would be burdensome for users as they'd have to constantly shuffle their Points back and forth.
2. Arbitrum at a glance
Arbitrum Rollup has a unique value proposition as it offers a combination of features that no other scaling solution achieves. Here we highlight its core attributes.
Decentralized. Arbitrum Rollup is as decentralized as Ethereum. Unlike some other Layer 2 scaling projects, Arbitrum Rollup doesn't have any centralized components or centralized operators who can censor users or delay transactions. Even in non-custodial systems, centralized components provide a risk as the operators are generally incentivized to increase their profit by extracting rent from users often in ways that severely degrade user experience. Even if centralized operators are altruistic, centralized components are subject to hacking, coercion, and potential liability.
Massive Scaling. Arbitrum achieves order of magnitude scaling over Ethereum's L1 smart contracts. Our software currently supports 453 transactions-per-second for basic transactions (at 1616 Ethereum gas per tx). We have a lot of room left to optimize (e.g. aggregating signatures), and over the next several months capacity will increase significantly. As described in detail below, Arbitrum can easily support and surpass Reddit's anticipated initial load, and its capacity will continue to improve as Reddit's capacity needs grow.
Low cost. The cost of running Arbitrum Rollup is quite low compared to L1 Ethereum and other scaling solutions such as those based on zero-knowledge proofs. Layer 2 fees are low, fixed, and predictable and should not be overly burdensome for Reddit to cover. Nobody needs to use special equipment or high-end machines. Arbitrum requires validators, which is a permissionless role that can be run on any reasonable on-line machine. Although anybody can act as a validator, in order to protect against a “tragedy of the commons” and make sure reputable validators are participating, we support a notion of “invited validators” that are compensated for their costs. In general, users pay (low) fees to cover the invited validators’ costs, but we imagine that Reddit may cover this cost for its users. See more on the costs and validator options below.
Ethereum Developer Experience. Not only does Arbitrum support EVM smart contracts, but the developer experience is identical to that of L1 Ethereum contracts and fully compatible with Ethereum tooling. Developers can port existing Solidity apps or write new ones using their favorite and familiar toolchains (e.g. Truffle, Buidler). There are no new languages or coding paradigms to learn.
Ethereum wallet compatibility. Just as in Ethereum, Arbitrum users need only hold keys, but do not have to store any coin history or additional data to protect or access their funds. Since Arbitrum transactions are semantically identical to Ethereum L1 transactions, existing Ethereum users can use their existing Ethereum keys with their existing wallet software such as Metamask.
Token interoperability. Users can easily transfer their ETH, ERC-20 and ERC-721 tokens between Ethereum and the Arbitrum Rollup chain. As we explain in detail below, it is possible to mint tokens in L2 that can subsequently be withdrawn and recognized by the L1 token contract.
Fast finality. Transactions complete with the same finality time as Ethereum L1 (and it's possible to get faster finality guarantees by trading away trust assumptions; see the Arbitrum Rollup whitepaper for details).
Non-custodial. Arbitrum Rollup is a non-custodial scaling solution, so users control their funds/points and neither Reddit nor anyone else can ever access or revoke points held by users.
Censorship Resistant. Since it's completely decentralized, and the Arbitrum protocol guarantees progress trustlessly, Arbitrum Rollup is just as censorship-proof as Ethereum.
Block explorer. The Arbitrum Rollup block explorer allows users to view and analyze transactions on the Rollup chain.
Limitations
Although this is a bake-off, we're not going to sugar coat anything. Arbitrum Rollup, like any Optimistic Rollup protocol, does have one limitation, and that's the delay on withdrawals.
As for the concrete length of the delay, we've done a good deal of internal modeling and have blogged about this as well. Our current modeling suggests a 3-hour delay is sufficient (but as discussed in the linked post there is a tradeoff space between the length of the challenge period and the size of the validators’ deposit).
Note that this doesn't mean that the chain is delayed for three hours. Arbitrum Rollup supports pipelining of execution, which means that validators can keep building new states even while previous ones are “in the pipeline” for confirmation. As the challenge delays expire for each update, a new state will be confirmed (read more about this here).
So activity and progress on the chain are not delayed by the challenge period. The only thing that's delayed is the consummation of withdrawals. Recall though that any single honest validator knows immediately (at the speed of L1 finality) which state updates are correct and can guarantee that they will eventually be confirmed, so once a valid withdrawal has been requested on-chain, every honest party knows that the withdrawal will definitely happen. There's a natural place here for a liquidity market in which a validator (or someone who trusts a validator) can provide withdrawal loans for a small interest fee. This is a no-risk business for them as they know which withdrawals will be confirmed (and can force their confirmation trustlessly no matter what anyone else does) but are just waiting for on-chain finality.
3. The recipe: How Arbitrum Rollup works
For a description of the technical components of Arbitrum Rollup and how they interact to create a highly scalable protocol with a developer experience that is identical to Ethereum, please refer to the following documents:
Arbitrum Rollup Whitepaper
Arbitrum academic paper (describes a previous version of Arbitrum)
4. Developer docs and APIs
For full details about how to set up and interact with an Arbitrum Rollup chain or validator, please refer to our developer docs, which can be found at https://developer.offchainlabs.com/.
Note that the Arbitrum version described on that site is older and will soon be replaced by the version we are entering in Reddit Bake-Off, which is still undergoing internal testing before public release.
5. Who are the validators?
As with any Layer 2 protocol, advancing the protocol correctly requires at least one validator (sometimes called block producers) that is honest and available. A natural question is: who are the validators?
Recall that the validator set for an Arbitrum chain is open and permissionless; anyone can start or stop validating at will. (A useful analogy is to full nodes on an L1 chain.) But we understand that even though anyone can participate, Reddit may want to guarantee that highly reputable nodes are validating their chain. Reddit may choose to validate the chain themselves and/or hire third-party validators.To this end, we have begun building a marketplace for validator-for-hire services so that dapp developers can outsource validation services to reputable nodes with high up-time. We've announced a partnership in which Chainlink nodes will provide Arbitrum validation services, and we expect to announce more partnerships shortly with other blockchain infrastructure providers.
Although there is no requirement that validators are paid, Arbitrum’s economic model tracks validators’ costs (e.g. amount of computation and storage) and can charge small fees on user transactions, using a gas-type system, to cover those costs. Alternatively, a single party such as Reddit can agree to cover the costs of invited validators.
6. Reddit Contract Support
Since Arbitrum contracts and transactions are byte-for-byte compatible with Ethereum, supporting the Reddit contracts is as simple as launching them on an Arbitrum chain.
Minting. Arbitrum Rollup supports hybrid L1/L2 tokens which can be minted in L2 and then withdrawn onto the L1. An L1 contract at address A can make a special call to the EthBridge which deploys a "buddy contract" to the same address A on an Arbitrum chain. Since it's deployed at the same address, users can know that the L2 contract is the authorized "buddy" of the L1 contract on the Arbitrum chain.
For minting, the L1 contract is a standard ERC-20 contract which mints and burns tokens when requested by the L2 contract. It is paired with an ERC-20 contract in L2 which mints tokens based on whatever programmer provided minting facility is desired and burns tokens when they are withdrawn from the rollup chain. Given this base infrastructure, Arbitrum can support any smart contract based method for minting tokens in L2, and indeed we directly support Reddit's signature/claim based minting in L2.
Batch minting. What's better than a mint cookie? A whole batch! In addition to supporting Reddit’s current minting/claiming scheme, we built a second minting design, which we believe outperforms the signature/claim system in many scenarios.
In the current system, Reddit periodically issues signed statements to users, who then take those statements to the blockchain to claim their tokens. An alternative approach would have Reddit directly submit the list of users/amounts to the blockchain and distribute the tokens to the users without the signature/claim process.
To optimize the cost efficiency of this approach, we designed an application-specific compression scheme to minimize the size of the batch distribution list. We analyzed the data from Reddit's previous distributions and found that the data is highly compressible since token amounts are small and repeated, and addresses appear multiple times. Our function groups transactions by size, and replaces previously-seen addresses with a shorter index value. We wrote client code to compress the data, wrote a Solidity decompressing function, and integrated that function into Reddit’s contract running on Arbitrum.
When we ran the compression function on the previous Reddit distribution data, we found that we could compress batched minting data down to to 11.8 bytes per minting event (averaged over a 6-month trace of Reddit’s historical token grants)compared with roughly 174 bytes of on-chain data needed for the signature claim approach to minting (roughly 43 for an RLP-encoded null transaction + 65 for Reddit's signature + 65 for the user's signature + roughly 8 for the number of Points) .
The relative benefit of the two approaches with respect to on-chain call data cost depends on the percentage of users that will actually claim their tokens on chain. With the above figures, batch minting will be cheaper if roughly 5% of users redeem their claims. We stress that our compression scheme is not Arbitrum-specific and would be beneficial in any general-purpose smart contract platform.
8. Benchmarks and costs
In this section, we give the full costs of operating the Reddit contracts on an Arbitrum Rollup chain including the L1 gas costs for the Rollup chain, the costs of computation and storage for the L2 validators as well as the capital lockup requirements for staking.
Arbitrum Rollup is still on testnet, so we did not run mainnet benchmarks. Instead, we measured the L1 gas cost and L2 workload for Reddit operations on Arbitrum and calculated the total cost assuming current Ethereum gas prices. As noted below in detail, our measurements do not assume that Arbitrum is consuming the entire capacity of Ethereum. We will present the details of our model now, but for full transparency you can also play around with it yourself and adjust the parameters, by copying the spreadsheet found here.
Our cost model is based on measurements of Reddit’s contracts, running unmodified (except for the addition of a batch minting function) on Arbitrum Rollup on top of Ethereum.
On the distribution of transactions and frequency of assertions. Reddit's instructions specify the following minimum parameters that submissions should support:
Over a 5 day period, your scaling PoC should be able to handle:
  • 100,000 point claims (minting & distributing points)
  • 25,000 subscriptions
  • 75,000 one-off points burning
  • 100,000 transfers
We provide the full costs of operating an Arbitrum Rollup chain with this usage under the assumption that tokens are minted or granted to users in batches, but other transactions are uniformly distributed over the 5 day period. Unlike some other submissions, we do not make unrealistic assumptions that all operations can be submitted in enormous batches. We assume that batch minting is done in batches that use only a few percent on an L1 block’s gas, and that other operations come in evenly over time and are submitted in batches, with one batch every five minutes to keep latency reasonable. (Users are probably already waiting for L1 finality, which takes at least that long to achieve.)
We note that assuming that there are only 300,000 transactions that arrive uniformly over the 5 day period will make our benchmark numbers lower, but we believe that this will reflect the true cost of running the system. To see why, say that batches are submitted every five minutes (20 L1 blocks) and there's a fixed overhead of c bytes of calldata per batch, the cost of which will get amortized over all transactions executed in that batch. Assume that each individual transaction adds a marginal cost of t. Lastly assume the capacity of the scaling system is high enough that it can support all of Reddit's 300,000 transactions within a single 20-block batch (i.e. that there is more than c + 300,000*t byes of calldata available in 20 blocks).
Consider what happens if c, the per-batch overhead, is large (which it is in some systems, but not in Arbitrum). In the scenario that transactions actually arrive at the system's capacity and each batch is full, then c gets amortized over 300,000 transactions. But if we assume that the system is not running at capacity--and only receives 300,000 transactions arriving uniformly over 5 days-- then each 20-block assertion will contain about 200 transactions, and thus each transaction will pay a nontrivial cost due to c.
We are aware that other proposals presented scaling numbers assuming that 300,000 transactions arrived at maximum capacity and was executed in a single mega-transaction, but according to our estimates, for at least one such report, this led to a reported gas price that was 2-3 orders of magnitude lower than it would have been assuming uniform arrival. We make more realistic batching assumptions, and we believe Arbitrum compares well when batch sizes are realistic.
Our model. Our cost model includes several sources of cost:
  • L1 gas costs: This is the cost of posting transactions as calldata on the L1 chain, as well as the overhead associated with each batch of transactions, and the L1 cost of settling transactions in the Arbitrum protocol.
  • Validator’s staking costs: In normal operation, one validator will need to be staked. The stake is assumed to be 0.2% of the total value of the chain (which is assumed to be $1 per user who is eligible to claim points). The cost of staking is the interest that could be earned on the money if it were not staked.
  • Validator computation and storage: Every validator must do computation to track the chain’s processing of transactions, and must maintain storage to keep track of the contracts’ EVM storage. The cost of computation and storage are estimated based on measurements, with the dollar cost of resources based on Amazon Web Services pricing.
It’s clear from our modeling that the predominant cost is for L1 calldata. This will probably be true for any plausible rollup-based system.
Our model also shows that Arbitrum can scale to workloads much larger than Reddit’s nominal workload, without exhausting L1 or L2 resources. The scaling bottleneck will ultimately be calldata on the L1 chain. We believe that cost could be reduced substantially if necessary by clever encoding of data. (In our design any compression / decompression of L2 transaction calldata would be done by client software and L2 programs, never by an L1 contract.)
9. Status of Arbitrum Rollup
Arbitrum Rollup is live on Ethereum testnet. All of the code written to date including everything included in the Reddit demo is open source and permissively licensed under the Apache V2 license. The first testnet version of Arbitrum Rollup was released on testnet in February. Our current internal version, which we used to benchmark the Reddit contracts, will be released soon and will be a major upgrade.
Both the Arbitrum design as well as the implementation are heavily audited by independent third parties. The Arbitrum academic paper was published at USENIX Security, a top-tier peer-reviewed academic venue. For the Arbitrum software, we have engaged Trail of Bits for a security audit, which is currently ongoing, and we are committed to have a clean report before launching on Ethereum mainnet.
10. Reddit Universe Arbitrum Rollup Chain
The benchmarks described in this document were all measured using the latest internal build of our software. When we release the new software upgrade publicly we will launch a Reddit Universe Arbitrum Rollup chain as a public demo, which will contain the Reddit contracts as well as a Uniswap instance and a Connext Hub, demonstrating how Community Points can be integrated into third party apps. We will also allow members of the public to dynamically launch ecosystem contracts. We at Offchain Labs will cover the validating costs for the Reddit Universe public demo.
If the folks at Reddit would like to evaluate our software prior to our public demo, please email us at [email protected] and we'd be more than happy to provide early access.
11. Even more scaling: Arbitrum Sidechains
Rollups are an excellent approach to scaling, and we are excited about Arbitrum Rollup which far surpasses Reddit's scaling needs. But looking forward to Reddit's eventual goal of supporting hundreds of millions of users, there will likely come a time when Reddit needs more scaling than any Rollup protocol can provide.
While Rollups greatly reduce costs, they don't break the linear barrier. That is, all transactions have an on-chain footprint (because all calldata must be posted on-chain), albeit a far smaller one than on native Ethereum, and the L1 limitations end up being the bottleneck for capacity and cost. Since Ethereum has limited capacity, this linear use of on-chain resources means that costs will eventually increase superlinearly with traffic.
The good news is that we at Offchain Labs have a solution in our roadmap that can satisfy this extreme-scaling setting as well: Arbitrum AnyTrust Sidechains. Arbitrum Sidechains are similar to Arbitrum Rollup, but deviate in that they name a permissioned set of validators. When a chain’s validators agree off-chain, they can greatly reduce the on-chain footprint of the protocol and require almost no data to be put on-chain. When validators can't reach unanimous agreement off-chain, the protocol reverts to Arbitrum Rollup. Technically, Arbitrum Sidechains can be viewed as a hybrid between state channels and Rollup, switching back and forth as necessary, and combining the performance and cost that state channels can achieve in the optimistic case, with the robustness of Rollup in other cases. The core technical challenge is how to switch seamlessly between modes and how to guarantee that security is maintained throughout.
Arbitrum Sidechains break through this linear barrier, while still maintaining a high level of security and decentralization. Arbitrum Sidechains provide the AnyTrust guarantee, which says that as long as any one validator is honest and available (even if you don't know which one will be), the L2 chain is guaranteed to execute correctly according to its code and guaranteed to make progress. Unlike in a state channel, offchain progress does not require unanimous consent, and liveness is preserved as long as there is a single honest validator.
Note that the trust model for Arbitrum Sidechains is much stronger than for typical BFT-style chains which introduce a consensus "voting" protocols among a small permissioned group of validators. BFT-based protocols require a supermajority (more than 2/3) of validators to agree. In Arbitrum Sidechains, by contrast, all you need is a single honest validator to achieve guaranteed correctness and progress. Notice that in Arbitrum adding validators strictly increases security since the AnyTrust guarantee provides correctness as long as any one validator is honest and available. By contrast, in BFT-style protocols, adding nodes can be dangerous as a coalition of dishonest nodes can break the protocol.
Like Arbitrum Rollup, the developer and user experiences for Arbitrum Sidechains will be identical to that of Ethereum. Reddit would be able to choose a large and diverse set of validators, and all that they would need to guarantee to break through the scaling barrier is that a single one of them will remain honest.
We hope to have Arbitrum Sidechains in production in early 2021, and thus when Reddit reaches the scale that surpasses the capacity of Rollups, Arbitrum Sidechains will be waiting and ready to help.
While the idea to switch between channels and Rollup to get the best of both worlds is conceptually simple, getting the details right and making sure that the switch does not introduce any attack vectors is highly non-trivial and has been the subject of years of our research (indeed, we were working on this design for years before the term Rollup was even coined).
12. How Arbitrum compares
We include a comparison to several other categories as well as specific projects when appropriate. and explain why we believe that Arbitrum is best suited for Reddit's purposes. We focus our attention on other Ethereum projects.
Payment only Rollups. Compared to Arbitrum Rollup, ZK-Rollups and other Rollups that only support token transfers have several disadvantages:
  • As outlined throughout the proposal, we believe that the entire draw of Ethereum is in its rich smart contracts support which is simply not achievable with today's zero-knowledge proof technology. Indeed, scaling with a ZK-Rollup will add friction to the deployment of smart contracts that interact with Community Points as users will have to withdraw their coins from the ZK-Rollup and transfer them to a smart contract system (like Arbitrum). The community will be best served if Reddit builds on a platform that has built-in, frictionless smart-contract support.
  • All other Rollup protocols of which we are aware employ a centralized operator. While it's true that users retain custody of their coins, the centralized operator can often profit from censoring, reordering, or delaying transactions. A common misconception is that since they're non-custodial protocols, a centralized sequencer does not pose a risk but this is incorrect as the sequencer can wreak havoc or shake down users for side payments without directly stealing funds.
  • Sidechain type protocols can eliminate some of these issues, but they are not trustless. Instead, they require trust in some quorum of a committee, often requiring two-third of the committee to be honest, compared to rollup protocols like Arbitrum that require only a single honest party. In addition, not all sidechain type protocols have committees that are diverse, or even non-centralized, in practice.
  • Plasma-style protocols have a centralized operator and do not support general smart contracts.
13. Concluding Remarks
While it's ultimately up to the judges’ palate, we believe that Arbitrum Rollup is the bakeoff choice that Reddit kneads. We far surpass Reddit's specified workload requirement at present, have much room to optimize Arbitrum Rollup in the near term, and have a clear path to get Reddit to hundreds of millions of users. Furthermore, we are the only project that gives developers and users the identical interface as the Ethereum blockchain and is fully interoperable and tooling-compatible, and we do this all without any new trust assumptions or centralized components.
But no matter how the cookie crumbles, we're glad to have participated in this bake-off and we thank you for your consideration.
About Offchain Labs
Offchain Labs, Inc. is a venture-funded New York company that spun out of Princeton University research, and is building the Arbitrum platform to usher in the next generation of scalable, interoperable, and compatible smart contracts. Offchain Labs is backed by Pantera Capital, Compound VC, Coinbase Ventures, and others.
Leadership Team
Ed Felten
Ed Felten is Co-founder and Chief Scientist at Offchain Labs. He is on leave from Princeton University, where he is the Robert E. Kahn Professor of Computer Science and Public Affairs. From 2015 to 2017 he served at the White House as Deputy United States Chief Technology Officer and senior advisor to the President. He is an ACM Fellow and member of the National Academy of Engineering. Outside of work, he is an avid runner, cook, and L.A. Dodgers fan.
Steven Goldfeder
Steven Goldfeder is Co-founder and Chief Executive Officer at Offchain Labs. He holds a PhD from Princeton University, where he worked at the intersection of cryptography and cryptocurrencies including threshold cryptography, zero-knowledge proof systems, and post-quantum signatures. He is a co-author of Bitcoin and Cryptocurrency Technologies, the leading textbook on cryptocurrencies, and he has previously worked at Google and Microsoft Research, where he co-invented the Picnic signature algorithm. When not working, you can find Steven spending time with his family, taking a nature walk, or twisting balloons.
Harry Kalodner
Harry Kalodner is Co-founder and Chief Technology Officer at Offchain Labs where he leads the engineering team. Before the company he attended Princeton as a Ph.D candidate where his research explored economics, anonymity, and incentive compatibility of cryptocurrencies, and he also has worked at Apple. When not up at 3:00am writing code, Harry occasionally sleeps.
submitted by hkalodner to ethereum [link] [comments]

[ Bitcoin ] Technical: Taproot: Why Activate?

Topic originally posted in Bitcoin by almkglor [link]
This is a follow-up on https://old.reddit.com/Bitcoin/comments/hqzp14/technical_the_path_to_taproot_activation/
Taproot! Everybody wants it!! But... you might ask yourself: sure, everybody else wants it, but why would I, sovereign Bitcoin HODLer, want it? Surely I can be better than everybody else because I swapped XXX fiat for Bitcoin unlike all those nocoiners?
And it is important for you to know the reasons why you, o sovereign Bitcoiner, would want Taproot activated. After all, your nodes (or the nodes your wallets use, which if you are SPV, you hopefully can pester to your wallet vendoimplementor about) need to be upgraded in order for Taproot activation to actually succeed instead of becoming a hot sticky mess.
First, let's consider some principles of Bitcoin.
I'm sure most of us here would agree that the above are very important principles of Bitcoin and that these are principles we would not be willing to remove. If anything, we would want those principles strengthened (especially the last one, financial privacy, which current Bitcoin is only sporadically strong with: you can get privacy, it just requires effort to do so).
So, how does Taproot affect those principles?

Taproot and Your /Coins

Most HODLers probably HODL their coins in singlesig addresses. Sadly, switching to Taproot would do very little for you (it gives a mild discount at spend time, at the cost of a mild increase in fee at receive time (paid by whoever sends to you, so if it's a self-send from a P2PKH or bech32 address, you pay for this); mostly a wash).
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash, so the Taproot output spends 12 bytes more; spending from a P2WPKH requires revealing a 32-byte public key later, which is not needed with Taproot, and Taproot signatures are about 9 bytes smaller than P2WPKH signatures, but the 32 bytes plus 9 bytes is divided by 4 because of the witness discount, so it saves about 11 bytes; mostly a wash, it increases blockweight by about 1 virtual byte, 4 weight for each Taproot-output-input, compared to P2WPKH-output-input).
However, as your HODLings grow in value, you might start wondering if multisignature k-of-n setups might be better for the security of your savings. And it is in multisignature that Taproot starts to give benefits!
Taproot switches to using Schnorr signing scheme. Schnorr makes key aggregation -- constructing a single public key from multiple public keys -- almost as trivial as adding numbers together. "Almost" because it involves some fairly advanced math instead of simple boring number adding, but hey when was the last time you added up your grocery list prices by hand huh?
With current P2SH and P2WSH multisignature schemes, if you have a 2-of-3 setup, then to spend, you need to provide two different signatures from two different public keys. With Taproot, you can create, using special moon math, a single public key that represents your 2-of-3 setup. Then you just put two of your devices together, have them communicate to each other (this can be done airgapped, in theory, by sending QR codes: the software to do this is not even being built yet, but that's because Taproot hasn't activated yet!), and they will make a single signature to authorize any spend from your 2-of-3 address. That's 73 witness bytes -- 18.25 virtual bytes -- of signatures you save!
And if you decide that your current setup with 1-of-1 P2PKH / P2WPKH addresses is just fine as-is: well, that's the whole point of a softfork: backwards-compatibility; you can receive from Taproot users just fine, and once your wallet is updated for Taproot-sending support, you can send to Taproot users just fine as well!
(P2WPKH and P2WSH -- SegWit v0 -- addresses start with bc1q; Taproot -- SegWit v1 --- addresses start with bc1p, in case you wanted to know the difference; in bech32 q is 0, p is 1)
Now how about HODLers who keep all, or some, of their coins on custodial services? Well, any custodial service worth its salt would be doing at least 2-of-3, or probably something even bigger, like 11-of-15. So your custodial service, if it switched to using Taproot internally, could save a lot more (imagine an 11-of-15 getting reduced from 11 signatures to just 1!), which --- we can only hope! --- should translate to lower fees and better customer service from your custodial service!
So I think we can say, very accurately, that the Bitcoin principle --- that YOU are in control of your money --- can only be helped by Taproot (if you are doing multisignature), and, because P2PKH and P2WPKH remain validly-usable addresses in a Taproot future, will not be harmed by Taproot. Its benefit to this principle might be small (it mostly only benefits multisignature users) but since it has no drawbacks with this (i.e. singlesig users can continue to use P2WPKH and P2PKH still) this is still a nice, tidy win!
(even singlesig users get a minor benefit, in that multisig users will now reduce their blockchain space footprint, so that fees can be kept low for everybody; so for example even if you have your single set of private keys engraved on titanium plates sealed in an airtight box stored in a safe buried in a desert protected by angry nomads riding giant sandworms because you're the frickin' Kwisatz Haderach, you still gain some benefit from Taproot)
And here's the important part: if P2PKH/P2WPKH is working perfectly fine with you and you decide to never use Taproot yourself, Taproot will not affect you detrimentally. First do no harm!

Taproot and Your Contracts

No one is an island, no one lives alone. Give and you shall receive. You know: by trading with other people, you can gain expertise in some obscure little necessity of the world (and greatly increase your productivity in that little field), and then trade the products of your expertise for necessities other people have created, all of you thereby gaining gains from trade.
So, contracts, which are basically enforceable agreements that facilitate trading with people who you do not personally know and therefore might not trust.
Let's start with a simple example. You want to buy some gewgaws from somebody. But you don't know them personally. The seller wants the money, you want their gewgaws, but because of the lack of trust (you don't know them!! what if they're scammers??) neither of you can benefit from gains from trade.
However, suppose both of you know of some entity that both of you trust. That entity can act as a trusted escrow. The entity provides you security: this enables the trade, allowing both of you to get gains from trade.
In Bitcoin-land, this can be implemented as a 2-of-3 multisignature. The three signatories in the multisgnature would be you, the gewgaw seller, and the escrow. You put the payment for the gewgaws into this 2-of-3 multisignature address.
Now, suppose it turns out neither of you are scammers (whaaaat!). You receive the gewgaws just fine and you're willing to pay up for them. Then you and the gewgaw seller just sign a transaction --- you and the gewgaw seller are 2, sufficient to trigger the 2-of-3 --- that spends from the 2-of-3 address to a singlesig the gewgaw seller wants (or whatever address the gewgaw seller wants).
But suppose some problem arises. The seller gave you gawgews instead of gewgaws. Or you decided to keep the gewgaws but not sign the transaction to release the funds to the seller. In either case, the escrow is notified, and if it can sign with you to refund the funds back to you (if the seller was a scammer) or it can sign with the seller to forward the funds to the seller (if you were a scammer).
Taproot helps with this: like mentioned above, it allows multisignature setups to produce only one signature, reducing blockchain space usage, and thus making contracts --- which require multiple people, by definition, you don't make contracts with yourself --- is made cheaper (which we hope enables more of these setups to happen for more gains from trade for everyone, also, moon and lambos).
(technology-wise, it's easier to make an n-of-n than a k-of-n, making a k-of-n would require a complex setup involving a long ritual with many communication rounds between the n participants, but an n-of-n can be done trivially with some moon math. You can, however, make what is effectively a 2-of-3 by using a three-branch SCRIPT: either 2-of-2 of you and seller, OR 2-of-2 of you and escrow, OR 2-of-2 of escrow and seller. Fortunately, Taproot adds a facility to embed a SCRIPT inside a public key, so you can have a 2-of-2 Taprooted address (between you and seller) with a SCRIPT branch that can instead be spent with 2-of-2 (you + escrow) OR 2-of-2 (seller + escrow), which implements the three-branched SCRIPT above. If neither of you are scammers (hopefully the common case) then you both sign using your keys and never have to contact the escrow, since you are just using the escrow public key without coordinating with them (because n-of-n is trivial but k-of-n requires setup with communication rounds), so in the "best case" where both of you are honest traders, you also get a privacy boost, in that the escrow never learns you have been trading on gewgaws, I mean ewww, gawgews are much better than gewgaws and therefore I now judge you for being a gewgaw enthusiast, you filthy gewgawer).

Taproot and Your Contracts, Part 2: Cryptographic Boogaloo

Now suppose you want to buy some data instead of things. For example, maybe you have some closed-source software in trial mode installed, and want to pay the developer for the full version. You want to pay for an activation code.
This can be done, today, by using an HTLC. The developer tells you the hash of the activation code. You pay to an HTLC, paying out to the developer if it reveals the preimage (the activation code), or refunding the money back to you after a pre-agreed timeout. If the developer claims the funds, it has to reveal the preimage, which is the activation code, and you can now activate your software. If the developer does not claim the funds by the timeout, you get refunded.
And you can do that, with HTLCs, today.
Of course, HTLCs do have problems:
Fortunately, with Schnorr (which is enabled by Taproot), we can now use the Scriptless Script constuction by Andrew Poelstra. This Scriptless Script allows a new construction, the PTLC or Pointlocked Timelocked Contract. Instead of hashes and preimages, just replace "hash" with "point" and "preimage" with "scalar".
Or as you might know them: "point" is really "public key" and "scalar" is really a "private key". What a PTLC does is that, given a particular public key, the pointlocked branch can be spent only if the spender reveals the private key of the given private key to you.
Another nice thing with PTLCs is that they are deniable. What appears onchain is just a single 2-of-2 signature between you and the developemanufacturer. It's like a magic trick. This signature has no special watermarks, it's a perfectly normal signature (the pledge). However, from this signature, plus some datta given to you by the developemanufacturer (known as the adaptor signature) you can derive the private key of a particular public key you both agree on (the turn). Anyone scraping the blockchain will just see signatures that look just like every other signature, and as long as nobody manages to hack you and get a copy of the adaptor signature or the private key, they cannot get the private key behind the public key (point) that the pointlocked branch needs (the prestige).
(Just to be clear, the public key you are getting the private key from, is distinct from the public key that the developemanufacturer will use for its funds. The activation key is different from the developer's onchain Bitcoin key, and it is the activation key whose private key you will be learning, not the developer's/manufacturer's onchain Bitcoin key).
So:
Taproot lets PTLCs exist onchain because they enable Schnorr, which is a requirement of PTLCs / Scriptless Script.
(technology-wise, take note that Scriptless Script works only for the "pointlocked" branch of the contract; you need normal Script, or a pre-signed nLockTimed transaction, for the "timelocked" branch. Since Taproot can embed a script, you can have the Taproot pubkey be a 2-of-2 to implement the Scriptless Script "pointlocked" branch, then have a hidden script that lets you recover the funds with an OP_CHECKLOCKTIMEVERIFY after the timeout if the seller does not claim the funds.)

Quantum Quibbles!

Now if you were really paying attention, you might have noticed this parenthetical:
(technical details: a Taproot output is 1 version byte + 32 byte public key, while a P2WPKH (bech32 singlesig) output is 1 version byte + 20 byte public key hash...)
So wait, Taproot uses raw 32-byte public keys, and not public key hashes? Isn't that more quantum-vulnerable??
Well, in theory yes. In practice, they probably are not.
It's not that hashes can be broken by quantum computes --- they're still not. Instead, you have to look at how you spend from a P2WPKH/P2PKH pay-to-public-key-hash.
When you spend from a P2PKH / P2WPKH, you have to reveal the public key. Then Bitcoin hashes it and checks if this matches with the public-key-hash, and only then actually validates the signature for that public key.
So an unconfirmed transaction, floating in the mempools of nodes globally, will show, in plain sight for everyone to see, your public key.
(public keys should be public, that's why they're called public keys, LOL)
And if quantum computers are fast enough to be of concern, then they are probably fast enough that, in the several minutes to several hours from broadcast to confirmation, they have already cracked the public key that is openly broadcast with your transaction. The owner of the quantum computer can now replace your unconfirmed transaction with one that pays the funds to itself. Even if you did not opt-in RBF, miners are still incentivized to support RBF on RBF-disabled transactions.
So the extra hash is not as significant a protection against quantum computers as you might think. Instead, the extra hash-and-compare needed is just extra validation effort.
Further, if you have ever, in the past, spent from the address, then there exists already a transaction indelibly stored on the blockchain, openly displaying the public key from which quantum computers can derive the private key. So those are still vulnerable to quantum computers.
For the most part, the cryptographers behind Taproot (and Bitcoin Core) are of the opinion that quantum computers capable of cracking Bitcoin pubkeys are unlikely to appear within a decade or two.
So:
For now, the homomorphic and linear properties of elliptic curve cryptography provide a lot of benefits --- particularly the linearity property is what enables Scriptless Script and simple multisignature (i.e. multisignatures that are just 1 signature onchain). So it might be a good idea to take advantage of them now while we are still fairly safe against quantum computers. It seems likely that quantum-safe signature schemes are nonlinear (thus losing these advantages).

Summary

I Wanna Be The Taprooter!

So, do you want to help activate Taproot? Here's what you, mister sovereign Bitcoin HODLer, can do!

But I Hate Taproot!!

That's fine!

Discussions About Taproot Activation

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What is Blockchain Technology?

What is Blockchain Technology?
The original article appeared here: https://www.securities.io/what-is-blockchain-technology/
Its been almost ten years since Satoshi Nakamoto first introduced Blockchain technology to the world in his 2008 Bitcoin Whitepaper. Since that time, these revolutionary networks have gained popularity in both the corporate and governmental sectors. This growth is easily explained when you consider that blockchain technology provides the world with some unique advantages that were previously unimaginable. Consequently, today, you can find blockchain technology in nearly every sector of the global economy.

What is Blockchain Technology?

A blockchain is a network of computers that share a distributed ledger across all network participants (nodes). This strategy is far different than say, fiat currencies that originate from a centralized authority figure. Importantly, this ledger keeps an unbroken chain of transactions since the birth of the network. This “chain” of transactions grows larger as new “blocks” of transactions are approved and added to it.
Bitcoin Whitepaper
In order to approve new transactions, each node works together with others to validate new blocks. Additionally, the nodes also validate the current state of the entire blockchain. In order for a new block of transactions to be added to the blockchain, they must receive approval from 51% of the network’s nodes. Nodes are also referred to as miners. In this manner, blockchain networks are decentralized networks that provide unmatched security to the world of digital assets.

Security via Decentralization

Decentralization is an important aspect of blockchain technology because it makes these revolutionary ledgers immutable and unalterable. In fact, since there is no centralized attack vector, hacking a blockchain is nearly impossible. The larger the blockchain network, the more secure the data on it remains.
For example, let’s look at the world’s largest blockchain, Bitcoin. Currently, the Bitcoin blockchain has over 10,000 active nodes located across the globe. This distribution means that in order for an attacker to alter even just one tiny piece of information on the blockchain, they would need to successfully hack 5,000+ computers at once.
While this task may not be impossible for the quantum computers of the future, it’s so unprofitable that it makes no sense to even attempt such a monumental task. Additionally, on top of successfully hacking 5000+ computers at once, an attacker would also need a supercomputer to recalculate the new blockchain transactions in time to introduce them into the network. It would literally be more affordable to create a new cryptocurrency from scratch.

Consensus Mechanisms

One of the reasons why blockchain networks are so secure is the integration of consensus mechanisms. Consensus mechanisms are cryptographic protocols that leverage the participants of a blockchain network in securing its data. In the case of Bitcoin, the Proof-of-Work (PoW) consensus mechanism is used.

Proof-of-Work (PoW)

The Proof-of-Work consensus mechanism was revolutionary to the world of cryptography when it was first introduced years prior by Adam Back in his Hashcash whitepaper. In the concept, Back describes the integration of a mathematical equation to the network’s security protocols. In this way, every computer can show “proof” of their work securing the network.

Miner Rewards

It’s important to understand that nodes receive a reward for their mining efforts. These rewards adjust automatically depending on the network’s difficulty and value. In the case of Bitcoin, miners originally received 50 Bitcoin for their efforts. Today, this seems like fortune, but back in 2009, Bitcoin was only worth pennies. As the value of the token rises and the network goes, the mining rewards shrink. Today, Bitcoin miners receive 6.5 BTC if they add the next block to the chain.

SHA-256

Notably, every node validates and secures the blockchain, but only one gets to add the next block of transactions to the network. To determine who the next miner is that gets to add this block, every computer competes in a mathematical race to figure out the PoW equation. In the case of Bitcoin, the equation is known as SHA-256. Importantly, the first SHA algorithm dates back to Hashcash. This early version of the equation was known as SHA-1.
Notably, the SHA-256 equation is so difficult that it’s easier and more efficient for your computer to just make random guesses rather than attempting to figure out the equation directly. The answer to the equation must begin with a predetermined amount of 0s. In the Bitcoin blockchain, the equation’s answer must start with four zeros. However, if the network’s congestion rises, so does the difficulty of these equations. This difficulty adjusts by the addition of another zero at the beginning of the required SHA-256 answer.
Similarly to traditional commodities such as gold, there are costs that are associated with the creation and introduction of these digital assets into the market. These random guesses utilize intense computational power. This power equates to real-world costs such as electricity bills. Studies have shown that securing the Bitcoin network can use more electricity than required by entire countries. Luckily, over 80% of Bitcoin’s power consumption comes from renewable sources such as solar or hydroelectric. This cost of mining also adds measurable value to each Bitcoin.

Miners

As Bitcoin began to gain in profitability, its network’s computing power expanded significantly. In the beginning, nodes, also known as miners, could mine for Bitcoin using nothing more than your home PC. Eventually, miners realized that graphic cards were far better at the repetitive guessing required to figure out the SHA-256 algorithm. This led to a computational race in the market.

ASIC

Eventually, large blockchain firms such as Bitmain introduced Application Specific Integrated Circuit (ASIC) miners into the equation. These purpose-built miners were thousands of times more efficient at guessing the SHA-256 algorithm than the GPUs and CPUs before them. Consequently, their introduction created a scenario in which the average miner now needed to invest thousands in mining equipment to stay relevant.

Mining Pools

Luckily, some creative minds in the field began to think of ways to level the playing field out again. They developed “mining pools.” A mining pool is a network of miners that all share computational power for the common goal of mining blockchain transactions. Importantly, mining pool participants receive a percentage of the reward based on their contributions to the network’s overall hash (computational power).
Importantly, over the last three years, there has been a push to move away from power-hungry consensus mechanisms such as PoW. This desire to secure blockchains in a more efficient manner has led to the development of some truly unique consensus mechanisms in the sector.

Proof-of-Stake (PoS)

The Proof-of-Stake mechanism does away with the difficult mathematical algorithms and instead utilizes a more psychological approach to securing the network. In a PoS blockchain, users don’t need to compete mathematically to add the next block to the blockchain. Instead, PoS users “stake” their coins via network wallets to secure the network. The way staking works is simple.
Keeping a certain amount of coins in your wallet allows you to participate in transaction validations. The more coins you stake, the more likely the chances are you get to add the next block of transactions to the network. In most PoS systems, a miner from those with the most tokens staked at the time receives the chance to add the blocks.
The advantages of a PoS consensus mechanism are immediately evident. For one, you don’t need to pour tons of resources into your network to keep it safe. Additionally, since nodes are chosen based on their amount of staked coins, there is never a scenario in which a node gains anything from validating incorrect transactions. Basically, a hacker would have to fully invest in the cryptocurrency prior to attacking the network. In this way, PoS systems create a huge deterrent to attackers.

The Future of Blockchain Technology

Blockchain technology has come a long way from its early days as a means to secure cryptocurrency networks. Today, blockchain technology has numerous uses across every type of industry imaginable. Specifically, blockchain programs have impacted the logistical, financial, and data security sectors in a major way.

Blockchain Technology Logistics

Blockchain logistical systems are more efficient and cost-effective to operate than traditional paper-based models. In fact, the immutable and unalterable nature of blockchain tech makes it ideally suited to logistical tasks. Soon, you may be able to ascertain much more information regarding the creation and delivery of your products thanks to these new-age systems emerging.

Fundraising

Blockchain technology has also altered the way in which businesses raise funds. In a traditional corporate crowdfunding strategy such as an IPO, companies must balance between cost-effectiveness and participation. The inability to process smaller transactions meant that for the longest time, companies had to turn away potential investors. Nowadays, blockchain technology enables businesses to easily automate these procedures via smart contracts.

Smart Contracts

Smart Contracts feature preprogrammed protocols that execute when they receive a certain amount of cryptocurrency sent to their address. These contracts live on the blockchain and enable remarkable functionality. For example, in the case of fundraising, a smart contract can automate processes such as the approval of investors and the distribution of funds.

Blockchain Technology Today

You can expect to see further expansion of the blockchain sector in the coming months as more governments and institutions explore its benefits. For now, the blockchain revolution is well underway.
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aantonop - YouTube Value of Bitcoin - YouTube Bitcoin Q&A - YouTube Introduction to Cryptography of Bitcoin, Explained! Bitcoin Q&A: The mining process

Bitcoin mining also relies on cryptography, albeit in a different way. Miners dedicate large amounts of computing power in order to solve cryptographic puzzles, in exchange for block rewards. The fact that countless miners are powering the network means that Bitcoin is decentralized – no single user can control it. If one user gains access to a quantum computer, they could produce hashes ... Quantum cryptography is used today in a small amount of applications. Some interesting research in this field shows that while quantum can break existing cryptography and make it useless, it actually introduces a different forum which could be used to fix the problem. I think it’s fascinating that the very thing that breaks cryptography also ... Quantum Cryptography Explained Bitcoin . Quantum Cryptography Explained . Apr 3, 2018 DTN Staff. twitter. pinterest. google plus. facebook. Introduction To Quantum Cryptography ... We can advise clients today on quantum security and we’ll start unveiling quantum-safe cryptography services on our public cloud next year. This is designed to better help organizations keep their data secured while it is in-transit within IBM Cloud. To accomplish this, we are enhancing TLS and SSL implementations in IBM Cloud services by using algorithms designed to be quantum-safe, and ... Therefore too early to be afraid of a quantum computer threat for Bitcoin. Today we still don't have a fully functioning quantum computer, while the post-quantum cryptography has already presented ...

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aantonop - YouTube

Whether or not it's worth investing in, the math behind Bitcoin is an elegant solution to some complex problems. Hosted by: Michael Aranda Special Thanks: Da... Please DONATE to my education fund, SUBSCRIBE, and LIKE for more cryptocurrency lecture videos. Bitcoin: 1369eCF1pqBfLjhsdmDFWwbbrTmVENUUyz Litecoin ... He is the author of two books: “Mastering Bitcoin,” published by O’Reilly Media and considered the best technical guide to bitcoin; “The Internet of Money,” a book about why bitcoin matters. aantonop's YouTube channel is THE place to find free, unbiased educational videos on all things Bitcoin and open blockchain. Subscribe & join the channel to ... He is the author of two books: “Mastering Bitcoin,” published by O’Reilly Media and considered the best technical guide to bitcoin; “The Internet of Money,” a book about why bitcoin matters.

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