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\section*{Handout 8 (Bitcoins)}

In my opinion Bitcoins are an elaborate Ponzi

scheme\footnote{\url{http://en.wikipedia.org/wiki/Ponzi_scheme}}---still

the ideas behind them are really beautiful and not too

difficult to understand. Since many colourful claims about

Bitcoins float around in the mainstream and not-so-mainstream

media, it will be instructive to re-examine such claims from a

more technically informed vantage point. For example, it is

often claimed that Bitcoins are anonymous and free from any

potential government meddling. It turns out that the first

claim ignores a lot of research in de-anonymising social

networks, and the second underestimates the persuasive means a

government has at its disposal. 

There are a lot of articles, blogposts, research papers

etc.~available about Bitcoins. Below I will follow closely the

very readable explanations from

\begin{center}

\url{http://www.michaelnielsen.org/ddi/how-the-bitcoin-protocol-actually-works/} \;\;and\smallskip\\

\url{http://www.imponderablethings.com/2013/07/how-bitcoin-works-under-hood.html}

\end{center}

\noindent The latter also contains a link to a nice youtube

video about the technical details behind Bitcoins. I will

also use some of their pictures.

Let us start with the question who invented Bitcoins? You

could not make up the answer, but we actually do not know who

the inventor is. All we know is that the first paper

\begin{center}

\url{https://bitcoin.org/bitcoin.pdf}

\end{center}

\noindent is signed by Satoshi Nakamoto, which however is

likely only a pen name. There is a lot of speculation who

could be the inventor, or inventors, but we simply do not

know. This part of Bitcoins is definitely anonymous so far. The paper

above is from the end of 2008; the first Bitcoin transaction

was made in January 2009. The rules in Bitcoin are set up so

that there will only ever be 21 Million Bitcoins with the

maximum reached around the year 2140. Currently there are

already 11 Million Bitcoins in `existence'. Contrast this with

traditional fiat currencies where money can be printed almost

at will. The smallest unit of a Bitcoin is called a Satoshi,

which is the $10^{-8}$th part of a Bitcoin. Remember a Penny

is the $10^{-2}$th part of a Pound.

The two main cryptographic building blocks of Bitcoins are

cryptographic hashing functions (SHA-256) and public-private

keys using the elliptic-curve encryption scheme for digital

signatures. Hashes are used to generate `fingerprints' of data

that ensure integrity (absence of tampering). Public-private

keys are used for signatures. For example sending a message,

say $msg$, together with the encrypted version

\[

msg, \{msg\}_{K^{priv}}

\]

\noindent allows everybody with access to the corresponding

public key $K^{pub}$ to verify that the message came from the

person who knew the private key. Signatures are used in

Bitcoins for verifying the addresses where the Bitcoins are

sent from. Addresses in Bitcoins are essentially the public

keys. There are $2^{160}$ possible addresses, which is such a

vast amount that there is not even a check for duplicates, or

already used addresses. If you start with a random number to

generate a public-private key pair it is very unlikely that

you step on somebody else's shoes. Compare this with the

email-addresses you wanted to register with, say

Gmail, but which are always already taken.

One major difference between Bitcoins and traditional banking

is that you do not have a place, or few places, that record the

balance on your account. Traditional banking involves a

central ledger which specifies the current balance in each

account, for example 

\begin{center}

\begin{tabular}{l|r}

account owner & balance\\\hline

Alice   & \pounds{10.01}\\

Bob     & \pounds{4.99}\\

Charlie & -\pounds{1.23}\\

Eve     & \pounds{0.00}

\end{tabular}

\end{center}

\noindent Bitcoins work differently in that there is no such

central ledger, but instead a public record of all

transactions ever made. This means spending money corresponds

to sending messages of the (oversimplified) form 

\begin{equation}

\{\text{I, Alice, am giving Bob one Bitcoin.}\}_{K^{priv}_{Alice}}

\end{equation}

\noindent These messages, called transactions, are the only

data that is ever stored in the Bitcoin system (we will come

to the precise details later on). The transactions are

encrypted with Alice's private key so that everybody,

including Bob, can use Alice's public key $K^{pub}_{Alice}$ to

verify that this message came really from Alice, or more

precisely from the person who knows $K^{priv}_{Alice}$. 

The problem with such messages in a distributed system is that

what happens if Bob receives 10, say, of these transactions?

Did Alice intend to send him 10 Bitcoins, or did the message

get duplicated by for example an attacker re-playing a sniffed

message? What is needed is a kind of serial number for such

transactions. This means transaction messages shoul look more like 

\begin{center}

$\{\text{I, Alice, am giving Bob Bitcoin \#1234567.}\}_{K^{priv}_{Alice}}$

\end{center}

\noindent There are two difficulties, however, that need to be

solved with serial numbers. One is who is assigning serial

numbers to Bitcoins and also how can Bob verify that Alice

actually owns this Bitcoin to pay him? In a system with a bank

as trusted third-party, Bob could do the following:

\begin{itemize}

\item Bob asks the bank whether the Bitcoin with that serial

      number belongs to Alice and Alice hasn't already spent

      this Bitcoin.

\item If yes, then Bob tells the bank he accepts this Bitcoin.

      The bank updates the records to show that the Bitcoin

      with that serial number is now in Bob’s possession and

      no longer belongs to Alice. 

\end{itemize}

\noindent But for this banks would need to be trusted and

would also be an easy target for any government interference,

for example. Think of the early days of music sharing where

the company Napster was the trusted third-party but also the single point of ``failure'' which

was taken offline by law enforcement. Bitcoins is more like a

system such as BitTorrent without a single central entity that

can be taken offline.\footnote{There is some Bitcoin

infrastructure that is not so immune from being taken offline:

for example Bitcoin exchanges, HQs of Bitcoin mining pools,

Bitcoin developers and so on.}

Bitcoins solve the problem of not being able to rely on a bank

by making everybody the ``bank''. Everybody who cares can have

the entire transaction history starting with the first

transaction made in January 2009. This history of transactions

is called the \emph{blockchain}. Bob, for example, can use his

copy of the blockchain for determining whether Alice owned the

Bitcoin he received, and if she did, he transmits the message

that he owns it now to every other participant on the Bitcoin

network. An illustration of a three-block segment of the

blockchain is (simplified) as follows

\begin{equation}

\includegraphics[scale=0.4]{../pics/bitcoinblockchain0.png}

\label{segment}

\end{equation}

\noindent The chain grows with time. Each block contains a

list of individual transactions, written txn in the picture

above, and also a reference to the previous block, written

prev. The data in a block (txn's and prev) is hashed so that

the reference and transactions in them cannot be tampered

with. This hash is also the unique serial number of each

block. Since this previous-block-reference is also part of the

hash, the whole chain is robust against tampering. I let you

think why this is the case?\ldots{}But does it actually

eliminate all possibilities of fraud?

We can check the consistency of the blockchain by checking

whether all the references and hashes are correctly recorded.

I have not tried it myself, but it is said that with the

current amount of data (appr.~12GB) it takes roughly a day to

check the consistency of the blockchain on a normal computer.

Fortunately this ``extended'' consistency check usually only

needs to be done once. Afterwards the blockchain only needs to

be updated consistently.

Recall I wrote earlier that Bitcoins do not maintain a ledger,

which lists all the current balances in each account. Instead

only transactions are recorded. While a current balance of an

account is not immediately available, it is possible to

extract from the blockchain a transaction graph that looks

like the picture shown in Figure~\ref{txngraph}. Each

rectangle represents a single transaction. Take for example

the rightmost lower transaction from Charles to Emily. This

transaction has as receiver the address of Emily and as the

sender the address of Charles. In this way no Bitcoins can

appear out of thin air (we will discuss later how Bitcoins are

actually generated). If Charles did not have a transaction of

at least the amount he wants to give Emily to his name

(i.e.~send to an address with his public-private key) then

there is no way he can make a payment to Emily. Equally, if

now Emily wants to pay for a coffee, say, with the Bitcoin she

received from Charles she can essentially only forward the

message she received. The only slight complication with this

setup in Bitcoins is that ``incoming'' Bitcoins can be

combined in a transaction and ``outgoing'' Bitcoins can be

split. For example in the leftmost upper transactions in

Figure~\ref{txngraph}, Fred makes a payment to Alice. But this

payment (or transaction) combines the Bitcoins that were send

by Jane to Fred and also by Juan to Fred. This allows you to

``consolidate'' your funds: if it were only possible to split

transactions, then the amounts would get smaller and smaller. 

In Bitcoins you have the ability to both combine incoming

transactions, but also to split outgoing transactions to

potentially more than one receiver. The latter is also needed.

Consider again the rightmost transactions in

Figure~\ref{txngraph} and suppose Alice is a coffeeshop owner

selling coffees for 1 Bitcoin. Charles received a transaction

from Zack over 5 Bitcoins, say. How does Charles pay for the

coffee? There is no explicit notion of \emph{change} in the

Bitcoin system. What Charles has to do instead is to make one

single transaction with 1 Bitcoin to Alice and with 4 Bitcoins

going back to himself, which then Charles can use to give to

Emily, for example.

\begin{figure}[t]

\begin{center}

\includegraphics[scale=0.4]{../pics/blockchain.png}

\end{center}

\caption{Transaction graph that is implicitly recorded in the

public blockchain.\label{txngraph}}

\end{figure}

Let us consider another example. Suppose Emily received 4

Bitcoins from Charles and independently received another

transaction (not shown in the picture) that sends 6 Bitcoins

to her. If she now wants to buy a coffee from Alice for 1

Bitcoin, she has two possibilities: She could just forward the

transaction from Charles over 4 Bitcoins to Alice split in

such a way that Alice receives 1 Bitcoin and Emily sends the

remaining 3 Bitcoins back to herself. In this case she would

now be in the possession of two unspend Bitcoin transactions,

one over 3 Bitcoins and the independent one over 6 Bitcoins.

Or, Emily could combine both transactions (one over 4 Bitcoins

from Charles and the independent one over 6 Bitcoins) and then

split this amount with 1 Bitcoin going to Alice and 9 Bitcoins

going back to herself. 

I think this is a good time for you to pause to let this

concept of transactions to really sink in\ldots{}You should

come to the conclusion that there is really no need for a central ledger and no

need for an account balance as familiar from traditional

banking. The closest what Bitcoin has to offer for the notion

of a balance in a bank account are the unspend transactions

that a person (more precisely a public-private key address)

received. That means transactions that can still be forwarded. 

After the pause also consider the fact that whatever

transaction is recorded in the blockchain will be in the

``historical record'' for the Bitcoin system. If a transaction

says 1 Bitcoin goes from address $A$ to address $B$, then this

is what will be---$B$ has then the possibility to spend the

corresponding Bitcoins, whether the transaction was done

fraudulently or not. There is no exception to this rule.

Interestingly this is also how Bitcoins can get lost: One

possibility is that you send Bitcoins to an address for which

nobody has generated a private key, for example because of a

typo in the address field---bad luck for fat

fingers\footnote{\url{http://en.wikipedia.org/wiki/Typographical_error}}

in the Bitcoin system. The reason is that nobody has a private

key for this erroneous address and consequently cannot forward

the transaction anymore. Another possibility is that you

forget your private key and you had messages forwarded to the

corresponding public key. Also in this case bad luck: you will

never be able to forward this message again, because you will

not be able to form a valid message that sends this to

somebody else (we will see the details of this later). But

this is also a way how you can get robbed of your Bitcoins. By

old-fashioned hacking-into-a-computer crime, for example, an

attacker might get hold of your private key and then quickly

forwards the Bitcoins that are in your name to an address the

attacker controls. You will never again have access to these

Bitcoins, because for the Bitcoin system they are assumed to

be spent. And remember with Bitcoins you cannot appeal to any

higher authority. Once the Bitcoins are gone, they are gone.

This is much different in traditional banking where at least

you can try to harass the bank to roll back the transaction. 

This brings us to back to problem of double spend. Suppose Bob

is a merchant. How can he make sure that Alice does not cheat

him? She could for example send a transaction to Bob. But also

forward the ``same'' transaction to Charlie, or even herself.

If Alice manages to get the second transaction into the

blockchain, Bob will be cheated out of his money. The problem

in such conflicting situations is how should the network

update their blockchain? You might end up with a picture like

this

\begin{center}

\includegraphics[scale=0.4]{../pics/bitcoindisagreement.png}

\end{center}

\noindent where Alice convinced some part of the ``world''

that she is still the owner of the Bitcoin and some other part

of the ``world'' thinks it's Bob's. How should such a

disagreement be resolved? This is actually the main hurdle

where Bitcoin really innovated. The answer is that Bob needs

to convince ``enough'' people on the network that the

transaction from Alice to him is legit. 

What does, however, ``enough'' mean in a distributed system?

If Alice sets up a network of a billion, say, puppy identities

and whenever Bob tries to convince, or validate, that he is

the rightful owner of the Bitcoin, then the puppy identities

agree. Bob would then have no reason to not give Alice her

coffee. But behind his back she has convinced everybody else

on the network that she is still the rightful owner of the

Bitcoin. After being outvoted, Bob would be a tad peeved. 

The reflex reaction to such a situation would be to make the

process of validating a transaction as cheap as possible. The

intention is that Bob will easily get enough peers to agree with him

that he is the rightful owner. But such a solution has always

the limitation of Alice setting up an even bigger network of

puppy identities. The really cool idea of Bitcoin is to go

into the other direction of making the process of transaction

validation (artificially) as expensive as possible, but reward

people for helping with the validation. This is really a novel

and counterintuitive idea that makes the whole system of

Bitcoins work so beautifully.

\subsubsection*{Proof-of-Work Puzzles}

In order to make the process of transaction validation

difficult, Bitcoin uses a kind of puzzle. Solving the puzzles

is called \emph{Bitcoin mining}, where whoever solves a puzzle

will be awarded some Bitcoins. At the beginning this was 50

Bitcoins, but the rules of Bitcoin are set up such that this

amount halves every 210,000 transactions or so. Currently you

will be awarded 25 Bitcoins for solving a puzzle. Because the

amount will halve again and then later again and again, around

the year 2140 it will go below the level of 1 Satoshi. In that

event no new Bitcoins will ever be created again and the

amount of Bitcoins stays fixed. There will be still an

incentive to help with validating transactions, because there

is the possibility in Bitcoins to offer a transaction fee to

whoever solves a puzzle. At the moment this fee is usually set

to 0, since the incentive for miners is the 25 Bitcoins that

are currently awarded for solving puzzles. 

What do the puzzles that miners have to solve look like? The

puzzles can be illustrated roughly as follows: Given a string,

say \code{"Hello, world!"}, what is the salt so that the hash

starts with a long run of zeros? Let us look at a concrete

example. Recall that Bitcoins use the hash-function SHA-256.

Suppose we call this hash function \code{h}, then we could try

the salt \code{0} as follows:

\begin{quote}

\code{h("Hello, world!0") =}\\

\mbox{}\quad\footnotesize\pcode{1312af178c253f84028d480a6adc1e25e81caa44c749ec81976192e2ec934c64}

\end{quote}

\noindent OK this does not have any zeros at all. We could 

next try the salt \code{1}: 

\begin{quote}

\code{h("Hello, world!1") =}\\

\mbox{}\quad\footnotesize\pcode{e9afc424b79e4f6ab42d99c81156d3a17228d6e1eef4139be78e948a9332a7d8}

\end{quote}

\noindent Again this hash value does not contain any leading 

zeros. We could now try out every salt until we reach

\begin{quote}

\code{h("Hello, world!4250") =}\\

\mbox{}\quad\footnotesize\pcode{0000c3af42fc31103f1fdc0151fa747ff87349a4714df7cc52ea464e12dcd4e9}

\end{quote}

\noindent where we have four leading zeros. If four zeros are

enough, then the puzzle would be solved with this salt. The

point is that we can very quickly check whether a salt solves

a puzzle, but it is hard to find one. Latest research suggest

it is an NP-problem. If we want the output hash value to begin

with 10 zeroes, say, then we will, on average, need to try

$16^{10} \approx 10^{12}$ different salts before we find a

suitable one. 

In Bitcoins the puzzles are not solved according to how many

leading zeros a hash-value has, but rather whether it is below

a \emph{target}. The hardness of the puzzle can actually be

controlled by changing the target according to the available

computational power available. I think the adjustment of the

hardness of the problems is done every 2060 blocks

average the Bitcoin network will most likely solve a puzzle

within 10 Minutes. 

\begin{center}

\includegraphics[scale=0.37]{../pics/blockchainsolving.png}

\end{center}

\noindent It could be solved quicker, but equally it could 

take longer, but on average after 10 Minutes somebody on the 

network will have found a solution. 

Remember that the puzzles are a kind of proof-of-work that

make the validation of transactions artificially expensive.

Consider the following picture with a blockchain and some

unconfirmed transactions. 

\begin{equation}

\includegraphics[scale=0.38]{../pics/bitcoin_unconfirmed.png}

\label{unconfirmed}

\end{equation}

\noindent The puzzle is stated as follows: There are some

unconfirmed transactions. Choosing some of them, the miner

(i.e.~the person/computer that tries to solve a puzzle) will

form a putative block to be added to the blockchain. This

putative block will contain the transactions and the reference

to the previous block. The serial number of such a block is

simply the hash of all the data. The puzzle can then be stated

as the ``string'' corresponding to the block and which salt is

needed in order to have the hashed value being below the

target. Other miners will choose different transactions and

therefore work on a slightly different putative block and

puzzle.

The intention of the proof-of-work puzzle is that the

blockchain is at every given moment linearly ordered, see the

picture shown in \eqref{unconfirmed}. If we don’t have such a

linear ordering at any given moment then it may not be clear

who owns which Bitcoins. Assume a miner David is lucky and

finds a suitable salt to confirm some transactions. Should he

celebrate? Not yet. Typically the blockchain will look as

follows

\begin{center}

\includegraphics[scale=0.65]{../pics/block_chain1.png}

\end{center}

\noindent But every so often there will be a fork

\begin{center}

\includegraphics[scale=0.65]{../pics/block_chain_fork.png}

\end{center}

\noindent What should be done in this case? Well, the tie is

broken if another block is solved, like so:

\begin{center}

\includegraphics[scale=0.4]{../pics/bitcoin_blockchain_branches.png}

\end{center}

\noindent The rule in Bitcoins is: If a fork occurs, people on

the network keep track of all forks (they can see). But at any

given time, miners only work to extend whichever fork is

longest in their copy of the block chain. Why should miners

work on the longest fork? Well their incentive is to mine

Bitcoins. If somebody else already solved a puzzle, then it

makes more sense to work on a new puzzle and obtain the

Bitcoins for solving that puzzle, rather than waste efforts on

a fork that is shorter and therefore less likely to be

``accepted''. Note that whoever solved a puzzle on the

``loosing'' fork will actually not get any Bitcoins as reward.

Tough luck.

\subsubsection*{Alice against the Rest of the World}

Let us see how the blockchain and the proof-of-work puzzles

avoid the problem of double spend. If Alice wants to cheat

Bob, she would need to pull off the following ploy:

\begin{center}

\includegraphics[scale=0.4]{../pics/bitcoin_blockchain_double_spend.png}

\end{center}

\noindent Alice makes a transaction to Bob for paying, for

example, for an online order. This transaction is confirmed,

or validated, in block 2. Bob ships the goods around block 4.