bitcoinbook/chapters/authorization-authentication.adoc

[[c_authorization_authentication]]
== Authorization and Authentication

When you receive bitcoins, you have to decide who will have permission
to spend them, called _authorization_.  You also have to decide how full
nodes will distinguish the authorized spenders from everyone else,
called _authentication_.  Your authorization instructions and the
spender proof of authentication will be checked by thousands of
independent full nodes which all need to come to the same conclusion
that a spend was authorized and authentic in order for the transaction
containing it to be valid.

The original description of Bitcoin used a public key for authorization.
Alice paid Bob by putting his public key in the output of a transaction.
Authentication came from Bob in the form a signature that committed to a
spending transaction, such as from Bob to Carol.

The actual version of Bitcoin that was originally released provided a
more flexible mechanism for both authorization and authentication.
Improvements since then have only increased that flexibility.  In this
chapter, we'll explore those features and see how they're most commonly
used.

[[tx_script]]
[role="pagebreak-before less_space_h1"]
=== Transaction Scripts and Script Language

((("transactions", "scripts and Script language",
id="Tsript06")))((("scripting", "transactions and",
id="Stransact06")))The original version of Bitcoin introduced a new
programming language called _Script_, a Forth-like stack-based
language.  Both the scriptPubKey placed in an output and the legacy
scriptSig used in a spending transaction are written in this scripting
language.

Script is a very simple language.  It requires minimal processing and
cannot easily do many of the fancy things modern programming languages
can do.

((("Pay-to-Public-Key-Hash (P2PKH)")))When legacy transactions were the
most commonly used type of transaction, the majority of transactions processed
through the Bitcoin network had the form "Payment to Bob's Bitcoin
address" and used a script called a Pay-to-Public-Key-Hash script.
However, Bitcoin transactions are not limited to the "Payment to Bob's
Bitcoin address" script. In fact, scripts can be written to express a
vast variety of complex conditions. In order to understand these more
complex scripts, we must first understand the basics of transaction
scripts and Script language.

In this section, we will demonstrate the basic components of the Bitcoin
transaction scripting language and show how it can be used to express
conditions for spending and how those conditions can be satisfied.

[TIP]
====
((("programmable money")))Bitcoin transaction validation is not based on
a static pattern, but instead is achieved through the execution of a
scripting language. This language allows for a nearly infinite variety
of conditions to be expressed.
====

==== Turing Incompleteness

((("Turing incompleteness")))The bitcoin transaction script language
contains many operators, but is deliberately limited in one important
way--there are no loops or complex flow control capabilities other than
conditional flow control. This ensures that the language is not _Turing
Complete_, meaning that scripts have limited complexity and predictable
execution times. Script is not a general-purpose language.
((("denial-of-service attacks")))((("denial-of-service attacks",
see="also security")))((("security", "denial-of-service attacks")))These
limitations ensure that the language cannot be used to create an
infinite loop or other form of "logic bomb" that could be embedded in a
transaction in a way that causes a denial-of-service attack against the
Bitcoin network. Remember, every transaction is validated by every full
node on the Bitcoin network. A limited language prevents the transaction
validation mechanism from being used as a vulnerability.

==== Stateless Verification

((("stateless verification")))The Bitcoin transaction script language is
stateless, in that there is no state prior to execution of the script,
or state saved after execution of the script. All the
information needed to execute a script is contained within the script
and the transaction executing the script.  A
script will predictably execute the same way on any system. If your
system verifies a script, you can be sure that every other system in the
Bitcoin network will also verify the script, meaning that a valid
transaction is valid for everyone and everyone knows this. This
predictability of outcomes is an essential benefit of the Bitcoin
system.

[[tx_lock_unlock]]
==== Script Construction

Bitcoin's legacy transaction validation engine relies on two parts of scripts
to validate transactions: a scriptPubKey and a scriptSig.

A scriptPubKey is a spending condition placed on an output:
it specifies the conditions that must be met to spend the output in the
future, such as who is authorized to spend the output and how they will
be authenticated.

A scriptSig is a script that satisfies the
conditions placed on an output by a scriptPubKey and allows the output
to be spent. ScriptSigs are part of every transaction input. Most
of the time in legacy transactions they contain a digital signature produced by the user's
wallet from his or her private key, but not all scriptSigs
must contain signatures.

Every bitcoin validating node will validate transactions by executing
the scriptPubKey and scriptSig together.  As we saw in
<<c_transactions>>, each input contains an outpoint which refers to a
previous transaction output.  The input also contains a scriptSig.  The
validation software will copy the scriptSig, retrieve the UTXO
referenced by the input, and copy the scriptPubKey from that UTXO. The
scriptSig and scriptPubKey are then executed in sequence. The input is
valid if the scriptSig satisfies the scriptPubKey conditions
(see <<script_exec>>). All the inputs are validated independently, as
part of the overall validation of the transaction.

Note that the steps above involve making copies of all data.  The
original data in the previous output and current input is never changed.
In particular, the previous output is invariable and unaffected by
failed attempts to spend it.  Only a valid transaction that correctly
satisfies the conditions of the scriptPubKey results in the output being
considered as "spent".

<<scriptSig_and_scriptPubKey>> is an example of the scriptPubKey and
scriptSig for the most common type of legacy Bitcoin transaction (a
payment to a public key hash), showing the combined script resulting
from the concatenation of the scripts prior to
validation.

[[scriptSig_and_scriptPubKey]]
.Combining scriptSig and scriptPubKey to evaluate a transaction script
image::../images/mbc2_0603.png["scriptSig_and_scriptPubKey"]

===== The script execution stack

Bitcoin's scripting language is called a stack-based language because it
uses a data structure called a _stack_. A stack is a very simple data
structure that can be visualized as a stack of cards. A stack allows two
operations: push and pop. Push adds an item on top of the stack. Pop
removes the top item from the stack. Operations on a stack can only act
on the topmost item on the stack. A stack data structure is also called
a Last-In-First-Out, or "LIFO" queue.

The scripting language executes the script by processing each item from
left to right. Numbers (data constants) are pushed onto the stack.
Operators push or pop one or more parameters from the stack, act on
them, and might push a result onto the stack. For example, +OP_ADD+ will
pop two items from the stack, add them, and push the resulting sum onto
the stack.

Conditional operators evaluate a condition, producing a boolean result
of TRUE or FALSE. For example, +OP_EQUAL+ pops two items from the stack
and pushes TRUE (TRUE is represented by the number 1) if they are equal
or FALSE (represented by zero) if they are not equal. Bitcoin
transaction scripts usually contain a conditional operator, so that they
can produce the TRUE result that signifies a valid transaction.

===== A simple script

Now let's apply what we've learned about scripts and stacks to some simple examples.

In <<simplemath_script>>, the script +2 3 OP_ADD 5 OP_EQUAL+
demonstrates the arithmetic addition operator +OP_ADD+, adding two
numbers and putting the result on the stack, followed by the conditional
operator +OP_EQUAL+, which checks that the resulting sum is equal to
+5+. For brevity, the +OP_+ prefix may sometimes be omitted in examples
in this book. For more details on the available script operators and
functions, see <<tx_script_ops>>.

Although most legacy scriptPubKeys refer to a public key hash (essentially, a
legacy Bitcoin address), thereby requiring proof of ownership to spend the
funds, the script does not have to be that complex. Any combination of
scriptPubKey and scriptSig that results in a TRUE value is valid. The
simple arithmetic we used as an example of the scripting language is
also a valid script.

Use part of the arithmetic example script as the scriptPubKey:

----
3 OP_ADD 5 OP_EQUAL
----

which can be satisfied by a transaction containing an input with the
scriptSig:

----
2
----

The validation software combines the scripts:

----
2 3 OP_ADD 5 OP_EQUAL
----

As we saw in the step-by-step example in <<simplemath_script>>, when
this script is executed, the result is +OP_TRUE+, making the transaction
valid. Not only have we shown a valid transaction output scriptPubKey, but
the resulting UTXO could be spent by anyone with the arithmetic skills
to know that the number 2 satisfies the script.

[TIP]
====
((("transactions", "valid and invalid")))Transactions are valid if the
top result on the stack is +TRUE+, which is any
non-zero value.
Transactions are invalid if the top value on the stack is +FALSE+ (the
value zero or an empty stack), the script
execution is halted explicitly by an operator (such as +VERIFY+,
+OP_RETURN+), or the script was not semantically valid (such as
containing an +OP_IF+ statement that was not terminated by an +OP_ENDIF+
opcode). See
<<tx_script_ops>> for details.
====

[[simplemath_script]]
.Bitcoin's script validation doing simple math
image::../images/mbc2_0604.png["TxScriptSimpleMathExample"]

[role="pagebreak-before"]
The following is a slightly more complex script, which calculates ++2 +
7 -- 3 + 1++. Notice that when the script contains several operators in
a row, the stack allows the results of one operator to be acted upon by
the next operator:

----
2 7 OP_ADD 3 OP_SUB 1 OP_ADD 7 OP_EQUAL
----

Try validating the preceding script yourself using pencil and paper.
When the script execution ends, you should be left with a +TRUE+ value
on the stack.

[[script_exec]]
===== Separate execution of scriptPubKey and scriptSig

In the original Bitcoin
client, scriptPubKey and scriptSig were concatenated and executed
in sequence. For security reasons, this was changed in 2010 because of
a vulnerability known as the +1 OP_RETURN+ bug.  In the current
implementation, the scripts are executed separately with the stack
transferred between the two executions, as described next.

First, the scriptSig executed using the stack execution
engine. If the scriptSig is executed without errors and has
no operations left over, the stack is copied and the
scriptPubKey is executed. If the result of executing scriptPubKey
with the stack data copied from scriptSig is +TRUE+,
the scriptSig has succeeded in resolving the conditions imposed
by the scriptPubKey and, therefore, the input is a valid authorization
to spend the UTXO. If any result other than +TRUE+ remains after
execution of the combined script, the input is invalid because it has
failed to satisfy the spending conditions placed on the output.

//SOMEDAY:implications of not being able to use script in scriptSig

[[p2pkh]]
==== Pay-to-Public-Key-Hash (P2PKH)

((("Pay-to-Public-Key-Hash (P2PKH)")))
A Pay-to-Public-Key-Hash or "P2PKH" script uses a scriptPubKey that
contains a hash which commits to a public key.  P2PKH is best known as a
the basis for a legacy Bitcoin address. An P2PKH output can be spent by
presenting a public key which matches the hash commitment and a digital
signature created by the corresponding private key (see
<<digital_sigs>>).  Let's look at an example of a P2PKH scriptPubKey:

----
OP_DUP OP_HASH160 <Key Hash> OP_EQUALVERIFY OP_CHECKSIG
----

The +Key Hash+ is the data that would be encoded into a legacy Base58Check
address.  Most applications would show the _public key hash_ in a script
using hexadecimal encoding and not the familiar Bitcoin
address Base58Check format that begins with a "1."

The preceding scriptPubKey can be satisfied with a scriptSig
of the form:

----
<Signature> <Public Key>
----

The two scripts together would form the following combined validation
script:

----
<Sig> <Pubkey> OP_DUP OP_HASH160 <Hash> OP_EQUALVERIFY OP_CHECKSIG
----

When executed, this combined script will evaluate to TRUE if, and only
if, the scriptSig matches the conditions set by the scriptPubKey.
In other words, the result will be TRUE if the scriptSig
has a valid signature from Bob's private key that corresponds to
the public key hash set as an encumbrance.

Figures pass:[<a data-type="xref" href="#P2PubKHash1"
data-xrefstyle="select: labelnumber">#P2PubKHash1</a>] and pass:[<a
data-type="xref" href="#P2PubKHash2" data-xrefstyle="select:
labelnumber">#P2PubKHash2</a>] show (in two parts) a step-by-step
execution of the combined script, which will prove this is a valid
transaction.((("", startref="Tsript06")))((("",
startref="Stransact06")))

[[P2PubKHash1]]
.Evaluating a script for a P2PKH transaction (part 1 of 2)
image::../images/mbc2_0605.png["Tx_Script_P2PubKeyHash_1"]

[[P2PubKHash2]]
.Evaluating a script for a P2PKH transaction (part 2 of 2)
image::../images/mbc2_0606.png["Tx_Script_P2PubKeyHash_2"]

[[multisig]]
=== Multisignature

((("transactions", "advanced", "multisignature
scripts")))((("transactions", "advanced", id="Tadv07")))((("scripting",
"multisignature scripts", id="Smulti07")))((("multisignature
scripts")))Multisignature scripts set a condition where N public keys
are recorded in the script and at least K of those must provide
signatures to spend the funds. This is also known as a K-of-N scheme,
where N is the total number of keys and K is the threshold of signatures
required for validation. For example, a 2-of-3 multisignature is one
where three public keys are listed as potential signers and at least two
of those must be used to create signatures for a valid transaction to
spend the funds.

[TIP]
====
Some Bitcoin documentation, including earlier editions of this book,
uses the term M-of-N for traditional multisignature.  However, it's hard
to tell "M" and "N" apart when they're spoken, so we use the alternative
K-of-N.  Both phrases refer to the same type of signature scheme.
====

At this time, _standard_ multisignature scriptPubKeys are limited to at most 3
listed public keys, meaning you can do anything from a 1-of-1 to a
3-of-3 multisignature or any combination within that range.
You may want to check the +IsStandard()+ function to see what is currently
accepted by the network. Note that the limit of 3 keys applies only to
standard (also known as "bare") multisignature scripts, not to
multisignature scripts wrapped in a Pay-to-Script-Hash (P2SH) script.
P2SH multisignature scripts are limited to 15 keys, allowing for up to
15-of-15 multisignature. We will learn about P2SH in <<p2sh>>.

The general form of a scriptPubKey setting an K-of-N multisignature
condition is:

----
K <Public Key 1> <Public Key 2> ... <Public Key N> N OP_CHECKMULTISIG
----

where N is the total number of listed public keys and K is the threshold
of required signatures to spend the output.

A scriptPubKey setting a 2-of-3 multisignature condition looks like
this:

----
2 <Public Key A> <Public Key B> <Public Key C> 3 OP_CHECKMULTISIG
----

The preceding scriptPubKey can be satisfied with a scriptSig
containing pairs of signatures and public keys:

----
<Signature B> <Signature C>
----

or any combination of two signatures from the private keys corresponding
to the three listed public keys.

The two scripts together would form the combined validation script:

----
<Signature B> <Signature C> 2 <Public Key A> <Public Key B> <Public Key C> 3 OP_CHECKMULTISIG
----

When executed, this combined script will evaluate to TRUE if, and only
if, the scriptSig script matches the conditions set by the scriptPubKey.
In this case, the condition is whether the scriptSig has
a valid signature from the two private keys that correspond to two of
the three public keys set as an encumbrance.

[[multisig_bug]]
==== An oddity in CHECKMULTISIG execution

((("scripting", "multisignature scripts", "OP_CHECKMULTISIG
oddity")))((("OP_CHECKMULTISIG oddity workaround")))There is an oddity in
++OP_CHECKMULTISIG++'s execution that requires a slight workaround. When
+OP_CHECKMULTISIG+ executes, it should consume K+N+2 items on the stack as
parameters. However, due to the oddity, +OP_CHECKMULTISIG+ will pop an extra
value or one value more than expected.

Let's look at this in greater detail using the previous validation
example:

----
<Signature B> <Signature C> 2 <Public Key A> <Public Key B> <Public Key C> 3 OP_CHECKMULTISIG
----

First, +OP_CHECKMULTISIG+ pops the top item, which is +N+ (in this example
"3"). Then it pops +N+ items, which are the public keys that can sign.
In this example, public keys A, B, and C. Then, it pops one item, which
is +K+, the quorum (how many signatures are needed). Here M = 2. At this
point, +OP_CHECKMULTISIG+ should pop the final +K+ items, which are the
signatures, and see if they are valid. However, unfortunately, an oddity in
the implementation causes +OP_CHECKMULTISIG+ to pop one more item (K+1
total) than it should. The extra item is called the _dummy stack
element_ and it is disregarded when checking the
signatures so it has no direct effect on +OP_CHECKMULTISIG+ itself.
However, the dummy element must be present because, if it is not present,
when +OP_CHECKMULTISIG+ attempts to pop on an empty stack, it will cause a
stack error and script failure (marking the transaction as invalid).
Because the dummy element is disregarded it can be anything, but
it became the custom early on to use +OP_0+, which later became a
relay policy rule and eventually a consensus rule (with the enforcement of BIP147).

Because popping the dummy element is part of the consensus rules, it must now be
replicated forever. Therefore the correct script validation would look
like this:

----
OP_0 <Signature B> <Signature C> 2 <Public Key A> <Public Key B> <Public Key C> 3 OP_CHECKMULTISIG
----

Thus the scriptSig actually used in multisig is not:

----
<Signature B> <Signature C>
----

but instead it is:

----
OP_0 <Signature B> <Signature C>
----

Some people believe this oddity was a bug in the original code for
Bitcoin, but a plausible alternative explanation exists.  Verifying
K-of-N signatures can require many more than K or N signature checking
operations.  Let's consider a simple example of 1-in-3, with the
following combined script:

----
<dummy> <Sig4> 1 <key0> <key1> <key2> <key3> <key4> 5 OP_CHECKMULTISIG
----

The signature is checked first against +key0+, then +key1+, and then
the other keys before it is finally compared to its corresponding
+key4+.  That means five signature checking operations need to be
performed even though there's only one signature.  One way to eliminate
this redundancy would have been to provide OP_CHECKMULTISIG a map
indicating which provided signature corresponds to which public key,
allowing the OP_CHECKMULTISIG operation to only perform exactly K
signature-checking operations.  It's possible that Bitcoin's original
developer added the extra element (which we now call the dummy stack
element) in the original version of Bitcoin so that they could add the
feature for allowing a map to be passed in a later soft fork.  However,
that feature was never implemented and the BIP147 update to the
consensus rules in 2017 makes it impossible to add that feature in the
future.

Only Bitcoin's original developer could tell us whether the dummy stack
element was the result of a bug or a plan for a future upgrade.  In this
book, we simply call it an oddity.

From now on, if you see a multisig script, you should expect
to see an extra +OP_0+ in the beginning, whose only purpose is as a
workaround to an oddity in the consensus rules.((("",
startref="Smulti07")))

[[p2sh]]
=== Pay-to-Script-Hash (P2SH)

((("transactions", "advanced", "Pay-to-Script-Hash")))((("scripting",
"Pay-to-Script-Hash", id="Spay07")))Pay-to-Script-Hash (P2SH) was
introduced in 2012 as a powerful new type of transaction that greatly
simplifies the use of complex transaction scripts. To explain the need
for P2SH, let's look at a practical example.

((("use cases", "import/export", id="mohamseven")))((("scripting",
"Pay-to-Script-Hash", "import/export example")))((("Pay-to-Script-Hash
(P2SH)", "import/export example")))In <<ch01_intro_what_is_bitcoin>> we
introduced Mohammed, an electronics importer based in Dubai. Mohammed's
company uses bitcoin's multisignature feature extensively for its
corporate accounts. Multisignature scripts are one of the most common
uses of bitcoin's advanced scripting capabilities and are a very
powerful feature. ((("accounts receivable (AR)")))Mohammed's company
uses a multisignature script for all customer payments, known in
accounting terms as "accounts receivable," or AR. With the
multisignature scheme, any payments made by customers are locked in such
a way that they require at least two signatures to release, from
Mohammed and one of his partners or from his attorney who has a backup
key. A multisignature scheme like that offers corporate governance
controls and protects against theft, embezzlement, or loss.

The resulting script is quite long and looks like this:

----
2 <Mohammed's Public Key> <Partner1 Public Key> <Partner2 Public Key> <Partner3 Public Key> <Attorney Public Key> 5 OP_CHECKMULTISIG
----

Although multisignature scripts are a powerful feature, they are
cumbersome to use. Given the preceding script, Mohammed would have to
communicate this script to every customer prior to payment. Each
customer would have to use special bitcoin wallet software with the
ability to create custom transaction scripts, and each customer would
have to understand how to create a transaction using custom scripts.
Furthermore, the resulting transaction would be about five times larger
than a simple payment transaction, because this script contains very
long public keys. The burden of that extra data would be
borne by the customer in the form of extra transaction fees. Finally, a large transaction
script like this would be carried in the UTXO set in every full
node, until it was spent. All of these issues make using complex scriptPubKeys
difficult in practice.

P2SH was developed to resolve these practical difficulties and to make
the use of complex scripts as easy as a payment to a Bitcoin address.
With P2SH payments, the complex script is replaced with a
commitment, the digest of a cryptographic hash. When a transaction attempting
to spend the UTXO is presented later, it must contain the script that
matches the commitment in addition to the data which satisfies the script. In simple terms,
P2SH means "pay to a script matching this hash, a script that will be
presented later when this output is spent."

((("redeemScripts")))((("scripting", "redeemScripts")))In P2SH
transactions, the script that is replaced by a hash is referred
to as the _redeemScript_ because it is presented to the system at
redemption time rather than as a scriptPubKey. <<without_p2sh>> shows
the script without P2SH and <<with_p2sh>> shows the same script encoded
with P2SH.

[[without_p2sh]]
.Complex script without P2SH
|=======
| ScriptPubKey | 2 PubKey1 PubKey2 PubKey3 PubKey4 PubKey5 5 OP_CHECKMULTISIG
| ScriptSig | Sig1 Sig2
|=======

[[with_p2sh]]
.Complex script as P2SH
|=======
| RedeemScript | 2 PubKey1 PubKey2 PubKey3 PubKey4 PubKey5 5 OP_CHECKMULTISIG
| ScriptPubKey | OP_HASH160 <20-byte hash of redeemScript> OP_EQUAL
| ScriptSig | Sig1 Sig2 <redeemScript>
|=======

As you can see from the tables, with P2SH the complex script that
details the conditions for spending the output (redeemScript) is not
presented in the scriptPubKey. Instead, only a hash of it is in the
scriptPubKey and the reedemScript itself is presented later, as part
of the scriptSig when the output is spent. This shifts the burden
in fees and complexity from the sender to the recipient (spender) of the
transaction.

Let's look at Mohammed's company, the complex multisignature script, and
the resulting P2SH scripts.

First, the multisignature script that Mohammed's company uses for all
incoming payments from customers:

----
2 <Mohammed's Public Key> <Partner1 Public Key> <Partner2 Public Key> <Partner3 Public Key> <Attorney Public Key> 5 OP_CHECKMULTISIG
----

If the placeholders are replaced by actual public keys (shown here as
520-bit numbers starting with 04) you can see that this script becomes
very long:

----
2
04C16B8698A9ABF84250A7C3EA7EEDEF9897D1C8C6ADF47F06CF73370D74DCCA01CDCA79DCC5C395D7EEC6984D83F1F50C900A24DD47F569FD4193AF5DE762C58704A2192968D8655D6A935BEAF2CA23E3FB87A3495E7AF308EDF08DAC3C1FCBFC2C75B4B0F4D0B1B70CD2423657738C0C2B1D5CE65C97D78D0E34224858008E8B49047E63248B75DB7379BE9CDA8CE5751D16485F431E46117B9D0C1837C9D5737812F393DA7D4420D7E1A9162F0279CFC10F1E8E8F3020DECDBC3C0DD389D99779650421D65CBD7149B255382ED7F78E946580657EE6FDA162A187543A9D85BAAA93A4AB3A8F044DADA618D087227440645ABE8A35DA8C5B73997AD343BE5C2AFD94A5043752580AFA1ECED3C68D446BCAB69AC0BA7DF50D56231BE0AABF1FDEEC78A6A45E394BA29A1EDF518C022DD618DA774D207D137AAB59E0B000EB7ED238F4D800 5 OP_CHECKMULTISIG
----

This entire script can instead be represented by a 20-byte cryptographic
hash, by first applying the SHA256 hashing algorithm and then applying
the RIPEMD160 algorithm on the result.

We use +libbitcoin-explorer+ (+bx+) on the command-line to produce the
script hash, as follows:

----
echo \
2 \
[04C16B8698A9ABF84250A7C3EA7EEDEF9897D1C8C6ADF47F06CF73370D74DCCA01CDCA79DCC5C395D7EEC6984D83F1F50C900A24DD47F569FD4193AF5DE762C587] \
[04A2192968D8655D6A935BEAF2CA23E3FB87A3495E7AF308EDF08DAC3C1FCBFC2C75B4B0F4D0B1B70CD2423657738C0C2B1D5CE65C97D78D0E34224858008E8B49] \
[047E63248B75DB7379BE9CDA8CE5751D16485F431E46117B9D0C1837C9D5737812F393DA7D4420D7E1A9162F0279CFC10F1E8E8F3020DECDBC3C0DD389D9977965] \
[0421D65CBD7149B255382ED7F78E946580657EE6FDA162A187543A9D85BAAA93A4AB3A8F044DADA618D087227440645ABE8A35DA8C5B73997AD343BE5C2AFD94A5] \
[043752580AFA1ECED3C68D446BCAB69AC0BA7DF50D56231BE0AABF1FDEEC78A6A45E394BA29A1EDF518C022DD618DA774D207D137AAB59E0B000EB7ED238F4D800] \
5 CHECKMULTISIG \
| bx script-encode | bx sha256 | bx ripemd160
54c557e07dde5bb6cb791c7a540e0a4796f5e97e
----

The series of commands above first encodes Mohammed's multisig
redeemScript as a serialized hex-encoded bitcoin Script. The next +bx+ command
calculates the SHA256 hash of that. The next +bx+ command hashes again
with RIPEMD160, producing the final script-hash:

The 20-byte hash of Mohammed's redeemScript is:

----
54c557e07dde5bb6cb791c7a540e0a4796f5e97e
----

A P2SH transaction locks the output to this hash instead of the longer
redeemScript, using a special scriptPubKey template:

----
OP_HASH160 54c557e07dde5bb6cb791c7a540e0a4796f5e97e OP_EQUAL
----

which, as you can see, is much shorter. Instead of "pay to this 5-key
multisignature script," the P2SH equivalent transaction is "pay to a
script with this hash." A customer making a payment to Mohammed's
company need only include this much shorter scriptPubKey in his
payment. When Mohammed and his partners want to spend this UTXO, they
must present the original redeemScript (the one whose hash locked the
UTXO) and the signatures necessary to unlock it, like this:

----
<Sig1> <Sig2> <2 PK1 PK2 PK3 PK4 PK5 5 OP_CHECKMULTISIG>
----

The two scripts are combined in two stages. First, the redeemScript is
checked against the scriptPubKey to make sure the hash matches:

----
<2 PK1 PK2 PK3 PK4 PK5 5 OP_CHECKMULTISIG> OP_HASH160 <redeemScript hash> OP_EQUAL
----

If the redeemScript hash matches, the redeemScript is executed:

----
<Sig1> <Sig2> 2 PK1 PK2 PK3 PK4 PK5 5 OP_CHECKMULTISIG
----

==== P2SH Addresses

((("scripting", "Pay-to-Script-Hash",
"addresses")))((("Pay-to-Script-Hash (P2SH)", "addresses")))((("bitcoin
improvement proposals", "Address Format for P2SH (BIP13)")))Another
important part of the P2SH feature is the ability to encode a script
hash as an address, as defined in BIP13. P2SH addresses are Base58Check
encodings of the 20-byte hash of a script, just like Bitcoin addresses
are Base58Check encodings of the 20-byte hash of a public key. P2SH
addresses use the version prefix "5," which results in
Base58Check-encoded addresses that start with a "3."

For example, Mohammed's complex script, hashed and Base58Check-encoded
as a P2SH address, becomes +39RF6JqABiHdYHkfChV6USGMe6Nsr66Gzw+. We can
confirm that with the +bx+ command:

----
echo \
'54c557e07dde5bb6cb791c7a540e0a4796f5e97e'\
 | bx address-encode -v 5
39RF6JqABiHdYHkfChV6USGMe6Nsr66Gzw
----

Now, Mohammed can give this "address" to his customers and they can use
almost any bitcoin wallet to make a simple payment, as if it were a
Bitcoin address. The 3 prefix gives them a hint that this is a special
type of address, one corresponding to a script instead of a public key,
but otherwise it works in exactly the same way as a payment to any other Bitcoin
address.

P2SH addresses hide all of the complexity so that the person making a
payment does not see the script.

==== Benefits of P2SH

((("scripting", "Pay-to-Script-Hash", "benefits
of")))((("Pay-to-Script-Hash (P2SH)", "benefits of")))The P2SH feature
offers the following benefits compared to the direct use of complex
scripts in outputs:

- Complex scripts are replaced by shorter commitments in the
  transaction output, making the transaction smaller.

- Scripts can be coded as an address, so the sender and the sender's
  wallet don't need complex engineering to implement P2SH.

- P2SH shifts the burden of constructing the script to the recipient,
  not the sender.

- P2SH shifts the burden in data storage for the long script from the
  output (which additionally to being stored on the blockchain is in the
  UTXO set) to the input (only stored on the blockchain).

- P2SH shifts the burden in data storage for the long script from the
  present time (payment) to a future time (when it is spent).

- P2SH shifts the transaction fee cost of a long script from the sender
  to the recipient, who has to include the long redeemScript to spend
  it.

==== RedeemScript and Validation

You are not able to put a P2SH inside a P2SH redeemScript, because the
P2SH specification is not recursive. Also, while it is technically
possible to include +OP_RETURN+ (see <<op_return>>) in a redeemScript, as
nothing in the rules prevents you from doing so, it is of no practical
use because executing +OP_RETURN+ during validation will cause the
transaction to be marked invalid.

Note that because the redeemScript is not presented to the network
until you attempt to spend a P2SH output, if you create an output with the
hash of an invalid redeemScript, you will not be able to spend
it.  The spending transaction, which includes the redeemScript,
will not be accepted because it is an invalid script. This creates a
risk, because you can send bitcoin to a P2SH address that cannot be spent later.

[WARNING]
====
((("warnings and cautions", "accidental bitcoin invalidation")))P2SH scriptPubKeys
contain the hash of a redeemScript, which gives no clues as to
the content of the redeemScript. The P2SH transaction will be
considered valid and accepted even if the redeemScript is invalid. You
might accidentally receive bitcoin in such a way that it cannot later be
spent.
====

[[op_return]]
=== Data Recording Output (OP_RETURN)

((("transactions", "advanced", "data recording output")))((("scripting",
"data recording output")))((("OP_RETURN operator")))((("data recording
(nonpayment data)")))((("nonpayment data")))((("blockchain (the)",
"nonpayment data recording")))((("digital notary services")))Bitcoin's
distributed and timestamped ledger, the blockchain, has potential uses
beyond payments. Many developers have tried to use the transaction
scripting language to take advantage of the security and resilience of
the system for applications such as digital notary services
and smart contracts. Early attempts to use bitcoin's
script language for these purposes involved creating transaction outputs
that recorded data on the blockchain; for example, to record a digital
fingerprint of a file in such a way that anyone could establish
proof-of-existence of that file on a specific date by reference to that
transaction.

((("blockchain bloat")))((("bloat")))((("unspent transaction outputs
(UTXO)")))((("UTXO sets")))The use of bitcoin's blockchain to store data
unrelated to bitcoin payments is a controversial subject. Many
people consider such use abusive and want to discourage it. Others
view it as a demonstration of the powerful capabilities of blockchain
technology and want to encourage such experimentation. Those who object
to the inclusion of nonpayment data argue that it causes "blockchain
bloat," burdening those running full Bitcoin nodes with carrying the
cost of disk storage for data that the blockchain was not intended to
carry. Moreover, such transactions create UTXO that cannot be spent,
using the destination legacy Bitcoin address as a freeform 20-byte field.
Because the address is used for data, it doesn't correspond to a private
key and the resulting UTXO can _never_ be spent; it's a fake payment.
These transactions that can never be spent are therefore never removed
from the UTXO set and cause the size of the UTXO database to forever
increase, or "bloat."

A compromise was reached
that allows the a scriptPubKey starting with +OP_RETURN+ to
add nonpayment data to a transaction output. However, unlike
the use of "fake" UTXOs, the +OP_RETURN+ operator creates an explicitly
_provably unspendable_ output, which does not need to be stored in the
UTXO set. +OP_RETURN+ outputs are recorded on the blockchain, so they
consume disk space and contribute to the increase in the blockchain's
size, but they are not stored in the UTXO set and therefore do not bloat
the UTXO memory pool and burden full nodes with the cost of more
expensive database operations.

+OP_RETURN+ scripts look like this:

----
OP_RETURN <data>
----

((("Proof of Existence")))((("DOCPROOF prefix")))The data portion
often represents a hash, such as the output
from the SHA256 algorithm (32 bytes). Some applications put a prefix in
front of the data to help identify the application. For example, the
http://proofofexistence.com[Proof of Existence] digital notarization
service uses the 8-byte prefix +DOCPROOF+, which is ASCII encoded as +44
4f 43 50 52 4f 4f 46+ in hexadecimal.

Keep in mind that there is no scriptSig that corresponds to
+OP_RETURN+ that could possibly be used to "spend" an +OP_RETURN+ output. The
whole point of an +OP_RETURN+ output is that you can't spend the money locked in that
output, and therefore it does not need to be held in the UTXO set as
potentially spendable: +OP_RETURN+ is _provably unspendable_. +OP_RETURN+ is
usually an output with a zero amount, because any bitcoins
assigned to such an output is effectively lost forever. If an +OP_RETURN+ is
referenced as an input in a transaction, the script validation engine
will halt the execution of the validation script and mark the
transaction as invalid. The execution of +OP_RETURN+ essentially causes the
script to "RETURN" with a +FALSE+ and halt. Thus, if you accidentally
reference a +OP_RETURN+ output as an input in a transaction, that
transaction is invalid.

[[locktime_limitations]]
==== Transaction locktime limitations

+nLockTime+ has the limitation that while it makes it possible to spend some outputs in the future, it does not make it impossible to spend them until that time. Let's explain that with the following example.

((("use cases", "buying coffee", id="alicesseven")))Alice signs a transaction spending one of her outputs to Bob's address, and sets the transaction +nLockTime+ to 3 months in the future. Alice sends that transaction to Bob to hold. With this transaction Alice and Bob know that:

* Bob cannot transmit the transaction to redeem the funds until 3 months have elapsed.
* Bob may transmit the transaction after 3 months.

However:

* Alice can create another transaction, double-spending the same inputs without a locktime. Thus, Alice can spend the same UTXO before the 3 months have elapsed.
* Bob has no guarantee that Alice won't do that.

It is important to understand the limitations of transaction +nLockTime+. The only guarantee is that Bob will not be able to redeem it before 3 months have elapsed. There is no guarantee that Bob will get the funds. To achieve such a guarantee, the timelock restriction must be placed on the UTXO itself and be part of the script, rather than on the transaction. This is achieved by the next form of timelock, called Check Lock Time Verify.

==== Check Lock Time Verify (OP_CLTV)

((("Check Lock Time Verify (OP_CLTV)", id="cltv07")))((("timelocks", "Check
Lock Time Verify (OP_CLTV)")))((("scripting", "timelocks", "Check Lock Time
Verify (OP_CLTV)")))((("bitcoin improvement proposals",
"OP_CHECKLOCKTIMEVERIFY (BIP65)")))In December 2015, a new form of
timelock was introduced to Bitcoin as a soft fork upgrade. Based on a
specification in BIP65, a new script operator called
_OP_CHECKLOCKTIMEVERIFY_ (_CLTV_) was added to the scripting language.
+OP_CLTV+ is a per-output timelock, rather than a per-transaction timelock
as is the case with +nLockTime+. This allows for much greater
flexibility in the way timelocks are applied.

In simple terms, by committing to the +OP_CLTV+ opcode in an
output, that output is restricted so that it can only be spent after the
specified time has elapsed.

[TIP]
====
While +nLockTime+ is a transaction-level timelock, +OP_CLTV+ is an
output-based timelock.
====

+OP_CLTV+ doesn't replace +nLockTime+, but rather restricts specific UTXO
such that they can only be spent in a future transaction with
+nLockTime+ set to a greater or equal value.

The +OP_CLTV+ opcode takes one parameter as input, expressed as a number in
the same format as +nLockTime+ (either a block height or Unix epoch
time). As indicated by the +VERIFY+ suffix, +OP_CLTV+ is the type of opcode
that halts execution of the script if the outcome is +FALSE+. If it
results in TRUE, execution continues.

In order to use +OP_CLTV+, you insert it into the redeemScript of the
output in the transaction that creates the output. For
example, if Alice is paying Bob's address, the output would normally
contain a P2PKH script like this:

----
OP_DUP OP_HASH160 <Bob's Public Key Hash> OP_EQUALVERIFY OP_CHECKSIG
----

To lock it to a time, say 3 months from now, the transaction would be a
P2SH transaction with a redeemScript like this:

----
<now + 3 months> OP_CHECKLOCKTIMEVERIFY OP_DROP OP_DUP OP_HASH160 <Bob's Public Key Hash> OP_EQUALVERIFY OP_CHECKSIG
----

where +<now {plus} 3 months>+ is a block height or time value estimated
3 months from the time the transaction is mined: current block height
{plus} 12,960 (blocks) or current Unix epoch time {plus} 7,760,000
(seconds). For now, don't worry about the +OP_DROP+ opcode that follows
+OP_CHECKLOCKTIMEVERIFY+; it will be explained shortly.

When Bob tries to spend this UTXO, he constructs a transaction that
references the UTXO as an input. He uses his signature and public key in
the scriptSig of that input and sets the transaction +nLockTime+
to be equal or greater to the timelock in the +OP_CHECKLOCKTIMEVERIFY+
Alice set. Bob then broadcasts the transaction on the Bitcoin network.

Bob's transaction is evaluated as follows. If the +OP_CHECKLOCKTIMEVERIFY+
parameter Alice set is less than or equal the spending transaction's
+nLockTime+, script execution continues (acts as if a _no
operation_ or OP_NOP opcode was executed). Otherwise, script
execution halts and the transaction is deemed invalid.

More precisely, +OP_CHECKLOCKTIMEVERIFY+ fails and halts execution, marking
the transaction invalid if (source: BIP65):

1. the stack is empty; or
2. the top item on the stack is less than 0; or
3. the lock-time type (height versus timestamp) of the top stack item and the +nLockTime+ field are not the same; or
4. the top stack item is greater than the transaction's +nLockTime+ field; or
5. the +nSequence+ field of the input is 0xffffffff.

[[timelock_conflicts]]
.Timelock conflicts
[WARNING]
====
+OP_CLTV+ and +nLockTime+ use the same format to describe timelocks, either
a block height or the time elapsed in seconds since Unix epoch.
Critically, when used together, the format of +nLockTime+ must match
that of +OP_CLTV+ in the outputs--they must both reference either
block height or time in seconds.

The above implies that a script can never be valid if it must execute
two different calls to +OP_CLTV+, one which uses a height and one which
uses a time.  It can be easy to make this mistake when writing advanced
scripts, so be sure to thoroughly test your scripts on a test network or
use a tool designed to prevent this issue, like a Miniscript compiler.

An additional implication is that only one variety of +OP_CLTV+ can be
used in any of the scripts of a transaction.  If the script for one
input uses the height variety and a different script for a different
input uses the time variety, there is no way to construct a valid transaction.
====

After execution, if +OP_CLTV+ is satisfied, the parameter that
preceded it remains as the top item on the stack and may need to be
dropped, with +OP_DROP+, for correct execution of subsequent script
opcodes. You will often see +OP_CHECKLOCKTIMEVERIFY+ followed by +OP_DROP+ in
scripts for this reason.

By using nLockTime in conjunction with +OP_CLTV+, the scenario described in
<<locktime_limitations>> changes. Alice can no longer spend the money
(because it's locked with Bob's key) and Bob cannot spend it before the
3-month locktime has expired.((("", startref="alicesseven")))

By introducing timelock functionality directly into the scripting
language, +OP_CLTV+ allows us to develop some very interesting complex
scripts.((("", startref="cltv07")))

The standard is defined in
https://github.com/bitcoin/bips/blob/master/bip-0065.mediawiki[BIP65
(OP_CHECKLOCKTIMEVERIFY)].

==== Relative Timelocks

+nLockTime+ and +OP_CLTV+ are ((("timelocks", "relative timelocks",
id="Trelative07")))((("scripting", "timelocks", "relative
timelocks")))((("relative timelocks", id="relativetime07")))both
_absolute timelocks_ in that they specify an absolute point in time. The
next two timelock features we will examine are _relative timelocks_ in
that they specify, as a condition of spending an output, an elapsed time
from the confirmation of the output in the blockchain.

Relative timelocks are useful because they allow
imposing a time
constraint on one transaction that is dependent on the elapsed time from
the confirmation of a previous transaction. In other words, the clock
doesn't start counting until the UTXO is recorded on the blockchain.
This functionality is especially useful in bidirectional state channels
and Lightning Networks, as we will see in <<state_channels>>.

Relative timelocks, like absolute timelocks, are implemented with both a
transaction-level feature and a script-level opcode. The
transaction-level relative timelock is implemented as a consensus rule
on the value of +nSequence+, a transaction field that is set in every
transaction input. Script-level relative timelocks are implemented with
the +OP_CHECKSEQUENCEVERIFY+ (OP_CSV) opcode.

((("bitcoin improvement proposals", "Relative lock-time using
consensus-enforced sequence numbers (BIP68)")))((("bitcoin improvement
proposals", "CHECKSEQUENCEVERIFY (BIP112)")))Relative timelocks are
implemented according to the specifications in
https://github.com/bitcoin/bips/blob/master/bip-0068.mediawiki[BIP68,
Relative lock-time using consensus-enforced sequence numbers] and
https://github.com/bitcoin/bips/blob/master/bip-0112.mediawiki[BIP112,
OP_CHECKSEQUENCEVERIFY].

BIP68 and BIP112 were activated in May 2016 as a soft fork upgrade to
the consensus rules.

==== Relative Timelocks with OP_CSV

((("scripting", "timelocks", "relative timelocks with
CHECKSEQUENCEVERIFY")))((("CHECKSEQUENCEVERIFY (CSV)")))Just like OP_CLTV
and +nLockTime+, there is a script opcode for relative timelocks that
leverages the +nSequence+ value in scripts. That opcode is
+OP_CHECKSEQUENCEVERIFY+, commonly referred to as +OP_CSV+ for short.

The +OP_CSV+ opcode when evaluated in an UTXO's script allows
spending only in a transaction whose input +nSequence+ value is greater
than or equal to the +OP_CSV+ parameter. Essentially, this restricts
spending the UTXO until a certain number of blocks or seconds have
elapsed relative to the time the UTXO was mined.

As with CLTV, the value in +OP_CSV+ must match the format in the
corresponding +nSequence+ value. If +OP_CSV+ is specified in terms of
blocks, then so must +nSequence+. If +OP_CSV+ is specified in terms of
seconds, then so must +nSequence+.

[[WARNING]]
====
A script executing multiple +OP_CSV+ opcodes must only use the same
variety, either time-based or height-based.  Mixing varieties will
produce an invalid script that can never be spent, the same problem we
saw with +OP_CLTV+ in <<timelock_conflicts>>.  However, +OP_CSV+ allows
any two valid inputs to be included in the same transaction, so the problem
of interaction across inputs that occurs with +OP_CLTV+ doesn't affect +OP_CSV+.
====

Relative timelocks with +OP_CSV+ are especially useful when several
(chained) transactions are created and signed, but not propagated--that
is, they're kept off the blockchain (_offchain_). A child transaction cannot be used until the
parent transaction has been propagated, mined, and aged by the time
specified in the relative timelock. One application of this use case can
be seen in <<state_channels>> and <<lightning_network>>.((("",
startref="relativetime07")))((("", startref="Trelative07")))

+OP_CSV+ is defined in detail in
https://github.com/bitcoin/bips/blob/master/bip-0112.mediawiki[BIP112,
CHECKSEQUENCEVERIFY].

=== Scripts with Flow Control (Conditional Clauses)

((("transactions", "advanced", "flow control scripts")))((("scripting",
"flow control scripts", id="Sflow07")))((("conditional clauses",
id="condition07")))((("flow control", id="flow07")))One of the more
powerful features of Bitcoin Script is flow control, also known as
conditional clauses. You are probably familiar with flow control in
various programming languages that use the construct +IF...THEN...ELSE+.
Bitcoin conditional clauses look a bit different, but are essentially
the same construct.

At a basic level, bitcoin conditional opcodes allow us to construct a
script that has two ways of being unlocked, depending on a
+TRUE+/+FALSE+ outcome of evaluating a logical condition. For example,
if x is +TRUE+, the executed code path is A and the ELSE code path is B.

Additionally, bitcoin conditional expressions can be "nested"
indefinitely, meaning that a conditional clause can contain another
within it, which contains another, etc. Bitcoin Script flow control can
be used to construct very complex scripts with hundreds
of possible execution paths. There is no limit to nesting, but
consensus rules impose a limit on the maximum size of a
script in bytes.

Bitcoin implements flow control using the +OP_IF+, +OP_ELSE+, +OP_ENDIF+, and
+OP_NOTIF+ opcodes. Additionally, conditional expressions can contain
boolean operators such as +OP_BOOLAND+, +OP_BOOLOR+, and +OP_NOT+.

At first glance, you may find the bitcoin's flow control scripts
confusing. That is because Bitcoin Script is a stack language. The same
way that +1 {plus} 1+ looks "backward" when expressed as +1 1 OP_ADD+, flow
control clauses in bitcoin also look "backward."

In most traditional (procedural) programming languages, flow control
looks like this:

.Pseudocode of flow control in most programming languages
----
if (condition):
  code to run when condition is true
else:
  code to run when condition is false
endif
code to run in either case
----

In a stack-based language like Bitcoin Script, the logical condition
comes before the +IF+, which makes it look "backward," like this:

.Bitcoin Script flow control
----
condition
IF
  code to run when condition is true
OP_ELSE
  code to run when condition is false
OP_ENDIF
code to run in either case
----

When reading Bitcoin Script, remember that the condition being evaluated
comes _before_ the +IF+ opcode.

==== Conditional Clauses with VERIFY Opcodes

((("VERIFY opcodes")))((("IF clauses")))((("opcodes", "VERIFY")))Another
form of conditional in Bitcoin Script is any opcode that ends in
+VERIFY+. The +VERIFY+ suffix means that if the condition evaluated is
not +TRUE+, execution of the script terminates immediately and the
transaction is deemed invalid.

((("guard clauses")))Unlike an +IF+ clause, which offers alternative
execution paths, the +VERIFY+ suffix acts as a _guard clause_,
continuing only if a precondition is met.

For example, the following script requires Bob's signature and a
pre-image (secret) that produces a specific hash. Both conditions must
be satisfied to unlock:

.A script with an +OP_EQUALVERIFY+  guard clause.
----
OP_HASH160 <expected hash> OP_EQUALVERIFY <Bob's Pubkey> OP_CHECKSIG
----

To spend this, Bob must present a
valid pre-image and a signature:

.Satisfying the above script
----
<Bob's Sig> <hash pre-image>
----

Without presenting the pre-image, Bob can't get to the part of the
script that checks for his signature.

[role="pagebreak-after"]
This script can be written with an +OP_IF+ instead:

.A script with an +IF+ guard clause
----
OP_HASH160 <expected hash> OP_EQUAL
OP_IF
   <Bob's Pubkey> OP_CHECKSIG
OP_ENDIF
----

Bob's authentication data identical:

.Satisfying the above script
----
<Bob's Sig> <hash pre-image>
----

The script with +OP_IF+ does the same thing as using an opcode with a
+VERIFY+ suffix; they both operate as guard clauses. However, the
+VERIFY+ construction is more efficient, using two fewer opcodes.

So, when do we use +VERIFY+ and when do we use +OP_IF+? If all we are
trying to do is to attach a precondition (guard clause), then +VERIFY+
is better. If, however, we want to have more than one execution path
(flow control), then we need an +OP_IF...OP_ELSE+ flow control clause.

==== Using Flow Control in Scripts

A very common use for flow control in Bitcoin Script is to construct a
script that offers multiple execution paths, each a different way
of redeeming the UTXO.

((("use cases", "buying coffee")))Let's look at a simple example, where
we have two signers, Alice and Bob, and either one is able to redeem.
With multisig, this would be expressed as a 1-of-2 multisig script. For
the sake of demonstration, we will do the same thing with an +OP_IF+
clause:

----
OP_IF
 <Alice's Pubkey>
OP_ELSE
 <Bob's Pubkey>
OP_ENDIF
OP_CHECKSIG
----

Looking at this redeemScript, you may be wondering: "Where is the
condition? There is nothing preceding the +IF+ clause!"

The condition is not part of the script. Instead, the condition
will be offered at spending time, allowing Alice and Bob to
"choose" which execution path they want.

.Alice satisfies the above script:
----
<Alice's Sig> OP_TRUE
----

The +OP_TRUE+ at the end serves as the condition (+TRUE+) that will make the
+OP_IF+ clause execute the first redemption path which puts the public
key on the stack for which Alice has a
signature.  The +OP_TRUE+ opcode, also known as +OP_1+, will put the
number 1 on the stack.

For Bob to redeem this, he would have to choose the second execution
path in +OP_IF+ by giving a +FALSE+ value.  The +OP_FALSE+ opcode, also
known as +OP_0+, pushes an empty byte array to the stack.

----
<Bob's Sig> OP_FALSE
----

Bob's scriptSig causes the +OP_IF+ clause
to execute the second (+OP_ELSE+) script, which requires Bob's signature.

Since +OP_IF+ clauses can be nested, we can create a "maze" of execution
paths. The scriptSig can provide a "map" selecting which
execution path is actually executed:

----
OP_IF
  subscript A
OP_ELSE
  OP_IF
    subscript B
  OP_ELSE
    subscript C
  OP_ENDIF
OP_ENDIF
----

In this scenario, there are three execution paths (+subscript A+, +subscript
B+, and +subscript C+). The scriptSig provides a path in the form of
a sequence of +TRUE+ or +FALSE+ values. To select path +subscript B+, for
example, the scriptSig must end in +OP_1 OP_0+ (+TRUE+, +FALSE+). These
values will be pushed onto the stack, so that the second value (+FALSE+)
ends up at the top of the stack. The outer +OP_IF+ clause pops the +FALSE+
value and executes the first +OP_ELSE+ clause. Then the +TRUE+ value moves
to the top of the stack and is evaluated by the inner (nested) +OP_IF+,
selecting the +B+ execution path.

Using this construct, we can build redeemScripts with tens or hundreds
of execution paths, each offering a different way to redeem the UTXO. To
spend, we construct an scriptSig that navigates the execution
path by putting the appropriate +TRUE+ and +FALSE+ values on the stack
at each flow control point.((("", startref="Sflow07")))((("",
startref="flow07")))((("", startref="condition07")))

=== Complex Script Example

((("transactions", "advanced", "example")))((("scripting", "complex
script example", id="Scomplex07")))In this section we combine many of
the concepts from this chapter into a single example.

((("use cases", "import/export", id="mohamseventwo")))Our example uses
the story of Mohammed, the company owner in Dubai who is operating an
import/export business.

((("transactions", "advanced", "multisignature
scripts")))((("scripting", "multisignature scripts", "import/export
example")))((("multisignature scripts")))In this example, Mohammed
wishes to construct a company capital account with flexible rules. The
scheme he creates requires different levels of authorization depending
on timelocks. The participants in the multisig scheme are Mohammed, his
two partners Saeed and Zaira, and their company lawyer Abdul. The three
partners make decisions based on a majority rule, so two of the three
must agree. However, in the case of a problem with their keys, they want
their lawyer to be able to recover the funds with one of the three
partner signatures. Finally, if all partners are unavailable or
incapacitated for a while, they want the lawyer to be able to manage the
account directly.

Here's the redeemScript that Mohammed designs to achieve this (line
number prefixed as XX):

[[variable_timelock_multisig]]
.Variable Multi-Signature with Timelock
----
01  OP_IF
02    OP_IF
03      2
04    OP_ELSE
05      <30 days> OP_CHECKSEQUENCEVERIFY OP_DROP
06      <Abdul the Lawyer's Pubkey> OP_CHECKSIGVERIFY
07      1
08    OP_ENDIF
09    <Mohammed's Pubkey> <Saeed's Pubkey> <Zaira's Pubkey> 3 OP_CHECKMULTISIG
10  OP_ELSE
11    <90 days> OP_CHECKSEQUENCEVERIFY OP_DROP
12    <Abdul the Lawyer's Pubkey> OP_CHECKSIG
13  OP_ENDIF
----

Mohammed's script implements three execution paths using nested
+OP_IF...OP_ELSE+ flow control clauses.

In the first execution path, this script operates as a simple 2-of-3
multisig with the three partners. This execution path consists of lines
3 and 9. Line 3 sets the quorum of the multisig to +2+ (2-of-3). This
execution path can be selected by putting +OP_TRUE OP_TRUE+ at the end of the
scriptSig:

.Spending data for the first execution path (2-of-3 multisig)
----
OP_0 <Mohammed's Sig> <Zaira's Sig> OP_TRUE OP_TRUE
----


[TIP]
====
The +OP_0+ at the beginning of this scriptSig is because of an oddity in
+OP_CHECKMULTISIG+ that pops an extra value from the stack. The extra value
is disregarded by the +OP_CHECKMULTISIG+, but it must be present or the
script fails. Pushing an empty byte array with +OP_0+ is a workaround to the oddity, as
described in <<multisig_bug>>.
====

The second execution path can only be used after 30 days have elapsed
from the creation of the UTXO. At that time, it requires the signature
of Abdul the lawyer and one of the three partners (a 1-of-3 multisig).
This is achieved by line 7, which sets the quorum for the multisig to
+1+. To select this execution path, the scriptSig would end in
+OP_FALSE OP_TRUE+:

.Spending data for the second execution path (Lawyer + 1-of-3)
----
0 <Saeed's Sig> <Abdul's Sig> OP_FALSE OP_TRUE
----

[TIP]
====
Why +OP_FALSE OP_TRUE+? Isn't that backward? Because the two values are pushed
on to the stack, with +FALSE+ pushed first, then +TRUE+ pushed second.
+TRUE+ is therefore popped _first_ by the first +OP_IF+ opcode.
====

Finally, the third execution path allows Abdul the lawyer to spend the
funds alone, but only after 90 days. To select this execution path, the
scriptSig has to end in +OP_FALSE+:

.ScriptSig for the third execution path (Lawyer only)
----
<Abdul's Sig> OP_FALSE
----

Try running the script on paper to see how it behaves on the stack.

A few more things to consider when reading this example. See if you can
find the answers:

- Why can't the lawyer redeem the third execution path at any time by
  selecting it with +OP_FALSE+ on the scriptSig?

- How many execution paths can be used 5, 35, and 105 days,
  respectively, after the UTXO is mined?

- Are the funds lost if the lawyer loses his key? Does your answer
  change if 91 days have elapsed?

- How do the partners "reset" the clock every 29 or 89 days to prevent
  the lawyer from accessing the funds?

- Why do some +OP_CHECKSIG+ opcodes in this script have the +VERIFY+ suffix
  while others don't?((("", startref="Scomplex07")))((("",
  startref="mohamseventwo")))

==== Segregated Witness Output and Transaction Examples

Let’s look at some of our example transactions and see how they would
change with Segregated Witness. We’ll first look at how a
Pay-to-Public-Key-Hash (P2PKH) payment is transformed with the
Segregated Witness program. Then, we’ll look at the Segregated Witness
equivalent for Pay-to-Script-Hash (P2SH) scripts. Finally, we’ll look at
how both of the preceding Segregated Witness programs can be embedded
inside a P2SH script.

[[p2wpkh]]
===== Pay-to-Witness-Public-Key-Hash (P2WPKH)

Let's start by looking at the example of a P2PKH
scriptPubKey:

.Example P2PKH scriptPubKey
----
OP_DUP OP_HASH160 ab68025513c3dbd2f7b92a94e0581f5d50f654e7 OP_EQUALVERIFY OP_CHECKSIG
----

With Segregated Witness, Alice would create a
Pay-to-Witness-Public-Key-Hash (P2WPKH) script.  If that script commits
to the same public key, it would look like this:

.Example P2WPKH output script
----
0 ab68025513c3dbd2f7b92a94e0581f5d50f654e7
----

As you can see, a P2WPKH scriptPubKey is much
simpler than the P2PKH equivalent. It consists of two values that are
pushed on to the script evaluation stack. To an old (nonsegwit-aware)
Bitcoin client, the two pushes would look like an output that anyone can
spend. To a newer, segwit-aware client, the first number (0)
is interpreted as a version number (the _witness version_) and the
second part (20 bytes) is a
_witness program_. The 20-byte witness program is simply the hash of the
public key, as in a P2PKH script

Now, let’s look at the corresponding transaction that Bob uses to spend
this output. For the original script, the spending transaction
would have to include a signature within the transaction input:

.Decoded transaction showing a P2PKH output being spent with a signature
----
[...]
“Vin” : [
"txid": "0627052b6f28912f2703066a912ea577f2ce4da4caa5a5fbd8a57286c345c2f2",
"vout": 0,
     	 "scriptSig": “<Bob’s scriptSig>”,
]
[...]
----

However, to spend the P2PKH output, the transaction has no
signature on that input. Instead, Bob’s transaction has an empty
+scriptSig+ and includes a witness field:

.Decoded transaction showing a P2WPKH output being spent with a witness field
----
[...]
“Vin” : [
"txid": "0627052b6f28912f2703066a912ea577f2ce4da4caa5a5fbd8a57286c345c2f2",
"vout": 0,
     	 "scriptSig": “”,
]
[...]
“witness”: “<Bob’s witness data>”
[...]
----

===== Wallet construction of P2WPKH

It is extremely important to note that P2WPKH witness programs should only be created by
the payee (recipient) and not converted by the sender from a known
public key, P2PKH script, or address. The sender has no way of knowing
if the recipient's wallet has the ability to construct segwit
transactions and spend P2WPKH outputs.

Additionally, P2WPKH outputs must be constructed from the hash of a
_compressed_ public key. Uncompressed public keys are nonstandard in
segwit and may be explicitly disabled by a future soft fork. If the hash
used in the P2WPKH came from an uncompressed public key, it may be
unspendable and you may lose funds. P2WPKH outputs should be created by
the payee's wallet by deriving a compressed public key from their
private key.

[WARNING]
====
P2WPKH should be constructed by the payee (recipient) by converting a
compressed public key to a P2WPKH hash. You should never transform a
P2PKH script, Bitcoin address, or uncompressed public key to a P2WPKH
witness script.
====

[[p2wsh]]
===== Pay-to-Witness-Script-Hash (P2WSH)

The ((("use cases", "import/export", id="mohamappd")))second type of
segwit v0 witness program corresponds to a Pay-to-Script-Hash (P2SH) script. We
saw this type of script in <<p2sh>>. In that example, P2SH was used by
Mohammed's company to express a multisignature script. Payments to
Mohammed's company were encoded with a script like this:

.Example P2SH scriptPubKey
----
OP_HASH160 54c557e07dde5bb6cb791c7a540e0a4796f5e97e OP_EQUAL
----

This P2SH script references the hash of a _redeemScript_ that defines a
2-of-3 multisignature requirement to spend funds. To spend this output,
Mohammed's company would present the redeemScript (whose hash matches
the script hash in the P2SH output) and the signatures necessary to
satisfy that redeemScript, all inside the transaction input:

.Decoded transaction showing a P2SH output being spent
----
[...]
“Vin” : [
"txid": "abcdef12345...",
"vout": 0,
     	 "scriptSig": “<SigA> <SigB> <2 PubA PubB PubC PubD PubE 5 OP_CHECKMULTISIG>”,
]
----

Now, let's look at how this entire example would be upgraded to segwit v0.
If Mohammed's customers were using a segwit-compatible wallet, they
would make a payment, creating a Pay-to-Witness-Script-Hash (P2WSH)
output that would look like this:

.Example P2WSH scriptPubKey
----
0 a9b7b38d972cabc7961dbfbcb841ad4508d133c47ba87457b4a0e8aae86dbb89
----

Again, as with the example of P2WPKH, you can see that the Segregated
Witness equivalent script is a lot simpler and reduces the template
overhead that you see in P2SH scripts. Instead, the Segregated Witness
scriptPubKey consists of two values pushed to the stack: a witness version
(0) and the 32-byte SHA256 hash of the witness script.

[TIP]
====
While P2SH uses the 20-byte +RIPEMD160(SHA256(script))+ hash, the P2WSH
witness program uses a 32-byte +SHA256(script)+ hash. This difference in
the selection of the hashing algorithm is deliberate
to provide stronger security to
P2WSH in certain use cases (128 bits of security in P2WSH versus 80 bits
of security in P2SH).  For details, see <<p2sh_collision_attacks>>.
====

Mohammed's company can spend the P2WSH output by presenting the
correct witness script and sufficient signatures to satisfy it. Both the
witness script and the signatures would be
included as part of the witness data. No data would be placed in the
scriptSig.

.Decoded transaction showing a P2WSH output being spent with witness data
----
[...]
“Vin” : [
"txid": "abcdef12345...",
"vout": 0,
     	 "scriptSig": “”,
]
[...]
“witness”: “<SigA> <SigB> <2 PubA PubB PubC PubD PubE 5 OP_CHECKMULTISIG>”
[...]
----

===== Differentiating between P2WPKH and P2WSH

In the previous two sections, we demonstrated two types of witness
programs: <<p2wpkh>> and <<p2wsh>>. Both types of witness programs
consist of the same version number followed by a data push. They
look very similar, but are interpreted very differently: one is
interpreted as a public key hash, which is satisfied by a signature and
the other as a script hash, which is satisfied by a witness script. The
critical difference between them is the length of the witness program:

- The witness program in P2WPKH is 20 bytes
- The witness program in P2WSH is 32 bytes

This is the one difference that allows a full node to differentiate between
the two types of witness programs. By looking at the length of the hash,
a node can determine what type of witness program it is, P2WPKH or
P2WSH.

==== Upgrading to Segregated Witness

As we can see from the previous examples, upgrading to Segregated
Witness is a two-step process. First, wallets must create segwit
type outputs. Then, these outputs can be spent by wallets that know how
to construct Segregated Witness transactions. In the examples, Alice's
wallet is able to create outputs paying
Segregated Witness scriptPubKeys. Bob's wallet is also segwit-aware and able
to spend those outputs.

Segregated Witness was implemented as a
backward-compatible upgrade, where _old and new clients can coexist_.
Wallet developers independently upgraded wallet software to add
segwit capabilities.
Legacy P2PKH and
P2SH continue to work for nonupgraded wallets. That leaves two
important scenarios, which are addressed in the next section:

- Ability of a sender's wallet that is not segwit-aware to make a
  payment to a recipient's wallet that can process segwit transactions

- Ability of a sender's wallet that is segwit-aware to recognize and
  distinguish between recipients that are segwit-aware and ones that are
  not, by their _addresses_.

===== Embedding Segregated Witness inside P2SH

Let's assume, for example, that Alice's wallet is not upgraded to
segwit, but Bob's wallet is upgraded and can handle segwit transactions.
Alice and Bob can use legacy non-segwit outputs. But Bob would
likely want to use segwit to reduce transaction fees, taking advantage
of the reduced cost of witness data.

In this case Bob's wallet can construct a P2SH address that contains a
segwit script inside it. Alice's wallet sees this as a "normal" P2SH
address and can make payments to it without any knowledge of segwit.
Bob's wallet can then spend this payment with a segwit transaction,
taking advantage of segwit and reducing transaction fees.

Both forms of witness scripts, P2WPKH and P2WSH, can be embedded in a
P2SH address. The first is noted as nested P2WPKH and the second is noted
as nested P2WSH.

===== Nested Pay-to-Witness-Public-Key-Hash

The first form of witness script we will examine is nested P2WPKH. This
is a Pay-to-Witness-Public-Key-Hash witness program, embedded inside a
Pay-to-Script-Hash script, so that it can be used by a wallet that is
not aware of segwit.

Bob's wallet constructs a P2WPKH witness program with Bob's public key.
This witness program is then hashed and the resulting hash is encoded as
a P2SH script. The P2SH script is converted to a Bitcoin address, one
that starts with a "3," as we saw in the <<p2sh>> section.

Bob's wallet starts with the P2WPKH witness program we saw earlier:

.Bob's P2WPKH witness program
----
0 ab68025513c3dbd2f7b92a94e0581f5d50f654e7
----

The P2WPKH witness program consists of the witness version and Bob's
20-byte public key hash.

Bob's wallet then hashes the preceding witness program, first with
SHA256, then with RIPEMD160, producing another 20-byte hash.

Let's use +bx+ on the command-line to replicate that:

.HASH160 of the P2WPKH witness program
----
echo \
'0 [ab68025513c3dbd2f7b92a94e0581f5d50f654e7]'\
 | bx script-encode | bx sha256 | bx ripemd160
3e0547268b3b19288b3adef9719ec8659f4b2b0b
----

Next, the redeemScript hash is converted to a Bitcoin address. Let's
use +bx+ on the command-line again:

.P2SH address
----
echo \
'3e0547268b3b19288b3adef9719ec8659f4b2b0b' \
| bx address-encode -v 5
37Lx99uaGn5avKBxiW26HjedQE3LrDCZru
----

Now, Alice's wallet can make a payment to
+37Lx99uaGn5avKBxiW26HjedQE3LrDCZru+, just as it would to any other
Bitcoin address.

To pay Bob, Alice's wallet would lock the output with a P2SH script:
----
OP_HASH160 3e0547268b3b19288b3adef9719ec8659f4b2b0b OP_EQUAL
----

Even though Alice's wallet has no support for segwit, the payment it
creates can be spent by Bob with a segwit transaction.((("",
startref="aliced")))

===== Nested Pay-to-Witness-Script-Hash

Similarly, a P2WSH witness program for a multisig script or other
complicated script can be embedded inside a P2SH script and address,
making it possible for any wallet to make payments that are segwit
compatible.

As we saw in <<p2wsh>>, Mohammed's ((("use cases",
"import/export")))company is using Segregated Witness payments to
multisignature scripts. To make it possible for any client to pay his
company, regardless of whether their wallets are upgraded for segwit,
Mohammed's wallet can embed the P2WSH witness program inside a P2SH
script.

First, Mohammed's wallet hashes the redeemScript with SHA256 (just
once). Let's use +bx+ to do that on the command-line:

.Mohammed's wallet creates a P2WSH witness program
----
echo \
2 \ [04C16B8698A9ABF84250A7C3EA7EEDEF9897D1C8C6ADF47F06CF73370D74DCCA01CDCA79DCC5C395D7EEC6984D83F1F50C900A24DD47F569FD4193AF5DE762C587] \
[04A2192968D8655D6A935BEAF2CA23E3FB87A3495E7AF308EDF08DAC3C1FCBFC2C75B4B0F4D0B1B70CD2423657738C0C2B1D5CE65C97D78D0E34224858008E8B49] \
[047E63248B75DB7379BE9CDA8CE5751D16485F431E46117B9D0C1837C9D5737812F393DA7D4420D7E1A9162F0279CFC10F1E8E8F3020DECDBC3C0DD389D9977965] \
[0421D65CBD7149B255382ED7F78E946580657EE6FDA162A187543A9D85BAAA93A4AB3A8F044DADA618D087227440645ABE8A35DA8C5B73997AD343BE5C2AFD94A5] \
[043752580AFA1ECED3C68D446BCAB69AC0BA7DF50D56231BE0AABF1FDEEC78A6A45E394BA29A1EDF518C022DD618DA774D207D137AAB59E0B000EB7ED238F4D800] \
5 CHECKMULTISIG \
| bx script-encode | bx sha256
9592d601848d04b172905e0ddb0adde59f1590f1e553ffc81ddc4b0ed927dd73
----

Next, the hashed redeemScript is turned into a P2WSH witness program:

----
0 9592d601848d04b172905e0ddb0adde59f1590f1e553ffc81ddc4b0ed927dd73
----

Then, the witness program itself is hashed with SHA256 and RIPEMD160,
producing a new 20-byte hash, as used in legacy P2SH. Let's use
+bx+ on the command-line to do that:

.The HASH160 of the P2WSH witness program
----
 echo \
'0 [9592d601848d04b172905e0ddb0adde59f1590f1e553ffc81ddc4b0ed927dd73]'\
 | bx script-encode | bx sha256 | bx ripemd160
86762607e8fe87c0c37740cddee880988b9455b2
----

Next, the wallet constructs a P2SH Bitcoin address from this hash.
Again, we use +bx+ to calculate on the command-line:

.P2SH Bitcoin address
----
echo \
'86762607e8fe87c0c37740cddee880988b9455b2'\
 | bx address-encode -v 5
3Dwz1MXhM6EfFoJChHCxh1jWHb8GQqRenG
----

Now, Mohammed's clients can make payments to this address even if they
don't support segwit. To send a payment to Mohammed, a wallet would
lock the output with the following P2SH script:

.P2SH script used to lock payments to Mohammed's multisig
----
OP_HASH160 86762607e8fe87c0c37740cddee880988b9455b2 OP_EQUAL
----

Mohammed's company can then construct segwit transactions to spend these
payments, taking advantage of segwit features including lower
transaction fees.

=== Merklized Alternative Script Trees (MAST)

Using +OP_IF+, you can authorize multiple different spending conditions,
but this approach has several undesirable aspects:

- _Weight (cost):_ every condition you add increases the size of the
  script, increasing the weight of the transaction and the amount of fee
  that will need to be paid in order to spend bitcoins protected by
  that script.

- _Limited size:_ even if you're willing to pay for extra conditions,
  there's a limit to the maximum number you can put in a script.  For
  example, legacy script is limited to 10,000 bytes, practically
  limiting you to a few hundred conditional branches at most.  Even if
  you could create a script as large as an entire block, it could still
  only contain about 20,000 useful branches.  That's a lot for simple
  payments but tiny compared to some imagined uses of Bitcoin.

- _Lack of privacy:_ every condition you add to your script becomes
  public knowledge when you spend bitcoins protected by that script.
  For example, Mohammed's lawyer and business partners will be able to
  see the entire script in <<variable_timelock_multisig>> whenever
  anyone spends from it.  That means their lawyer, even if he's not
  needed for signing, will be able to track all of their transactions.

However, Bitcoin already uses a data structure known as a merkle tree
that allows verifying an element is a member of a set without
needing to identify every other member of the set.

We'll learn more about merkle trees in <<merkle_trees>>, but the
essential information is that members of the set of information we want
(e.g. authorization conditions of any length) can be passed into a hash
function to create a short commitment (called a _leaf_ of the merkle
tree).  Each of those leaves is then paired with another leaf
and hashed again, creating a commitment to the leaves, called a
_branch_ commitment.  A commitment to a pair of branches can be created
the same way.  This step is repeated for the branches until only one
identifier remains, called the _merkle root_.  Using our example script
from <<variable_timelock_multisig>>, we construct a merkle tree for each
of the three authorization conditions in <<diagram_mast1>>.

[[diagram_mast1]]
.A MAST with three sub-scripts
image::../images/mast1.dot.png["A MAST with three sub-scripts"]

We can now create a compact membership proof that proves a particular
authorization condition is a member of the merkle tree without
disclosing any details about the other members of the merkle tree.  See
<<diagram_mast2>> and note that the nodes with diagonal corners can be
computed from other data provided by the user, so they don't need to be
specified at spend time.

[[diagram_mast2]]
.A MAST membership proof for one of the sub-scripts
image::../images/mast2.dot.png["A MAST membership proof for one of the sub-scripts"]

The hash digests used to create the commitments are each 32-bytes, so
proving the above spend is authorized (using a merkle tree and the
particular conditions) and authenticated (using signatures) uses 383
bytes.  By comparison, the same spend without a merkle tree (i.e.
providing all possible authorization conditions) uses 412 bytes.

Saving 29 bytes (7%) in this example doesn't fully
capture the potential savings.  The binary-tree nature of a merkle tree
means that you only need an additional 32-byte commitment every time
you double the number of members in the set (in this case, authorization
conditions).  In this case, with three conditions, we need to use three
commitments (one of them being the merkle root, which will need to be
included in the authorization data); we could also have four
commitments for the same cost.  An extra commitment would give us up to
eight conditions.  With just 16 commitments--512 bytes of commitments--we could have
over 32,000 authorization conditions, far more that could be effectively
used in an entire block of +OP_IF+ statements.  With 128 commitments
(4,096 bytes), the number of conditions we could create in theory far
exceeds the number of conditions that all the computers in the world
could create.

It's commonly the case that not every authorization condition is equally
as likely to be used.  In the our example case, we expect Mohammed and
his partners to spend their money frequently; the time delayed
conditions only exist in case something goes wrong.  We can re-structure
our tree with this knowledge in <<diagram_mast3>>.

[[diagram_mast3]]
.A MAST with the most-expected script in the best position
image::../images/mast3.dot.png["A MAST with the most-expected script in the best position"]

Now we only need to provide two commitments for the common case (saving 32
bytes), although we still need three commitments for the less common cases.
If you know (or can guess) the probabilities of
using the different authorization conditions, you can use the Huffman
algorithm to place them into a maximally-efficient tree; see BIP341 for
details.

Regardless of how the tree is constructed, we can see in the above
examples that we're only revealing the actual authorization conditions
that get used.  The other conditions remain private.  It even remains
private the number of conditions: a tree could have a single condition
or a trillion conditions--there's no way for someone looking only at the
onchain data for a single transaction to tell.

Except for increasing the complexity of Bitcoin slightly, there are no
significant downsides of MAST for Bitcoin and there were two solid
proposals for it, BIP114 and BIP116, before an improved approach was
discovered, which we'll see in <<taproot>>.

[[pay_to_contract]]
=== Pay to Contract (P2C)

As we saw in <<public_child_key_derivation>>, the math of Elliptic Curve
Cryptography (ECC) allows Alice to use a private key to derive a public
key that she gives to Bob.  He can add an arbitrary value to that public
key create a derived public key.  If he gives that arbitrary value to Alice, she can
add it to her private key to derive the private key for the derived
public key.  In short, Bob can create child public keys for which only
Alice can create the corresponding private keys.  This is useful for
BIP32-style Hierarchical Deterministic (HD) wallet recovery, but it can
also serve another use.

Let's imagine Bob wants to buy something from Alice but he also wants to
be able prove later what he paid for in case there's any dispute.  Alice
and Bob agree on the name of the item or service being sold, e.g.
"Alice's podcast episode #123", and transform that description into a
number by hashing it and interpreting the hash digest as a number.  Bob
adds that number to Alice's public key and pays it.  The process is
called _key tweaking_ and the number is known as a _tweak_.

Alice can spend the funds by tweaking her private key using the same
number (tweak).

Later, Bob can prove to anyone what he paid Alice by revealing her
underlying key and the description they used.  Anyone can verify that
the public key which was paid equals the underlying key plus the
hash commitment to the description.  If Alice admits that key is hers,
then she received the payment.  If Alice spent the funds, this further
proves she knew the description at the time she signed the spending
transaction, since she could only create a valid signature for the
tweaked public key if she knew the tweak (the description).

If neither Alice or Bob decided to publicly reveal the description they
use, the payment between them looks like any other payment.  There's no
privacy loss.

Because P2C is private by default, we can't know how often it is used
for its original purpose--in theory every payment could be using it,
although we consider that unlikely.  However, P2C is widely used today
in a slightly different form, which we'll see in <<taproot>>.

=== Scriptless Multisignature

In <<multisig>>, we looked at scripts which require signatures from
multiple keys.  However, there's another way to require cooperation from
multiple keys, which is also confusingly called _multisignature_.  To
distinguish between the two types in this section, we'll call the
version involving `OP_CHECKSIG`-style opcodes _script multisignatures_
and the other version _scriptless multisignatures_.

Scriptless multisignatures involves each participant creating their own
secret the same way they create a private key.  We'll call this secret a
_partial private key_, although we should note that it's the same length
as a regular full private key.  From the partial private key, each
participant derives a partial public key using the same algorithm used
for regular public keys we described in <<public_key_derivation>>.  Each
participant shares their partial public keys with all the other
participants and then combines all of the keys together to create the
scriptless multisignature public key.

This combined public key looks the same as any other Bitcoin public key.
A third party can't distinguish between a multi-party public key and an
ordinary key generated by a single user.

To spend bitcoins protected by the scriptless multisignature public key,
each participant generates a partial signature.  The partial signatures
are then combined to create a regular full signature.  There are
many known methods for creating and combining the partial signatures;
we'll look at this topic more in <<c_signatures>>.  Similar to the public
keys for scriptless multisignatures, the signatures generated by this
process look the same as any other Bitcoin signature.  Third parties
can't determine whether a signature was created by a single person or a
million people cooperating with each other.

Scriptless multisignatures are smaller and more private than scripted
multisignatures.  For scripted multisignatures, the number of bytes
placed in a transaction increases for every key and signature involved.
For scriptless multisignatures, the size is constant--a million
participants each providing their own partial key and partial signature
puts exactly the same amount of data in a transaction as an individual
using a single key and signature.  The story is the same for privacy:
because each new key or signature adds data to a transaction, scripted
multisignatures disclose data about how many keys and signatures are
being used--which may make it easy to figure out which transactions were
created by which group of participants.  But, because every scriptless
multisignatures looks identical to every other scriptless
multisignature and every single-signature, no privacy-reducing data is
leaked.

There are two downsides of scriptless multisignatures.  The first is
that all known secure algorithms for creating them for Bitcoin require more
rounds of interaction or more careful management of state (or both) than
scripted multisignature.  This can be challenging in cases where
signatures are being generated by nearly stateless hardware signing
devices and the keys are physically distributed.  For example, if you
keep a hardware signing device in a bank safe deposit box, you would
need to visit that box once to create a scripted multisignature but
possibly two or three times for a scriptless multisignature.

The other downside is that threshold signing doesn't reveal who signed.
In scripted threshold signing, Alice, Bob, and Carol agree (for example)
that any two of them signing will be sufficient to spend the funds.
If Alice and Bob sign, this requires putting signatures from each of
them on chain, proving to anyone who knows their keys that they signed
and Carol didn't.  In scriptless threshold signing, a signature from
Alice and Bob is indistinguishable from a signature between Alice and
Carol or Bob and Carol.  This is beneficial for privacy, but it means
that, even if Carol claims she didn't sign, she can't
prove that she didn't, which may be bad for accountability and
auditability.

For many users and use cases, the always reduced sized and increased
privacy of multisignatures outweighs its occasional challenges for
creating and auditing signatures.

[[taproot]]
=== Taproot

One reason people choose to use Bitcoin is that it's possible to create
contracts with highly predictable outcomes.  Legal contracts enforced by
a court of law depend in part on decisions by the judges and jurors
involved in the case.  By contrast, Bitcoin contracts often require
actions by their participants but are otherwise enforced by thousands of
full nodes all running functionally identical code.  When given the same
contract and the same input, every full node will always produce the
same result.  Any deviation would mean that Bitcoin was broken.
Human judges and juries can be much more flexible than software, but
when that flexibility isn't wanted or needed, the predictability of
Bitcoin contracts is a major asset.

If all of the participants in a contract recognize that its outcome has
become completely predictable, there's not actually any need for them to
continue using the contract.  They could just do whatever the contract
compels them to do and then terminate the contract.  In society, this
is how most contracts terminate: if the interested parties are
satisfied, they never take the contract before a judge or jury.  In
Bitcoin, it means that any contract which will use a significant amount
of block space to settle should also provide a clause that allows it to
instead be settled by mutual satisfaction.

In MAST and with scriptless multisignatures, a mutual satisfaction
clause is easy to design.  We simply make one of the top leaves of the
script tree a scriptless multisignature between all interested parties.
We already saw a complex contract between several parties with a
simple mutual satisfaction clause in <<diagram_mast3>>.  We could make
that more optimized by switching from scripted multisignature to
scriptless multisignature.

That's reasonably efficient and private.  If the mutual satisfaction
clause is used, we only need to provide a single merkle branch and all
we reveal is that a signature was involved (it could be from one person
or it could be from thousands of different participants).  But
developers in 2018 realized that we could do better if we also used
pay-to-contract.

In our previous description of pay-to-contract in <<pay_to_contract>>,
we tweaked a public key to commit to the text of an agreement between
Alice and Bob.  We can instead commit to the program code of a contract
by committing to the root of a MAST.  The public key we tweak
is a regular Bitcoin public key, meaning it could require a signature
from a single person or it could require a signature from multiple
people (or it could be created in a special way to make it impossible to
generate a signature for it).  That means we can satisfy the contract
either with a single signature from all interested parties or by
revealing the MAST branch we want to use.  That commitment tree
involving both a public key and a MAST is shown in [[diagram_taproot]].

[[diagram_taproot1]]
.A taproot with the public key committing to a merkle root
image::../images/taproot1.dot.png["A taproot with the public key committing to a merkle root"]

This makes the mutual satisfaction clause using a multisignature
extremely efficient and very private.  It's even more private than it
may appear because any transaction created by a single user who wants it
to be satisfied by a single signature (or a multisignature generated by
multiple different wallets they control) looks identical onchain to a
mutual-satisfaction spend.

When spending is possible using just the key, such for a single signature
or scriptless multisignature, that is called _keypath spending_.  When
the tree of scripts is used, that is called _scriptpath spending_.
For keypath spending, the data that gets put onchain is the public key
(in a witness program) and the signature (in the witness data).

For scriptpath spending, the onchain data also includes the public key,
which is placed in a witness program and called the _taproot output key_
in this context.  The witness data includes the following information:

1. A version number

2. The underlying key--the key which existed before being tweaked by the
   merkle root to produce the taproot output key.  This underlying key
   is called the _taproot internal key_

3. The script to execute

4. One 32-byte hash for each merkle branch connecting the script to the merkle root

5. Any data necessary to satisfy the script (such as signatures or hash preimages)

// Source for 33 bytes: https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-February/017622.html

We're only aware of one significant described downside of taproot:
contracts whose participants want to use MAST but who don't want a
mutual satisfaction clause have to include a taproot internal key on the
blockchain, adding about 33 bytes of overhead. Given that almost
all contracts are expected to benefit from a mutual satisfaction clause,
or other multisignature clause that uses the top-level public key, and
all users benefit from the increased anonymity set of outputs looking
similar to each other, that rare overhead was not considered important
by most users who participated in taproot's activation.

Support for taproot was added to Bitcoin in a soft fork which activated
in November 2021.

=== Tapscript

Taproot enables MAST but only with a slightly different version of the
Bitcoin Script language than previously used, the new version being
called _tapscript_.  The major differences include:

Scripted multisignature changes::
  The old +OP_CHECKMULTSIG+ and +OP_CHECKMULTISIGVERIFY+ opcodes are
  removed.  Those opcodes don't combine well with one of the other
  changes in the taproot soft fork, the ability to use schnorr signatures
  with batch validation.  A new +OP_CHECKSIGADD+ opcode is provided
  instead.  When it successfully verifies a signature, this new opcode
  increments a counter by one, making it possible to conveniently count
  how many signatures passed, which can be compared against the desired number
  of successful signatures to reimplement the same behavior as
  +OP_CHECKMULTISIG+.

Changes to all signatures::
  All signature operations in tapscript use the schnorr signature
  algorithm as defined in BIP340.  We'll explore schnorr signatures more
  in <<c_signatures>>.
+
Additionally, any signature-checking operation which is not expected
  to succeed must be fed the value +OP_FALSE+ (also called +OP_0+)
  instead of an actual signature.  Providing anything else to a failed
  signature-checking operation will cause the entire script to fail.
  This also helps support batch validation of schnorr signatures.

OP_SUCCESSx opcodes::
  Opcodes in previous versions of script which were unusable are now
  redefined to be cause an entire script to succeed if they are used.
  This allows future soft forks to redefine them as not succeeding, which
  is a restriction and so is possible to do in a soft fork.  (The
  opposite, to define a not-succeeding operation as a success can only
  be done in a hard fork, which is a much more challenging upgrade
  path.)

Although we've looked at authorization and authentication in depth in
this chapter, we've skipped over one very important part of how Bitcoin
authenticates spenders: its signatures.  We'll look at that next in
<<c_signatures>>.