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<body class="article">
<div id="header">
</div>
<div id="content">
<div class="sect1">
<h2 id="ch04_keys_addresses">Keys and Addresses</h2>
<div class="sectionbody">
<div class="paragraph">
<p>Alice wants to pay Bob, but the thousands of Bitcoin full nodes who
will verify her transaction don’t know who Alice or Bob are—​and we want
to keep it that way to protect their privacy. Alice needs to
communicate that Bob should receive some of her bitcoins without tying
any aspect of that transaction to Bob’s real-world identity or to other
Bitcoin payments that Bob receives. The method Alice uses must ensure
that only Bob can further spend the bitcoins he receives.</p>
</div>
<div class="paragraph">
<p>The original Bitcoin paper describes a very simple scheme for achieving
those goals, shown in <a href="#pay-to-pure-pubkey">Transaction chain from original Bitcoin paper.</a>.</p>
</div>
<div id="pay-to-pure-pubkey" class="imageblock">
<div class="content">
<img src="images/mbc3_aain01.png" alt="Transaction chain from original Bitcoin paper">
</div>
<div class="title">Figure 1. Transaction chain from original Bitcoin paper.</div>
</div>
<div class="paragraph">
<p>A receiver like Bob
accepts bitcoins to a public key in a transaction that is signed by the
spender (like Alice). The bitcoins that Alice is spending had been
previously received to one of her public keys, and she uses the
corresponding private key to generate her signature. Full nodes can
verify that Alice’s signature commits to the output of a hash function
that itself commits to Bob’s public key and other transaction details.</p>
</div>
<div class="paragraph">
<p>We’ll examine public keys, private keys, signatures, and hash functions
in this chapter, and then use all of them together to describe
the addresses used by modern Bitcoin software.</p>
</div>
<div class="sect2">
<h3 id="_public_key_cryptography">Public Key Cryptography</h3>
<div class="paragraph">
<p>Public key
cryptography was invented in the 1970s and is a mathematical foundation
for modern computer and information security.</p>
</div>
<div class="paragraph">
<p>Since the invention of public key cryptography, several suitable
mathematical functions, such as prime number exponentiation and elliptic
curve multiplication, have been discovered. These mathematical functions
are easy to calculate in
one direction and infeasible to calculate in the opposite direction
using the computers and algorithms available today.
Based on these mathematical functions, cryptography enables the creation
of unforgeable digital signatures. Bitcoin uses
elliptic curve addition and multiplication as the basis for its cryptography.</p>
</div>
<div class="paragraph">
<p>In Bitcoin, we can use public key cryptography to create a key pair that
controls access to bitcoins. The key pair consists of a private key
and a public key derived from the private key. The public key is used to
receive funds, and the private key is used to sign transactions to spend
the funds.</p>
</div>
<div class="paragraph">
<p>There is a mathematical relationship between the public and the private
key that allows the private key to be used to generate signatures on
messages. These signatures can be validated against the public key without
revealing the private key.</p>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>In some wallet
implementations, the private and public keys are stored together as a
<em>key pair</em> for convenience. However, the public key can be calculated
from the private key, so storing only the private key is also possible.</p>
</div>
</td>
</tr>
</table>
</div>
<div class="paragraph">
<p>A Bitcoin wallet contains a collection of key
pairs, each consisting of a private key and a public key. The private
key (<em>k</em>) is a number, usually derived from a number picked at random.
From the private key, we
use elliptic curve multiplication, a one-way cryptographic function, to
generate a public key (<em>K</em>).</p>
</div>
<div class="sidebarblock less_space pagebreak-before">
<div class="content">
<div class="title">Why Use Asymmetric Cryptography (Public/Private Keys)?</div>
<div class="paragraph">
<p>Why is asymmetric
cryptography used in Bitcoin? It’s not used to "encrypt" (make secret)
the transactions. Rather, a useful property of asymmetric cryptography
is the ability to generate <em>digital signatures</em>. A private key can be
applied to a transaction to produce a
numerical signature. This signature can only be produced by someone with
knowledge of the private key. However, anyone with access to the public
key and the transaction can use them to <em>verify</em> the
signature. This useful property of asymmetric cryptography makes it
possible for anyone to verify every signature on every transaction,
while ensuring that only the owners of private keys can produce valid
signatures.</p>
</div>
</div>
</div>
<div class="sect3">
<h4 id="private_keys">Private Keys</h4>
<div class="paragraph">
<p>A
private key is simply a number, picked at random. Control
over the private key is the root of user control over all funds
associated with the corresponding Bitcoin public key. The private key is
used to create signatures that are used to spend bitcoins by proving
control of funds used in a transaction. The private key must remain
secret at all times because revealing it to third parties is equivalent
to giving them control over the bitcoins secured by that key. The private
key must also be backed up and protected from accidental loss because
if it’s lost, it cannot be recovered and the funds secured by it are
forever lost too.</p>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>A Bitcoin private key is just a number. You can pick your private keys
randomly using just a coin, pencil, and paper: toss a coin 256 times and
you have the binary digits of a random private key you can use in a
Bitcoin wallet. The public key can then be generated from the private
key. Be careful, though, as any process that’s less than completely
random can significantly reduce the security of your private key and the
bitcoins it controls.</p>
</div>
</td>
</tr>
</table>
</div>
<div class="paragraph">
<p>The first and most important step in generating keys is to find a secure
source of randomness (which computer scientists call <em>entropy</em>). Creating a Bitcoin key is almost
the same as "Pick a number between 1 and 2<sup>256</sup>." The exact method you
use to pick that number does not matter as long as it is not predictable
or repeatable. Bitcoin software uses cryptographically secure random
number generators to produce 256 bits of entropy.</p>
</div>
<div class="paragraph less_space pagebreak-before">
<p>More precisely, the private key can be any number between 0 and <em>n</em> -
1 inclusive, where <em>n</em> is a constant (<em>n</em> = 1.1578 × 10<sup>77</sup>, slightly less
than 2<sup>256</sup>) defined as the order of the elliptic curve used in Bitcoin
(see <a href="#elliptic_curve">Elliptic Curve Cryptography Explained</a>). To create such a key, we randomly pick a
256-bit number and check that it is less than <em>n</em>. In programming terms,
this is usually achieved by feeding a larger string of random bits,
collected from a cryptographically secure source of randomness, into the
SHA256 hash algorithm, which will conveniently produce a 256-bit value
that can be interpreted as a number.
If the result is less than <em>n</em>, we have a suitable private key.
Otherwise, we simply try again with another random number.</p>
</div>
<div class="admonitionblock warning">
<table>
<tr>
<td class="icon">
<div class="title">Warning</div>
</td>
<td class="content">
<div class="paragraph">
<p>Do not write your own code to create a random
number or use a "simple" random number generator offered by your
programming language. Use a cryptographically secure pseudorandom number
generator (CSPRNG) with a seed from a source of sufficient entropy.
Study the documentation of the random number generator library you
choose to make sure it is cryptographically secure. Correct
implementation of the CSPRNG is critical to the security of the keys.</p>
</div>
</td>
</tr>
</table>
</div>
<div class="paragraph">
<p>The following is a randomly generated private key (<em>k</em>) shown in
hexadecimal format (256 bits shown as 64 hexadecimal digits, each 4
bits):</p>
</div>
<div class="listingblock">
<div class="content">
<pre>1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD</pre>
</div>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>The size of Bitcoin’s private key space (2<sup>256</sup>) is an unfathomably
large number. It is approximately 10<sup>77</sup> in decimal. For comparison, the
visible universe is estimated to contain 10<sup>80</sup> atoms.</p>
</div>
</td>
</tr>
</table>
</div>
</div>
<div class="sect3">
<h4 id="elliptic_curve">Elliptic Curve Cryptography Explained</h4>
<div class="paragraph">
<p>Elliptic curve cryptography (ECC) is a type of asymmetric
or public key cryptography based on the discrete logarithm problem as
expressed by addition and multiplication on the points of an elliptic
curve.</p>
</div>
<div class="paragraph">
<p><a href="#ecc-curve">An elliptic curve.</a> is an example of an elliptic curve, similar to that used
by Bitcoin.</p>
</div>
<div id="ecc-curve" class="imageblock width-50">
<div class="content">
<img src="images/mbc3_0402.png" alt="ecc-curve">
</div>
<div class="title">Figure 2. An elliptic curve.</div>
</div>
<div class="paragraph">
<p>Bitcoin uses a specific elliptic curve and set of mathematical
constants, as defined in a standard called secp256k1, established by
the National Institute of Standards and Technology (NIST). The
secp256k1 curve is defined by the following function, which produces
an elliptic curve:</p>
</div>
<div class="stemblock">
<div class="content">
\[\begin{equation}
{y^2 = (x^3 + 7)}~\text{over}~(\mathbb{F}_p)
\end{equation}\]
</div>
</div>
<div class="paragraph">
<p>or</p>
</div>
<div class="stemblock">
<div class="content">
\[\begin{equation}
{y^2 \mod p = (x^3 + 7) \mod p}
\end{equation}\]
</div>
</div>
<div class="paragraph">
<p>The <em>mod p</em> (modulo prime number <em>p</em>) indicates that this curve is over a
finite field of prime order <em>p</em>, also written as \(\(
\mathbb{F}_p\)\), where <em>p</em> = 2<sup>256</sup> – 2<sup>32</sup> – 2<sup>9</sup> – 2<sup>8</sup> – 2<sup>7</sup> – 2<sup>6</sup> –
2<sup>4</sup> – 1, a very large prime number.</p>
</div>
<div class="paragraph">
<p>Because this curve is defined over a finite field of prime order instead
of over the real numbers, it looks like a pattern of dots scattered in
two dimensions, which makes it difficult to visualize. However, the math
is identical to that of an elliptic curve over real numbers. As an
example, <a href="#ecc-over-F17-math">Elliptic curve cryptography: visualizing an elliptic curve over F(p), with p=17.</a> shows the same elliptic curve over a much
smaller finite field of prime order 17, showing a pattern of dots on a
grid. The secp256k1 Bitcoin elliptic curve can be thought of as a much
more complex pattern of dots on a unfathomably large grid.</p>
</div>
<div id="ecc-over-F17-math" class="imageblock">
<div class="content">
<img src="images/mbc3_0403.png" alt="ecc-over-F17-math">
</div>
<div class="title">Figure 3. Elliptic curve cryptography: visualizing an elliptic curve over F(p), with p=17.</div>
</div>
<div class="paragraph">
<p>So, for example, the following is a point P with coordinates (x, y) that
is a point on the secp256k1 curve:</p>
</div>
<div class="listingblock">
<div class="content">
<pre class="highlight"><code class="language-python" data-lang="python">P =
(55066263022277343669578718895168534326250603453777594175500187360389116729240,
32670510020758816978083085130507043184471273380659243275938904335757337482424)</code></pre>
</div>
</div>
<div class="paragraph">
<p><a href="#example_4_1">Using Python to confirm that this point is on the elliptic curve</a> shows how you can check this yourself using Python.</p>
</div>
<div id="example_4_1" class="exampleblock">
<div class="title">Example 1. Using Python to confirm that this point is on the elliptic curve</div>
<div class="content">
<div class="listingblock">
<div class="content">
<pre class="highlight"><code class="language-pycon" data-lang="pycon">Python 3.10.6 (main, Nov 14 2022, 16:10:14) [GCC 11.3.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
> p = 115792089237316195423570985008687907853269984665640564039457584007908834671663
> x = 55066263022277343669578718895168534326250603453777594175500187360389116729240
> y = 32670510020758816978083085130507043184471273380659243275938904335757337482424
> (x ** 3 + 7 - y**2) % p
0</code></pre>
</div>
</div>
</div>
</div>
<div class="paragraph less_space pagebreak-before">
<p>In elliptic curve math, there is a point called the "point at infinity,"
which roughly corresponds to the role of zero in addition. On computers,
it’s sometimes represented by x = y = 0 (which doesn’t satisfy the
elliptic curve equation, but it’s an easy separate case that can be
checked).</p>
</div>
<div class="paragraph">
<p>There is also a + operator, called "addition," which has some
properties similar to the traditional addition of real numbers that
gradeschool children learn. Given two points P<sub>1</sub> and P<sub>2</sub> on the
elliptic curve, there is a third point P<sub>3</sub> = P<sub>1</sub> + P<sub>2</sub>, also on the
elliptic curve.</p>
</div>
<div class="paragraph">
<p>Geometrically, this third point P<sub>3</sub> is calculated by drawing a line
between P<sub>1</sub> and P<sub>2</sub>. This line will intersect the elliptic curve in
exactly one additional place. Call this point P<sub>3</sub>' = (x, y). Then
reflect in the x-axis to get P<sub>3</sub> = (x, –y).</p>
</div>
<div class="paragraph">
<p>There are a couple of special cases that explain the need for the "point
at infinity."</p>
</div>
<div class="paragraph">
<p>If P<sub>1</sub> and P<sub>2</sub> are the same point, the line "between" P<sub>1</sub> and P<sub>2</sub>
should extend to be the tangent on the curve at this point P<sub>1</sub>. This
tangent will intersect the curve in exactly one new point. You can use
techniques from calculus to determine the slope of the tangent line.
These techniques curiously work, even though we are restricting our
interest to points on the curve with two integer coordinates!</p>
</div>
<div class="paragraph">
<p>In some cases (i.e., if P<sub>1</sub> and P<sub>2</sub> have the same x values but
different y values), the tangent line will be exactly vertical, in which
case P<sub>3</sub> = "point at infinity."</p>
</div>
<div class="paragraph">
<p>If P<sub>1</sub> is the "point at infinity," then P<sub>1</sub> + P<sub>2</sub> = P<sub>2</sub>. Similarly,
if P<sub>2</sub> is the point at infinity, then P<sub>1</sub> + P<sub>2</sub> = P<sub>1</sub>. This shows
how the point at infinity plays the role of zero.</p>
</div>
<div class="paragraph">
<p>It turns out that + is associative, which means that (A +
B) + C = A + (B + C). That means we can write A
+ B + C without parentheses and without ambiguity.</p>
</div>
<div class="paragraph">
<p>Now that we have defined addition, we can define multiplication in the
standard way that extends addition. For a point P on the elliptic curve,
if k is a whole number, then kP = P + P + P + …​ + P (k times). Note
that k is sometimes confusingly called an "exponent" in this case.</p>
</div>
</div>
<div class="sect3">
<h4 id="public_key_derivation">Public Keys</h4>
<div class="paragraph">
<p>The public key is calculated from
the private key using elliptic curve multiplication, which is
irreversible: <em>K</em> = <em>k</em> × <em>G</em>, where <em>k</em> is the private key, <em>G</em> is a
constant point called the <em>generator point</em>, and <em>K</em> is the resulting
public key. The reverse operation, known as "finding the discrete
logarithm"—calculating <em>k</em> if you know <em>K</em>—is as difficult as trying
all possible values of <em>k</em> (i.e., a brute-force search). Before we
demonstrate how to generate a public key from a private key, let’s look
at elliptic curve cryptography in a bit more detail.</p>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>Elliptic curve multiplication is a type of function that cryptographers
call a "trap door" function: it is easy to do in one direction
(multiplication) and impossible to do in the reverse direction
(division). Someone with a private key can easily create the public
key and then share it with the world knowing that no one can reverse the
function and calculate the private key from the public key. This
mathematical trick becomes the basis for unforgeable and secure digital
signatures that prove control over bitcoin funds.</p>
</div>
</td>
</tr>
</table>
</div>
<div class="paragraph">
<p>Starting with a private key in the
form of a randomly generated number <em>k</em>, we multiply it by a
predetermined point on the curve called the <em>generator point</em> <em>G</em> to
produce another point somewhere else on the curve, which is the
corresponding public key <em>K</em>. The generator point is specified as part
of the secp256k1 standard and is always the same for all keys in
bitcoin:</p>
</div>
<div class="stemblock">
<div class="content">
\[\begin{equation}
{K = k \times G}
\end{equation}\]
</div>
</div>
<div class="paragraph">
<p>where <em>k</em> is the private key, <em>G</em> is the generator point, and <em>K</em> is the
resulting public key, a point on the curve. Because the generator point
is always the same for all Bitcoin users, a private key <em>k</em> multiplied
with <em>G</em> will always result in the same public key <em>K</em>. The relationship
between <em>k</em> and <em>K</em> is fixed but can only be calculated in one
direction, from <em>k</em> to <em>K</em>. That’s why a Bitcoin public key (<em>K</em>) can be
shared with anyone and does not reveal the user’s private key (<em>k</em>).</p>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>A private key can be converted into a public key, but a public key
cannot be converted back into a private key because the math only works
one way.</p>
</div>
</td>
</tr>
</table>
</div>
<div class="paragraph">
<p>Implementing the elliptic curve multiplication, we take the private key
<em>k</em> generated previously and multiply it with the generator point <em>G</em> to
find the public key <em>K</em>:</p>
</div>
<div class="listingblock">
<div class="content">
<pre class="highlight"><code class="language-python" data-lang="python">K = 1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD × G</code></pre>
</div>
</div>
<div class="paragraph">
<p>Public key <em>K</em> is defined as a point <em>K</em> = (<em>x</em>, <em>y</em>):</p>
</div>
<div class="stemblock">
<div class="content">
\[\begin{equation}
K = (x, y)
\end{equation}\]
</div>
</div>
<div class="paragraph">
<p>where,</p>
</div>
<div class="listingblock">
<div class="content">
<pre>x = F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A
y = 07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB</pre>
</div>
</div>
<div class="paragraph">
<p>To visualize multiplication of a point with an integer, we will use the
simpler elliptic curve over real numbers—remember, the math is
the same. Our goal is to find the multiple <em>kG</em> of the generator point
<em>G</em>, which is the same as adding <em>G</em> to itself, <em>k</em> times in a row. In
elliptic curves, adding a point to itself is the equivalent of drawing a
tangent line on the point and finding where it intersects the curve
again, then reflecting that point on the x-axis.</p>
</div>
<div class="paragraph">
<p><a href="#ecc_illustrated">Elliptic curve cryptography: visualizing the multiplication of a point G by an integer k on an elliptic curve.</a> shows the process for deriving <em>G</em>, <em>2G</em>, <em>4G</em>, as a
geometric operation on the curve.</p>
</div>
<div class="admonitionblock tip">
<table>
<tr>
<td class="icon">
<div class="title">Tip</div>
</td>
<td class="content">
<div class="paragraph">
<p>Many Bitcoin implementations use
the <a href="https://oreil.ly/wD60m">libsecp256k1 cryptographic
library</a> to do the elliptic curve math.</p>
</div>
</td>
</tr>
</table>
</div>
<div id="ecc_illustrated" class="imageblock">
<div class="content">
<img src="images/mbc3_0404.png" alt="ecc_illustrated">
</div>
<div class="title">Figure 4. Elliptic curve cryptography: visualizing the multiplication of a point G by an integer k on an elliptic curve.</div>
</div>
</div>
</div>
<div class="sect2">
<h3 id="_output_and_input_scripts">Output and Input Scripts</h3>
<div class="paragraph">
<p>Although the illustration from the original Bitcoin paper, <a href="#pay-to-pure-pubkey">Transaction chain from original Bitcoin paper.</a>,
shows public keys (pubkeys) and signatures (sigs) being used directly,
the first version of Bitcoin instead had payments sent to a field called
<em>output script</em> and had spends of those bitcoins authorized by a field called <em>input script</em>.
These fields allow additional operations to be performed in addition to
(or instead of) verifying that a signature corresponds to a public key.
For example, an output script can contain two public keys and require two
corresponding signatures be placed in the spending input script.</p>
</div>
<div class="paragraph">
<p>Later, in <a href="#tx_script">[tx_script]</a>, we’ll learn about scripts in detail. For now,
all we need to understand is that bitcoins are received to an
output script that acts like a public key, and bitcoin spending is
authorized by an input script that acts like a signature.</p>
</div>
</div>
<div class="sect2">
<h3 id="p2pk">IP Addresses: The Original Address for Bitcoin (P2PK)</h3>
<div class="paragraph">
<p>We’ve established that Alice can pay Bob by assigning some of her
bitcoins to one of Bob’s public keys. But how does Alice get one of
Bob’s public keys? Bob could just give her a copy, but let’s look again
at the public key we worked with in <a href="#public_key_derivation">Public Keys</a>. Notice
that it’s quite long. Imagine Bob trying to read that to Alice over the
phone:</p>
</div>
<div class="listingblock">
<div class="content">
<pre>x = F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A
y = 07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB</pre>
</div>
</div>
<div class="paragraph">
<p>Instead of direct public key entry, the earliest version of Bitcoin
software allowed a spender to enter the receiver’s IP address, as shown in <a href="#bitcoin_01_send">Early send screen for Bitcoin via The Internet Archive.</a>. This
feature was later removed—​there are many problems
with using IP addresses—​but a quick description of it will help us
better understand why certain features may have been added to the
Bitcoin protocol.</p>
</div>
<div id="bitcoin_01_send" class="imageblock">
<div class="content">
<img src="images/mbc3_0405.png" alt="Early Bitcoin send screen">
</div>
<div class="title">Figure 5. Early send screen for Bitcoin via <a href="https://oreil.ly/IDV1a">The Internet Archive</a>.</div>
</div>
<div class="paragraph">
<p>If Alice entered Bob’s IP address in Bitcoin 0.1, her full node would
establish a connection with his full node and receive a new public key
from Bob’s wallet that his node had never previously given anyone. This
being a new public key was important to ensure that different
transactions paying Bob couldn’t be connected together by someone
looking at the blockchain and noticing that all of the transactions paid
the same public key.</p>
</div>