bitcoinlib · PyPI

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bitcoin address generator c++

A simple c++ program that could help you to generate address from any Branches of the original Bitcoin system. - SK-Yang/Bitcoin-address-gen. Keys, Addresses You may have heard that bitcoin is based on cryptography, which into a public key, and finally, generate a bitcoin address from the public key. The C++ code in Example 4-3 shows the complete step-by-step process, from. Example: Create wallet and generate new address (key) to receive bitcoins from bitcoinlib.wallets import Wallet, wallet_delete from bitcoinlib.mnemonic import Mnemonic passphrase This library requires a Microsoft Visual C++ Compiler. bitcoin address generator c++

Mastering Bitcoin by

Chapter 4. Keys, Addresses, Wallets

Ownership of bitcoin is established through digital keys, bitcoin addresses, and digital signatures. The digital keys are not actually stored in the network, but are instead created and stored by users in a file, or simple database, called a wallet. The digital keys in a user’s wallet are completely independent of the bitcoin protocol and can be generated and managed by the user’s wallet software without reference to the blockchain or access to the Internet. Keys enable many of the interesting properties of bitcoin, including de-centralized trust and control, ownership attestation, and the cryptographic-proof security model.

Every bitcoin transaction requires a valid signature to be included in the blockchain, which can only be generated with valid digital keys; therefore, anyone with a copy of those keys has control of the bitcoin in that account. Keys come in pairs consisting of a private (secret) key and a public key. Think of the public key as similar to a bank account number and the private key as similar to the secret PIN, or signature on a check that provides control over the account. These digital keys are very rarely seen by the users of bitcoin. For the most part, they are stored inside the wallet file and managed by the bitcoin wallet software.

In the payment portion of a bitcoin transaction, the recipient’s public key is represented by its digital fingerprint, called abitcoin address, which is used in the same way as the beneficiary name on a check (i.e., “Pay to the order of”). In most cases, a bitcoin address is generated from and corresponds to a public key. However, not all bitcoin addresses represent public keys; they can also represent other beneficiaries such as scripts, as we will see later in this chapter. This way, bitcoin addresses abstract the recipient of funds, making transaction destinations flexible, similar to paper checks: a single payment instrument that can be used to pay into people’s accounts, pay into company accounts, pay for bills, or pay to cash. The bitcoin address is the only representation of the keys that users will routinely see, because this is the part they need to share with the world.

In this chapter we will introduce wallets, which contain cryptographic keys. We will look at how keys are generated, stored, and managed. We will review the various encoding formats used to represent private and public keys, addresses, and script addresses. Finally, we will look at special uses of keys: to sign messages, to prove ownership, and to create vanity addresses and paper wallets.

Public Key Cryptography and Cryptocurrency

Public key cryptography was invented in the 1970s and is a mathematical foundation for computer and information security.

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 practically irreversible, meaning that they are easy to calculate in one direction and infeasible to calculate in the opposite direction. Based on these mathematical functions, cryptography enables the creation of digital secrets and unforgeable digital signatures. Bitcoin uses elliptic curve multiplication as the basis for its public key cryptography.

In bitcoin, we use public key cryptography to create a key pair that controls access to bitcoins. The key pair consists of a private key and—derived from it—a unique public key. The public key is used to receive bitcoins, and the private key is used to sign transactions to spend those bitcoins.

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. This signature can be validated against the public key without revealing the private key.

When spending bitcoins, the current bitcoin owner presents her public key and a signature (different each time, but created from the same private key) in a transaction to spend those bitcoins. Through the presentation of the public key and signature, everyone in the bitcoin network can verify and accept the transaction as valid, confirming that the person transferring the bitcoins owned them at the time of the transfer.


In most wallet implementations, the private and public keys are stored together as a key pair for convenience. However, the public key can be calculated from the private key, so storing only the private key is also possible.

A bitcoin wallet contains a collection of key pairs, each consisting of a private key and a public key. The private key (k) is a number, usually picked at random. From the private key, we use elliptic curve multiplication, a one-way cryptographic function, to generate a public key (K). From the public key (K), we use a one-way cryptographic hash function to generate a bitcoin address (A). In this section, we will start with generating the private key, look at the elliptic curve math that is used to turn that into a public key, and finally, generate a bitcoin address from the public key. The relationship between private key, public key, and bitcoin address is shown in Figure 4-1.

Figure 4-1. Private key, public key, and bitcoin address

A private key is simply a number, picked at random. Ownership and control over the private key is the root of user control over all funds associated with the corresponding bitcoin address. The private key is used to create signatures that are required to spend bitcoins by proving ownership 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.


The 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.

Generating a private key from a random number

The first and most important step in generating keys is to find a secure source of entropy, or randomness. Creating a bitcoin key is essentially the same as “Pick a number between 1 and 2256.” The exact method you use to pick that number does not matter as long as it is not predictable or repeatable. Bitcoin software uses the underlying operating system’s random number generators to produce 256 bits of entropy (randomness). Usually, the OS random number generator is initialized by a human source of randomness, which is why you may be asked to wiggle your mouse around for a few seconds. For the truly paranoid, nothing beats dice, pencil, and paper.

More accurately, the private key can be any number between and , where n is a constant (n = 1.158 * 1077, slightly less than 2256) defined as the order of the elliptic curve used in bitcoin (see Elliptic Curve Cryptography Explained). To create such a key, we randomly pick a 256-bit number and check that it is less than . 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 that will conveniently produce a 256-bit number. If the result is less than , we have a suitable private key. Otherwise, we simply try again with another random number.


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 pseudo-random 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.

The following is a randomly generated private key (k) shown in hexadecimal format (256 binary digits shown as 64 hexadecimal digits, each 4 bits):



The size of bitcoin’s private key space, 2256 is an unfathomably large number. It is approximately 1077 in decimal. The visible universe is estimated to contain 1080 atoms.

To generate a new key with the Bitcoin Core client (see Chapter 3), use the command. For security reasons it displays the public key only, not the private key. To ask bitcoind to expose the private key, use the command. The command shows the private key in a Base58 checksum-encoded format called the Wallet Import Format (WIF), which we will examine in more detail in Private key formats. Here’s an example of generating and displaying a private key using these two commands:

$ bitcoind getnewaddress 1J7mdg5rbQyUHENYdx39WVWK7fsLpEoXZy $ bitcoind dumpprivkey 1J7mdg5rbQyUHENYdx39WVWK7fsLpEoXZy KxFC1jmwwCoACiCAWZ3eXa96mBM6tb3TYzGmf6YwgdGWZgawvrtJ

The command opens the wallet and extracts the private key that was generated by the command. It is not otherwise possible for bitcoind to know the private key from the public key, unless they are both stored in the wallet.


The command is not generating a private key from a public key, as this is impossible. The command simply reveals the private key that is already known to the wallet and which was generated by the getnewaddress command.

You can also use the command-line sx tools (see Libbitcoin and sx Tools) to generate and display private keys with the sx command :

$ sx newkey 5J3mBbAH58CpQ3Y5RNJpUKPE62SQ5tfcvU2JpbnkeyhfsYB1Jcn

The public key is calculated from the private key using elliptic curve multiplication, which is irreversible: where k is the private key, G is a constant point called the generator point and K is the resulting public key. The reverse operation, known as “finding the discrete logarithm”—calculating k if you know K—is as difficult as trying all possible values of , 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.

Elliptic Curve Cryptography Explained

Elliptic curve cryptography 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.

Figure 4-2 is an example of an elliptic curve, similar to that used by bitcoin.

Figure 4-2. An elliptic curve

Bitcoin uses a specific elliptic curve and set of mathematical constants, as defined in a standard called, established by the National Institute of Standards and Technology (NIST). The curve is defined by the following function, which produces an elliptic curve:


The mod p (modulo prime number p) indicates that this curve is over a finite field of prime order p, also written as , where p = 2256 – 232 – 29 – 28 – 27 – 26 – 24 – 1, a very large prime number.

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 as that of an elliptic curve over the real numbers. As an example, Figure 4-3 shows the same elliptic curve over a much smaller finite field of prime order 17, showing a pattern of dots on a grid. The bitcoin elliptic curve can be thought of as a much more complex pattern of dots on a unfathomably large grid.

Figure 4-3. Elliptic curve cryptography: visualizing an elliptic curve over F(p), with p=17

So, for example, the following is a point P with coordinates (x,y) that is a point on the curve. You can check this yourself using Python:

P = (55066263022277343669578718895168534326250603453777594175500187360389116729240, 32670510020758816978083085130507043184471273380659243275938904335757337482424)

In elliptic curve math, there is a point called the “point at infinity,” which roughly corresponds to the role of 0 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).

There is also a + operator, called “addition,” which has some properties similar to the traditional addition of real numbers that grade school children learn. Given two points P1 and P2 on the elliptic curve, there is a third point P3 = P1 + P2, also on the elliptic curve.

Geometrically, this third point P3 is calculated by drawing a line between P1 and P2. This line will intersect the elliptic curve in exactly one additional place. Call this point P3' = (x, y). Then reflect in the x-axis to get P3 = (x, –y).

There are a couple of special cases that explain the need for the “point at infinity.”

If P1 and P2 are the same point, the line “between” P1 and P2 should extend to be the tangent on the curve at this point P1. 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!

In some cases (i.e., if P1 and P2 have the same x values but different y values), the tangent line will be exactly vertical, in which case P3 = “point at infinity.”

If P1 is the “point at infinity,” then the sum P1 + P2 = P2. Similary, if P2 is the point at infinity, then P1 + P2 = P1. This shows how the point at infinity plays the role of 0.

It turns out that + is associative, which means that (A+B)(B+C). That means we can write A+B+C without parentheses without any ambiguity.

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.

Starting with a private key in the form of a randomly generated number k, we multiply it by a predetermined point on the curve called thegenerator pointG to produce another point somewhere else on the curve, which is the corresponding public key K. The generator point is specified as part of the standard and is always the same for all keys in bitcoin:

where k is the private key, G is the generator point, and K 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 k multiplied with G will always result in the same public key K. The relationship between k and K is fixed, but can only be calculated in one direction, from k to K. That’s why a bitcoin address (derived from K) can be shared with anyone and does not reveal the user’s private key (k).


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.

Implementing the elliptic curve multiplication, we take the private key k generated previously and multiply it with the generator point G to find the public key K:

K = 1E99423A4ED27608A15A2616A2B0E9E52CED330AC530EDCC32C8FFC6A526AEDD * G

Public Key K is defined as a point :

K = (x, y) where, x = F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A y = 07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB

To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over the real numbers—remember, the math is the same. Our goal is to find the multiple kG of the generator point G. That is the same as adding G to itself, k 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.

Figure 4-4 shows the process for deriving G, 2G, 4G, as a geometric operation on the curve.


Most bitcoin implementations use theOpenSSL cryptographic library to do the elliptic curve math. For example, to derive the public key, the function is used.

Figure 4-4. Elliptic curve cryptography: Visualizing the multiplication of a point G by an integer k on an elliptic curve

A bitcoin address is a string of digits and characters that can be shared with anyone who wants to send you money. Addresses produced from public keys consist of a string of numbers and letters, beginning with the digit “1”. Here’s an example of a bitcoin address:


The bitcoin address is what appears most commonly in a transaction as the “recipient” of the funds. If we were to compare a bitcoin transaction to a paper check, the bitcoin address is the beneficiary, which is what we write on the line after “Pay to the order of.” On a paper check, that beneficiary can sometimes be the name of a bank account holder, but can also include corporations, institutions, or even cash. Because paper checks do not need to specify an account, but rather use an abstract name as the recipient of funds, that makes paper checks very flexible as payment instruments. Bitcoin transactions use a similar abstraction, the bitcoin address, to make them very flexible. A bitcoin address can represent the owner of a private/public key pair, or it can represent something else, such as a payment script, as we will see in Pay-to-Script-Hash (P2SH). For now, let’s examine the simple case, a bitcoin address that represents, and is derived from, a public key.

The bitcoin address is derived from the public key through the use of one-way cryptographic hashing. A “hashing algorithm” or simply “hash algorithm” is a one-way function that produces a fingerprint or “hash” of an arbitrary-sized input. Cryptographic hash functions are used extensively in bitcoin: in bitcoin addresses, in script addresses, and in the mining proof-of-work algorithm. The algorithms used to make a bitcoin address from a public key are the Secure Hash Algorithm (SHA) and the RACE Integrity Primitives Evaluation Message Digest (RIPEMD), specifically SHA256 and RIPEMD160.

Starting with the public key K, we compute the SHA256 hash and then compute the RIPEMD160 hash of the result, producing a 160-bit (20-byte) number:

where K is the public key and A is the resulting bitcoin address.


A bitcoin address is not the same as a public key. Bitcoin addresses are derived from a public key using a one-way function.

Bitcoin addresses are almost always presented to users in an encoding called “Base58Check” (see Base58 and Base58Check Encoding), which uses 58 characters (a Base58 number system) and a checksum to help human readability, avoid ambiguity, and protect against errors in address transcription and entry. Base58Check is also used in many other ways in bitcoin, whenever there is a need for a user to read and correctly transcribe a number, such as a bitcoin address, a private key, an encrypted key, or a script hash. In the next section we will examine the mechanics of Base58Check encoding and decoding, and the resulting representations. Figure 4-5 illustrates the conversion of a public key into a bitcoin address.

Figure 4-5. Public key to bitcoin address: conversion of a public key into a bitcoin address

Base58 and Base58Check Encoding

In order to represent long numbers in a compact way, using fewer symbols, many computer systems use mixed-alphanumeric representations with a base (or radix) higher than 10. For example, whereas the traditional decimal system uses the 10 numerals 0 through 9, the hexadecimal system uses 16, with the letters A through F as the six additional symbols. A number represented in hexadecimal format is shorter than the equivalent decimal representation. Even more compact, Base-64 representation uses 26 lower-case letters, 26 capital letters, 10 numerals, and two more characters such as “+” and “/” to transmit binary data over text-based media such as email. Base-64 is most commonly used to add binary attachments to email. Base58 is a text-based binary-encoding format developed for use in bitcoin and used in many other cryptocurrencies. It offers a balance between compact representation, readability, and error detection and prevention. Base58 is a subset of Base64, using the upper- and lowercase letters and numbers, but omitting some characters that are frequently mistaken for one another and can appear identical when displayed in certain fonts. Specifically, Base58 is Base64 without the 0 (number zero), O (capital o), l (lower L), I (capital i), and the symbols “\+” and “/”. Or, more simply, it is a set of lower and capital letters and numbers without the four (0, O, l, I) just mentioned.

Example 4-1. bitcoin’s Base58 alphabet

To add extra security against typos or transcription errors, Base58Check is a Base58 encoding format, frequently used in bitcoin, which has a built-in error-checking code. The checksum is an additional four bytes added to the end of the data that is being encoded. The checksum is derived from the hash of the encoded data and can therefore be used to detect and prevent transcription and typing errors. When presented with a Base58Check code, the decoding software will calculate the checksum of the data and compare it to the checksum included in the code. If the two do not match, that indicates that an error has been introduced and the Base58Check data is invalid. For example, this prevents a mistyped bitcoin address from being accepted by the wallet software as a valid destination, an error that would otherwise result in loss of funds.

To convert data (a number) into a Base58Check format, we first add a prefix to the data, called the “version byte,” which serves to easily identify the type of data that is encoded. For example, in the case of a bitcoin address the prefix is zero (0x00 in hex), whereas the prefix used when encoding a private key is 128 (0x80 in hex). A list of common version prefixes is shown in Table 4-1.

Next, we compute the “double-SHA” checksum, meaning we apply the SHA256 hash-algorithm twice on the previous result (prefix and data):

checksum = SHA256(SHA256(prefix+data))

From the resulting 32-byte hash (hash-of-a-hash), we take only the first four bytes. These four bytes serve as the error-checking code, or checksum. The checksum is concatenated (appended) to the end.

The result is composed of three items: a prefix, the data, and a checksum. This result is encoded using the Base58 alphabet described previously. Figure 4-6 illustrates the Base58Check encoding process.

Figure 4-6. Base58Check encoding: a Base58, versioned, and checksummed format for unambiguously encoding bitcoin data

In bitcoin, most of the data presented to the user is Base58Check-encoded to make it compact, easy to read, and easy to detect errors. The version prefix in Base58Check encoding is used to create easily distinguishable formats, which when encoded in Base58 contain specific characters at the beginning of the Base58Check-encoded payload. These characters make it easy for humans to identify the type of data that is encoded and how to use it. This is what differentiates, for example, a Base58Check-encoded bitcoin address that starts with a 1 from a Base58Check-encoded private key WIF format that starts with a 5. Some example version prefixes and the resulting Base58 characters are shown in Table 4-1.

Table 4-1. Base58Check version prefix and encoded result examples
Type Version prefix (hex) Base58 result prefix

Bitcoin Address



Pay-to-Script-Hash Address



Bitcoin Testnet Address


m or n

Private Key WIF


5, K or L

BIP38 Encrypted Private Key



BIP32 Extended Public Key



Let’s look at the complete process of creating a bitcoin address, from a private key, to a public key (a point on the elliptic curve), to a double-hashed address and finally, the Base58Check encoding. The C++ code in Example 4-2 shows the complete step-by-step process, from private key to Base58Check-encoded bitcoin address. The code example uses the libbitcoin library introduced in Alternative Clients, Libraries, and Toolkits for some helper functions.

Example 4-2. Creating a Base58Check-encoded bitcoin address from a private key

The code uses a predefined private key so that it produces the same bitcoin address every time it is run, as shown in Example 4-3.

Example 4-3. Compiling and running the addr code
g++ -o addr addr.cpp pkg-config --cflags --libs libbitcoin./addr Public key: 0202a406624211f2abbdc68da3df929f938c3399dd79fac1b51b0e4ad1d26a47aa Address: 1PRTTaJesdNovgne6Ehcdu1fpEdX7913CK

Both private and public keys can be represented in a number of different formats. These representations all encode the same number, even though they look different. These formats are primarily used to make it easy for people to read and transcribe keys without introducing errors.

The private key can be represented in a number of different formats, all of which correspond to the same 256-bit number. Table 4-2 shows three common formats used to represent private keys.

Table 4-2. Private key representations (encoding formats)



64 hexadecimal digits



Base58Check encoding: Base58 with version prefix of 128 and 32-bit checksum


K or L

As above, with added suffix 0x01 before encoding

Table 4-3 shows the private key generated in these three formats.

Table 4-3. Example: Same key, different formats
Format Private Key







All of these representations are different ways of showing the same number, the same private key. They look different, but any one format can easily be converted to any other format.

Decode from Base58Check to hex

The sx tools package (See Libbitcoin and sx Tools) makes it easy to write shell scripts and command-line “pipes” that manipulate bitcoin keys, addresses, and transactions. You can use sx tools to decode the Base58Check format on the command line.

We use the command:

$ sx base58check-decode 5J3mBbAH58CpQ3Y5RNJpUKPE62SQ5tfcvU2JpbnkeyhfsYB1Jcn 1e99423a4ed27608a15a2616a2b0e9e52ced330ac530edcc32c8ffc6a526aedd 128

The result is the hexadecimal key, followed by the Wallet Import Format (WIF) version prefix 128.

Encode from hex to Base58Check

To encode into Base58Check (the opposite of the previous command), we provide the hex private key, followed by the Wallet Import Format (WIF) version prefix 128:

$ sx base58check-encode 1e99423a4ed27608a15a2616a2b0e9e52ced330ac530edcc32c8ffc6a526aedd 128 5J3mBbAH58CpQ3Y5RNJpUKPE62SQ5tfcvU2JpbnkeyhfsYB1Jcn

Encode from hex (compressed key) to Base58Check encoding

To encode into Base58Check as a “compressed” private key (see Compressed private keys), we add the suffix to the end of the hex key and then encode as above:

$ sx base58check-encode 1e99423a4ed27608a15a2616a2b0e9e52ced330ac530edcc32c8ffc6a526aedd01 128 KxFC1jmwwCoACiCAWZ3eXa96mBM6tb3TYzGmf6YwgdGWZgawvrtJ

The resulting WIF-compressed format starts with a “K”. This denotes that the private key within has a suffix of “01” and will be used to produce compressed public keys only (see Compressed public keys).

Public keys are also presented in different ways, most importantly as either compressed or uncompressed public keys.

As we saw previously, the public key is a point on the elliptic curve consisting of a pair of coordinates . It is usually presented with the prefix followed by two 256-bit numbers, one for the x coordinate of the point, the other for the y coordinate. The prefix is used to distinguish uncompressed public keys from compressed public keys that begin with a or a .

Here’s the public key generated by the private key we created earlier, shown as the coordinates and :

x = F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A y = 07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB

Here’s the same public key shown as a 520-bit number (130 hex digits) with the prefix followed by and then coordinates, as :

K = 04F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A<?pdf-cr?>07CF33DA18BD734C600B96A72BBC4749D5141C90EC8AC328AE52DDFE2E505BDB

Compressed public keys were introduced to bitcoin to reduce the size of transactions and conserve disk space on nodes that store the bitcoin blockchain database. Most transactions include the public key, required to validate the owner’s credentials and spend the bitcoin. Each public key requires 520 bits (prefix \+ x \+ y), which when multiplied by several hundred transactions per block, or tens of thousands of transactions per day, adds a significant amount of data to the blockchain.

As we saw in the section Public Keys, a public key is a point (x,y) on an elliptic curve. Because the curve expresses a mathematical function, a point on the curve represents a solution to the equation and, therefore, if we know the x coordinate we can calculate the y coordinate by solving the equation y2 mod p = (x3 + 7) mod p. That allows us to store only the x coordinate of the public key point, omitting the y coordinate and reducing the size of the key and the space required to store it by 256 bits. An almost 50% reduction in size in every transaction adds up to a lot of data saved over time!

Whereas uncompressed public keys have a prefix of , compressed public keys start with either a or a prefix. Let’s look at why there are two possible prefixes: because the left side of the equation is y2, that means the solution for y is a square root, which can have a positive or negative value. Visually, this means that the resulting y coordinate can be above the x-axis or below the x-axis. As you can see from the graph of the elliptic curve in Figure 4-2, the curve is symmetric, meaning it is reflected like a mirror by the x-axis. So, while we can omit the y coordinate we have to store the sign of y (positive or negative), or in other words, we have to remember if it was above or below the x-axis because each of those options represents a different point and a different public key. When calculating the elliptic curve in binary arithmetic on the finite field of prime order p, the y coordinate is either even or odd, which corresponds to the positive/negative sign as explained earlier. Therefore, to distinguish between the two possible values of y, we store a compressed public key with the prefix if the is even, and if it is odd, allowing the software to correctly deduce the y coordinate from the x coordinate and uncompress the public key to the full coordinates of the point. Public key compression is illustrated in Figure 4-7.

Figure 4-7. Public key compression

Here’s the same public key generated previously, shown as a compressed public key stored in 264 bits (66 hex digits) with the prefix indicating the y coordinate is odd:

K = 03F028892BAD7ED57D2FB57BF33081D5CFCF6F9ED3D3D7F159C2E2FFF579DC341A

This compressed public key corresponds to the same private key, meaning that it is generated from the same private key. However, it looks different from the uncompressed public key. More importantly, if we convert this compressed public key to a bitcoin address using the double-hash function () it will produce a different bitcoin address. This can be confusing, because it means that a single private key can produce a public key expressed in two different formats (compressed and uncompressed) that produce two different bitcoin addresses. However, the private key is identical for both bitcoin addresses.

Compressed public keys are gradually becoming the default across bitcoin clients, which is having a significant impact on reducing the size of transactions and therefore the blockchain. However, not all clients support compressed public keys yet. Newer clients that support compressed public keys have to account for transactions from older clients that do not support compressed public keys. This is especially important when a wallet application is importing private keys from another bitcoin wallet application, because the new wallet needs to scan the blockchain to find transactions corresponding to these imported keys. Which bitcoin addresses should the bitcoin wallet scan for? The bitcoin addresses produced by uncompressed public keys, or the bitcoin addresses produced by compressed public keys? Both are valid bitcoin addresses, and can be signed for by the private key, but they are different addresses!

To resolve this issue, when private keys are exported from a wallet, the Wallet Import Format that is used to represent them is implemented differently in newer bitcoin wallets, to indicate that these private keys have been used to produce compressed public keys and therefore compressed bitcoin addresses. This allows the importing wallet to distinguish between private keys originating from older or newer wallets and search the blockchain for transactions with bitcoin addresses corresponding to the uncompressed, or the compressed, public keys, respectively. Let’s look at how this works in more detail, in the next section.

Ironically, the term “compressed private key” is misleading, because when a private key is exported as WIF-compressed it is actually one byte longer than an “uncompressed” private key. That is because it has the added 01 suffix, which signifies it comes from a newer wallet and should only be used to produce compressed public keys. Private keys are not compressed and cannot be compressed. The term “compressed private key” really means “private key from which compressed public keys should be derived,” whereas “uncompressed private key” really means “private key from which uncompressed public keys should be derived.” You should only refer to the export format as “WIF-compressed” or “WIF” and not refer to the private key as “compressed” to avoid further confusion.

Remember, these formats are not used interchangeably. In a newer wallet that implements compressed public keys, the private keys will only ever be exported as WIF-compressed (with a K or L prefix). If the wallet is an older implementation and does not use compressed public keys, the private keys will only ever be exported as WIF (with a 5 prefix). The goal here is to signal to the wallet importing these private keys whether it must search the blockchain for compressed or uncompressed public keys and addresses.

If a bitcoin wallet is able to implement compressed public keys, it will use those in all transactions. The private keys in the wallet will be used to derive the public key points on the curve, which will be compressed. The compressed public keys will be used to produce bitcoin addresses and those will be used in transactions. When exporting private keys from a new wallet that implements compressed public keys, the Wallet Import Format is modified, with the addition of a one-byte suffix to the private key. The resulting Base58Check-encoded private key is called a “Compressed WIF” and starts with the letter K or L, instead of starting with “5” as is the case with WIF-encoded (non-compressed) keys from older wallets.

Table 4-4 shows the same key, encoded in WIF and WIF-compressed formats.

Table 4-4. Example: Same key, different formats
Format Private Key










“Compressed private keys” is a misnomer! They are not compressed; rather, the WIF-compressed format signifies that they should only be used to derive compressed public keys and their corresponding bitcoin addresses. Ironically, a “WIF-compressed” encoded private key is one byte longer because it has the added 01 suffix to distinguish it from an “uncompressed” one.

Implementing Keys and Addresses in Python

The most comprehensive bitcoin library in Python is pybitcointools by Vitalik Buterin. In Example 4-4, we use the pybitcointools library (imported as “bitcoin”) to generate and display keys and addresses in various formats.

Example 4-4. Key and address generation and formatting with the pybitcointools library

Example 4-5 shows the output from running this code.

Example 4-5. Running
$ python key-to-address-ecc-example.

Example 4-6is another example, using the Python ECDSA library for the elliptic curve math and without using any specialized bitcoin libraries.

Example 4-6. A script demonstrating elliptic curve math used for bitcoin keys
\ \

Example 4-7 shows the output produced by running this script.

Example 4-7. Installing the Python ECDSA library and running the script
$ # Install Python PIP package manager $ sudo apt-get install python-pip $ # Install the Python ECDSA library $ sudo pip install ecdsa $ # Run the script $ python Secret: 38090835015954358862481132628887443905906204995912378278060168703580660294000 EC point: (70048853531867179489857750497606966272382583471322935454624595540007269312627, 105262206478686743191060800263479589329920209527285803935736021686045542353380) BTC public key: 029ade3effb0a67d5c8609850d797366af428f4a0d5194cb221d807770a1522873

Wallets are containers for private keys, usually implemented as structured files or simple databases. Another method for making keys isdeterministic key generation. Here you derive each new private key, using a one-way hash function from a previous private key, linking them in a sequence. As long as you can re-create that sequence, you only need the first key (known as a seed or master key) to generate them all. In this section we will examine the different methods of key generation and the wallet structures that are built around them.


Bitcoin wallets contain keys, not coins. Each user has a wallet containing keys. Wallets are really keychains containing pairs of private/public keys (see Private and Public Keys). Users sign transactions with the keys, thereby proving they own the transaction outputs (their coins). The coins are stored on the blockchain in the form of transaction-ouputs (often noted as vout or txout).

Nondeterministic (Random) Wallets

In the first bitcoin clients, wallets were simply collections of randomly generated private keys. This type of wallet is called a Type-0 nondeterministic wallet. For example, the Bitcoin Core client pregenerates 100 random private keys when first started and generates more keys as needed, using each key only once. This type of wallet is nicknamed “Just a Bunch Of Keys,” or JBOK, and such wallets are being replaced with deterministic wallets because they are cumbersome to manage, back up, and import. The disadvantage of random keys is that if you generate many of them you must keep copies of all of them, meaning that the wallet must be backed up frequently. Each key must be backed up, or the funds it controls are irrevocably lost if the wallet becomes inaccessible. This conflicts directly with the principle of avoiding address re-use, by using each bitcoin address for only one transaction. Address re-use reduces privacy by associating multiple transactions and addresses with each other. A Type-0 nondeterministic wallet is a poor choice of wallet, especially if you want to avoid address re-use because that means managing many keys, which creates the need for frequent backups. Although the Bitcoin Core client includes a Type-0 wallet, using this wallet is discouraged by developers of Bitcoin Core. Figure 4-8 shows a nondeterministic wallet, containing a loose collection of random keys.

Deterministic (Seeded) Wallets

Deterministic, or “seeded” wallets are wallets that contain private keys that are all derived from a common seed, through the use of a one-way hash function. The seed is a randomly generated number that is combined with other data, such as an index number or “chain code” (see Hierarchical Deterministic Wallets (BIP0032/BIP0044)) to derive the private keys. In a deterministic wallet, the seed is sufficient to recover all the derived keys, and therefore a single backup at creation time is sufficient. The seed is also sufficient for a wallet export or import, allowing for easy migration of all the user’s keys between different wallet implementations.

Figure 4-8. Type-0 nondeterministic (random) wallet: a collection of randomly generated keys

Mnemonic codes are English word sequences that represent (encode) a random number used as a seed to derive a deterministic wallet. The sequence of words is sufficient to re-create the seed and from there re-create the wallet and all the derived keys. A wallet application that implements deterministic wallets with mnemonic code will show the user a sequence of 12 to 24 words when first creating a wallet. That sequence of words is the wallet backup and can be used to recover and re-create all the keys in the same or any compatible wallet application. Mnemonic code words make it easier for users to back up wallets because they are easy to read and correctly transcribe, as compared to a random sequence of numbers.

Mnemonic codes are defined in Bitcoin Improvement Proposal 39 (see [bip0039]), currently in Draft status. Note that BIP0039 is a draft proposal and not a standard. Specifically, there is a different standard, with a different set of words, used by the Electrum wallet and predating BIP0039. BIP0039 is used by the Trezor wallet and a few other wallets but is incompatible with Electrum’s implementation.

BIP0039 defines the creation of a mnemonic code and seed as a follows:

  1. Create a random sequence (entropy) of 128 to 256 bits.
  2. Create a checksum of the random sequence by taking the first few bits of its SHA256 hash.
  3. Add the checksum to the end of the random sequence.
  4. Divide the sequence into sections of 11 bits, using those to index a dictionary of 2048 predefined words.
  5. Produce 12 to 24 words representing the mnemonic code.

Table 4-5 shows the relationship between the size of entropy data and the length of mnemonic codes in words.

Table 4-5. Mnemonic codes: entropy and word length
Entropy (bits) Checksum (bits) Entropy+checksum Word length





















The mnemonic code represents 128 to 256 bits, which are used to derive a longer (512-bit) seed through the use of the key-stretching function PBKDF2. The resulting seed is used to create a deterministic wallet and all of its derived keys.

Tables 4-6 and 4-7 show some examples of mnemonic codes and the seeds they produce.

Table 4-6. 128-bit entropy mnemonic code and resulting seed

Entropy input (128 bits)


Mnemonic (12 words)

army van defense carry jealous true garbage claim echo media make crunch

Seed (512 bits)

3338a6d2ee71c7f28eb5b882159634cd46a898463e9d2d0980f8e80dfbba5b0fa0291e5fb88 8a599b44b93187be6ee3ab5fd3ead7dd646341b2cdb8d08d13bf7

Table 4-7. 256-bit entropy mnemonic code and resulting seed

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