@ -7,7 +7,7 @@ Ownership of bitcoin is established through _digital keys_, _bitcoin addresses_
The digital keys within each user's wallet allow the user to sign transactions, thereby providing cryptographic proof of the ownership of the bitcoins sourced by the transaction. Keys come in pairs consisting of a private (secret) and public key. Think of the public key as similar to a bank account number and the private key as similar to the secret PIN number, or signature on a cheque, 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 transactions, the public key is represented by its digital fingeprint called a _bitcoin address_ which is used as the beneficiary name on a cheque (ie. "Pay to the order of"). In most cases a bitcoin address is a generated from and corresponds to a public key. However, like a beneficiary name on a cheque, some bitcoin addresses do not represent a public key and instead 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 cheques: a single payment instrument that can be used to pay into people's accounts, company accounts, pay for bills or pay to cash. The bitcoin address is the only part of the wallet that users will routinely see.
In transactions, the public key is represented by its digital fingeprint called a _bitcoin address_ which is used as the beneficiary name on a cheque (ie. "Pay to the order of"). In most cases a bitcoin address is generated from and corresponds to a public key. However, like a beneficiary name on a cheque, some bitcoin addresses do not represent a public key and instead 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 cheques: a single payment instrument that can be used to pay into people's accounts, company accounts, pay for bills or pay to cash. The bitcoin address is the only part of the wallet that users will routinely see.
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.
@ -18,7 +18,7 @@ In this chapter we will introduce wallets, which contain cryptographic keys. We
((("public key")))
Public key cryptography was invented in the 1970s and is mathematics applied to computer 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 special relationship between the public key and private key that allows the private key to be used to generate a signature. This signature can be validated against the public key without revealing the private key. When spending bitcoins, the current bitcoin owner presents their public key and a signature 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 that transaction as valid, meaning the person transfering the bitcoin owned them at the time of the transfer.
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 special relationship between the public and the private key that allows the private key to be used to generate a signature. This signature can be validated against the public key without revealing the private key. When spending bitcoins, the current bitcoin owner presents their public key and a signature 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 that transaction as valid, meaning the person transfering the bitcoins, owned them at the time of the transfer.
[TIP]
====
@ -27,7 +27,7 @@ In most implementations, the private and public keys are stored together as a _k
=== Keys
Your 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 below:
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 below:
[[k_to_K_to_A]]
.Private Key, Public Key and Bitcoin Address
@ -168,7 +168,7 @@ Bitcoin addresses are almost always presented to users in an encoding called "Ba
[[base58]]
====== Base58
Base58Check is a format developed for the use of bitcoin and used in many other crypto-currencies. It offers a balance between compact representation, readbility, disambiguity and error detection and prevention. In order to represent long numbers in a compact way, many computer systems use mixed-alphanumeric representations with a base (or radix) higher than 10. For example, whereas the traditional decimal system uses the ten numerals 0 through 9, the hexadecimal system uses six additional symbols, the letters A through F. A number represented in hexadecimal format is shorter than the equivalent decimal representation. Even more compact, base64 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. Base64 is most commonly used to add binary attachments to email. Base58 is a subset of base64, which uses the upper and lower case letters and numbers but ommits 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) mentioned above.
Base58Check is a format developed for the use of bitcoin and used in many other crypto-currencies. It offers a balance between compact representation, readbility, disambiguity and error detection and prevention. In order to represent long numbers in a compact way, many computer systems use mixed-alphanumeric representations with a base (or radix) higher than 10. For example, whereas the traditional decimal system uses the ten numerals 0 through 9, the hexadecimal system uses six additional symbols, the letters A through F. A number represented in hexadecimal format is shorter than the equivalent decimal representation. Even more compact, base64 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. Base64 is most commonly used to add binary attachments to email. Base58 is a subset of base64, which uses the upper and lower case letters and numbers but ommits 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) mentioned above.
[[base58alphabet]]
----
@ -183,7 +183,7 @@ To add extra security against typos or transcription errors, Base58Check is a fo
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).
Next compute the checksum by "double-SHA", 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 last four bytes. These four bytes serve as the error-checking code, or checksum. The checksum is concatenated (appended) to the end.
Next compute the checksum by "double-SHA", 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 last four bytes. These four bytes serve as the error-checking code or checksum. The checksum is concatenated (appended) to the end.
The result of the above is now a prefix, the data and a checksum, concatenated (bytewise). This result is encoded using the base-58 alphabet described in the section above.
The resulting WIF-compressed format, starts with a "K". This denotes that the private key within has a suffic of "01" and will be used to produce compressed public keys only (See <<compressed_pubkey>> below)
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_pubkey>> below)
===== Public Key Formats
@ -271,28 +271,28 @@ K = 04 32 5D 52 E3 B7 ... CD 90 C2
===== Compressed Public Keys
Compressed public keys were introduced to bitcoin to reduce the size of transaction and conserve disk space on nodes that store the bitcoin blockchain database. Most transactions include a signature and public key, required to spend the funds and "transfer out" of one bitcoin address and into another. Each public key requires 513 bytes (prefix \+ x \+ y) to store, which when multiplied by several hundred transactions per block, or tens of thousands of transactions per day can add a significant amount of data to the blockchain.
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 a signature and a public key, required to spend the funds and "transfer out" of one bitcoin address and into another. Each public key requires 513 bytes (prefix \+ x \+ y) to store, which when multiplied by several hundred transactions per block, or tens of thousands of transactions per day can add a significant amount of data to the blockchain.
As we saw in the section <<pubkey>> above, a public key is a point (x,y) on an elliptic curve. Since 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 euqation y^2 mod p = (x^3 + 7) mod p. That allows us to store only the x-coordinate of the public key point, ommitting the y-coordinate and reducing the size of the key and the space required to store it by 256 bits. A 50% reduction in size in every transaction adds up to a lot of data saved over time!
As we saw in the section <<pubkey>> above, a public key is a point (x,y) on an elliptic curve. Since 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 y^2^ mod p = (x^3^ + 7) mod p. That allows us to store only the x-coordinate of the public key point, ommitting the y-coordinate and therefore reducing the size of the key and the space required to store it by 256 bits. A 50% reduction in size in every transaction adds up to a lot of data saved over time!
There's one small detail to consider: since the left side of the equation is y^2, 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, the curve is symmetric, meaning it is reflected like a mirror by the x-axis. So, while we can ommit 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, as 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 ycoordinate is either even or odd, which corresponds to the positive/negative sign as explained above. Therefore, to distinguish between the two possible values of y, we store a +compressed public key+ with the prefix +02+ if the +y+ is even (same as above-the-axis), and +03+ if it is odd (same as below the axis), allowing the software to correctly deduce the y-coordinate from the x-coordinate.
There's one small detail to consider: since the left side of the equation is y^2, 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, the curve is symmetric, meaning it is reflected like a mirror by the x-axis. So, while we can ommit 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, as 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 above. Therefore, to distinguish between the two possible values of y, we store a +compressed public key+ with the prefix +02+ if the +y+ is even (same as above-the-axis), and +03+ if it is odd (same as below-the-axis), allowing the software to correctly deduce the y-coordinate from the x-coordinate.
Here's the same public key above, shown as a +compressed public key+ stored in 264-bits (66 hex digits) with the prefix +02+ indicating the +y+ coordinate is even:
Here's the same public key above, shown as a +compressed public key+ stored in 264-bits (66 hex digits) with the prefix +02+ indicating the y-coordinate is even:
.Compressed Public Key K shown in hex (66 hex digits) as +K = {02 or 03} x+
----
K = 02 32 5D 52 E3 B7 ... E5 D3 78
----
The compressed public key, above, 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 (RIPEMD160(SHA256(K))) 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) which produce two different bitcoin addresses. But the private key is identical and therefore can produce valid signatures that authorize spending funds from _either_ bitcoin address.
The compressed public key, above, 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 (RIPEMD160(SHA256(K))), 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) which produce two different bitcoin addresses. But the private key is identical and therefore can produce valid signatures that authorize spending funds from _either_ bitcoin address.
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 and older clients which 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, both can be signed for by the private key, but they are different 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 and older clients which 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 compressed, or the uncompressed public keys. Let's look at how this works in more detail, in the next section.
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 compressed, or the uncompressed public keys. Let's look at how this works in more detail in the next section.
===== Compressed Private Keys
If a bitcoin wallet is able to implement compressed public keys, then 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 +01+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.
If a bitcoin wallet is able to implement compressed public keys, then 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 +01+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.
Here's the same key, encoded in WIF and WIF-compressed formats:
@ -326,7 +326,7 @@ Wallets contain keys, not coins. The coins are stored on the blockchain in the f
===== Non-Deterministic (Random) Wallets
The first and most important step in generating keys is to find a secure source of entropy, or randomness. The private key is a 256-bit number, which must be selected at random. Creating a bitcoin key is essentially the same as "Pick a number between 1 and 2^256^". The exact method you use to pick that number does not matter as long as it is not predictable or repeatable. Bitcoin software will use the underlying operating system's random number generators to produce 256 bits of entropy. 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.
The first and most important step in generating keys is to find a secure source of entropy or randomness. The private key is a 256-bit number, which must be selected at random. Creating a bitcoin key is essentially the same as "Pick a number between 1 and 2^256^". The exact method you use to pick that number does not matter as long as it is not predictable or repeatable. Bitcoin software will use the underlying operating system's random number generators to produce 256 bits of entropy. 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.
Minh T. Nguyen
10 years ago