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Base58

Representing numbers with a set of characters that's easy to share.

  1 2 3 4 5 6 7 8 9
A B C D E F G H   J K L M N   P Q R S T U V W X Y Z
a b c d e f g h i j k   m n o p q r s t u v w x y z

Base58 is a set of characters you can use to represent big numbers in a shorter and more user-friendly format.

Try it! - Base58 Converter

What does base58 mean?

The “base” refers to the number of characters you use to represent a number.

Base Characters
2 (binary) 01
10 (decimal) 0123456789
16 (hexadecimal) 0123456789abcdef
58 123456789ABCDEFGH JKLMN PQRSTUVWXYZabcdefghijk mnopqrstuvwxyz

So in everyday life, we are used to working with base10 numbers (using the digits 0-9).

But if you’re a computer, it’s easy enough to use extra characters to represent numbers:

base10(9999) = 9999
base16(9999) = 270f
base58(9999) = 3yQ

All of these “numbers” have the same valuethey just use different sets of characters (bases) to represent it.

The more characters you have in your base, the less of them you will need to use to represent big numbers. So the bigger your base, the shorter your “number” can be.

Why base58?

Because 58 is the number of characters you are left with when you use all the characters in the alphanumeric alphabet (62), but remove all the easily mistakable characters like 0, O, l and I.

alphanumeric = 0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz
base58       =  123456789ABCDEFGH JKLMN PQRSTUVWXYZabcdefghijk mnopqrstuvwxyz

So base58 has two advantages:

  1. It gives you a large set of characters, so you can represent large numbers in a shorter format.
  2. It leaves out awkward characters, to save you from making mistakes when transcribing.
A pesky O/0

Base58 Encode

To convert an integer (base10) to base58, you use the modulus1 function to work out which characters you need to represent the original number.

Basically, you keep dividing your number by 58, taking the remainder at each step of the way to get the next character index, finishing when there are no remainders left.

base10 = 123456789

123456789 % 58 = 19
  2128565 % 58 = 23
    36699 % 58 = 43
      632 % 58 = 52
       10 % 58 = 10

base58 = [10][52][43][23][19]
base58 = BukQL

Base58 Encode (simple)

# A simple function that converts an _integer_ to base58:

def int_to_base58(i)

  @characters = %w[
      1 2 3 4 5 6 7 8 9
    A B C D E F G H   J K L M N   P Q R S T U V W X Y Z
    a b c d e f g h i j k   m n o p q r s t u v w x y z
]

  # create an empty string (in preparation to hold the new characters)
  buffer = ''

  # keep finding the remainder until our starting number hits zero
  while i > 0
    # find the remainder after dividing by 58 (% = modulus)
    remainder = i % 58

    # add the base58 character to the start of the string
    buffer = @characters[remainder] + buffer

    # divide our integer by 58, and repeat...
    i = i / 58
  end

  return buffer

end

puts int_to_base58(123456789) #=> BukQL

Base58 Decode

To convert a base58 number in to base10, you take each character index and multiply it with how many 58s that position in the number represents.

Then you just add all these values together.

base58 = BukQL

L = 19 * 58^0 = 19
Q = 23 * 58^1 = 1334
k = 43 * 58^2 = 144652
u = 52 * 58^3 = 10145824
B = 10 * 58^4 = 113164960

base10 = 19 + 1334 + 144652 + 10145824 + 113164960
base10 = 123456789

Base58 Decode (simple)

def base58_to_int(base58)

  @characters = %w[
      1 2 3 4 5 6 7 8 9
    A B C D E F G H   J K L M N   P Q R S T U V W X Y Z
    a b c d e f g h i j k   m n o p q r s t u v w x y z
    ]
  
  # create an integer to hold the result
  total = 0

  # reverse the base58 string so we can read characters from right to left
  base58 = base58.reverse
  
  # run through each character, including the index so we know how many character we've read
  base58.each_char.with_index do |char, i|
  
    # get the index number for this character
    char_i = @characters.index(char)
    
    # work out how many 58s this character represents (increment the power for each character)
    value  = char_i * (58**i)
    
    # add to total
    total = total + value
  end

  return total

end

puts base58_to_int("BukQL") #=> 123456789

Base58 in Bitcoin

Base58 is used when you want to convert commonly used things in bitcoin in to an easier-to-share format. For example:

  1. WIF Private Keys
    • A private key is like a “master password”, and you can use it when you want to import bitcoins in to a new wallet. For this occassion, there is such a thing as a WIF Private Key, which is basically a private key in base58.
  2. Addresses
    • A public key is the “public” counterpart to a private key, and you use them when you want to send bitcoins to someone, so it’s expected that you’re going to type one out from time to time. However, public keys are quite lengthy, so we convert them to Addresses instead, which makes use of base58 in the final step of the conversion.

Leading Zeros (0x00…)

Bitcoin Base58 also converts every byte of zeros (0x00) at the start of a number to a 1.

You see, putting zeros at the start of a number does not increase the size of the number (e.g. 0x12 is the same as 0x0012), so any zeros at the start of a number would be “lost” in a conversion to base58.

Therefore, to ensure that leading zeros appear in the result, the bitcoin base58 encoding includes an additional step to convert all leading 0x00’s to 1’s

Code

These code snippets do the base58 conversion used in Bitcoin. They convert to and from hexadecimal, because that’s the most common format we work with.

Ruby

module Base58

  @chars = %w[
      1 2 3 4 5 6 7 8 9
    A B C D E F G H   J K L M N   P Q R S T U V W X Y Z
    a b c d e f g h i j k   m n o p q r s t u v w x y z
]
  @base = @chars.length

  def self.encode(hex)
    i = hex.to_i(16)
    buffer = String.new

    while i > 0
      remainder = i % @base
      i = i / @base
      buffer = @chars[remainder] + buffer
    end

    # add '1's to the start based on number of leading bytes of zeros
    leading_zero_bytes = (hex.match(/^([0]+)/) ? $1 : '').size / 2

    ("1"*leading_zero_bytes) + buffer
  end
  
  def self.decode(base58)
    total = 0 # integer to hold conversion to decimal

    # run through each character
    base58.reverse.each_char.with_index do |char, i|
      char_i = @chars.index(char) # get the index number for this character
      value  = (58**i) * char_i   # work out how many 58s this character represents
      total = total + value     # add to total
    end

    # convert this integer to hex
    hex = total.to_s(16)

    # add leading 00s for every leading 1
    leading_1s = (base58.match(/^([1]+)/) ? $1 : '').size

    ("00"*leading_1s) + hex
  end

end

puts Base58.encode('0093ce48570b55c42c2af816aeaba06cfee1224faebb6127fe') #=> 1EUXSxuUVy2PC5enGXR1a3yxbEjNWMHuem
puts Base58.decode('1EUXSxuUVy2PC5enGXR1a3yxbEjNWMHuem') #=> 0093ce48570b55c42c2af816aeaba06cfee1224faebb6127fe

PHP

<?php

// Base58 Characters
$chars = str_split("123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz");

// -------------
// Base58 Encode
// -------------

$hex = "00662ad25db00e7bb38bc04831ae48b4b446d1269817d515b6";
$num = gmp_init($hex, 16); // convert to an integer
$base58 = "";

// Keep dividing by 58 and taking the remainder as the character
while ($num > 0) {
    $rem = gmp_mod($num, 58); // remainder (what we get the character for)
    $num = gmp_div($num, 58); // quotient  (keep dividing the number to get remainders)
    $base58 = $chars[intval($rem)].$base58; // add base58 char to the start
}

// Convert leading 00s in hex to leading 1s
$count = intval(strspn($hex, "0") / 2); // how many leading 0s, then divide by 2
$leading = str_repeat("1", $count);

// Result
$result = $leading.$base58;
echo $result.PHP_EOL; // 1AKDDsfTh8uY4X3ppy1m7jw1fVMBSMkzjP

// -------------
// Base58 Decode
// -------------
$base58 = "1AKDDsfTh8uY4X3ppy1m7jw1fVMBSMkzjP";
$int = gmp_init(0); // integer to hold result

// Convert to decimal
$base58a = str_split(strrev($base58));   // create an array we can loop through
foreach ($base58a as $i => $c) {         // run through each character
    $multiple = gmp_pow(58, $i);         // how many 58s this position holds (e.g. 58^0, 58^1, 58^2...)
    $index = array_search($c, $chars);   // get index number for base58 char (e.g. B=10)
    $value = gmp_mul($index, $multiple); // multiply to get number of 58s this character is representing
    $int = $int + $value;                // add to total
}

// Convert to hexadecimal
$gmp = gmp_init(strval($int), 10); // create gmp number from bit string (base 10) NOTE: gmp_init takes strings
$hex = gmp_strval($gmp, 16); // convert to hex string representation
if (strlen($hex) % 2 !== 0) { // return even number of characters (hex2bin prefers it)
    $hex = '0'.$hex;
}

// Convert leading 1s in base58 to leading 00s
$count = strspn($base58, "1"); // how many leading 0s, then divide by 2
$leading = str_repeat("00", $count);

// Result
$result = $leading.$hex;
echo $result.PHP_EOL; // 00662ad25db00e7bb38bc04831ae48b4b446d1269817d515b6

Notes

Modulus (%)

The modulus (%) function is a sister of the divide (/) function. It gives you the remainder of a division:

7 % 6 = 1
7 % 5 = 2
7 % 4 = 3
7 % 3 = 1

Source

Resources

Further Reading

Thanks

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