Recovery Phrases
Recovery phrases (also known as mnemonics) provide a way for humans to easily read and write down arbitrary large numbers which would otherwise be very difficult to remember.
They use a predefined dictionary of simple words (available in many different languages) which map uniquely back and forth to a binary code.
Seeds / Entropy
The recovery phrase is normally used to encode a wallet's "seed" - a secret random number which is 28 decimal digits long, or more!
A seed is also called just entropy1, implying that it is a sequence of bytes which has been generated using high-quality randomness methods.
All keys belonging to a wallet address-derivation from the wallet seed.
See also the definition of "entropy" from information theory.
Encoding
The process for encoding recovery phrases is described in BIP-0039 § Generating the mnemonic. Below is a reformulation of this specification.
The allowed size of entropy is 96-256 bits and must be multiple of 32 bits (4 bytes).
A checksum is appended to the initial entropy by taking the first $|ent| / 32$ bits of the SHA-256 hash of it, where $|ent|$ designates the entropy size in bits.
Then, the concatenated result is split into groups of 11 bits, each encoding a number from 0 to 2047 serving as an index into a known dictionary (see below).
| Sentence Length | Entropy Size | Checksum Size |
|---|---|---|
| 9 words | 96 bits (12 bytes) | 3 bits |
| 12 words | 128 bits (16 bytes) | 4 bits |
| 15 words | 160 bits (20 bytes) | 5 bits |
| 18 words | 192 bits (24 bytes) | 6 bits |
| 21 words | 224 bits (28 bytes) | 7 bits |
| 24 words | 256 bits (32 bytes) | 8 bits |
Dictionaries
Cardano uses the same dictionaries as defined in BIP-0039.
Example
This is an English recovery phrase, ordered left-to-right, then top-to-bottom.
write maid rib
female drama awake
release inhale weapon
crush mule jump
sound erupt stereo
It is 15 words long, so $15\times11 = 165$ bits of information, which is split into a 160 bit seed and 5 bit checksum.
Using the dictionary, these words resolve to:
2036 1072 1479
679 529 129
1449 925 1986
424 1162 967
1662 615 1708
Which is:
Seed:
01111100111 11100100110 01111101011
01010100111 01000010001 00010000001
10110101001 01110011101 11111000010
00110101000 10010001010 01111000111
11001111110 01001100111 110101
Checksum: 01100
Seed (base16): fe90c2e3aa7422206d4b9df846a2453c7cfc99f5
Checksum (base16): 0c