Entropy to Mnemonic Math

Converting a random number into a list of words is a precise mathematical process. It involves creating a random number, adding a checksum to prevent errors, and then splitting that number into 11-bit chunks.

1. Step 1: Generate Entropy

The process starts with a source of randomness called Entropy.

• For a 12-word phrase, you need 128 bits of entropy.

• For a 24-word phrase, you need 256 bits of entropy.

2. Step 2: Add the Checksum

To ensure the words aren't written down incorrectly, a checksum is added to the end of the entropy.

1. Hash the entropy using SHA256.

2. Take the first few bits of that hash (1 bit for every 32 bits of entropy).

   • 12 words $\rightarrow$ 4 bits of checksum.
   • 24 words $\rightarrow$ 8 bits of checksum.

3. Append these bits to the end of your entropy.

3. Step 3: Split into 11-bit Chunks

Now you have a long string of bits (e.g., 128 entropy + 4 checksum = 132 bits).

• Split this string into groups of 11 bits.

• $132 \div 11 = 12$ groups.

• Each group of 11 bits represents a number between 0 and 2047.

4. Step 4: Map to the Wordlist

Each of those numbers is used as an index into the BIP39 Wordlist.

• If your first 11 bits are 00000000000, the first word is abandon (index 0).

• If your bits are 11111111111, the word is zoo (index 2047).

Why 11 bits?

The number 11 was chosen because $2^{11} = 2048$. This allows for a wordlist of exactly 2048 words, which is large enough to prevent guessing but small enough to be manageable in code and on paper.

Input Entropy	Checksum Bits	Total Bits	Mnemonic Words
128 Bits	4	132	12 Words
160 Bits	5	165	15 Words
192 Bits	6	198	18 Words
224 Bits	7	231	21 Words
256 Bits	8	264	24 Words

In the next section, we will review the BIP39 English Wordlist.