Bit Grouping: The 5-Bit Transformation

The core mathematical engine of Bech32 is the conversion of standard 8-bit bytes into 5-bit chunks. This process is often called "Squashing" or "Bit Grouping."

1. The 8-to-5 Problem

Computers store data in bytes (8 bits). However, the Bech32 alphabet only has 32 characters, which can only represent 5 bits of information ($2^5 = 32$).

• To encode a 20-byte hash into Bech32, we must take the 160 bits (20 bytes $\times$ 8) and redistribute them into groups of 5.

• Total Groups: $160 \div 5 = 32$ groups.

2. The Conversion Process

Imagine we have three bytes (24 bits): 11111111 00000000 10101010. To convert to 5-bit chunks, we simply regroup them:

1. 11111 (Group 1)

2. 11100 (Group 2)

3. 00000 (Group 3)

4. 01010 (Group 4)

5. 1010? (Group 5 - needs padding)

3. Padding

Since the total number of bits might not be a perfect multiple of 5, the algorithm adds zero-bits at the end to complete the final 5-bit group.

• If we have 160 bits, it divides perfectly (32 groups).

• If we have a Taproot hash (256 bits), $256 \div 5 = 51$ with a remainder of 1. We would need 52 groups to hold all the data.

4. Why this is efficient?

Because 5 and 8 are both small integers, this bit-shifting can be done using simple bitwise operators (\u003c\u003c, \u003e\u003e, \u0026).

• Unlike Base58, which requires "BigInteger" division (calculating values across the entire 20-byte string at once), Bech32 can be processed 1 byte at a time.

• This makes Bech32 ideal for low-power devices like hardware wallets and embedded systems.

Data Type	Bit Count	5-Bit Groups
PK Hash	160 Bits	32 Groups
Script Hash	256 Bits	52 Groups
Taproot Key	256 Bits	52 Groups

In the final section, we will build a Python Bech32 Auditor.