Introduction

An 8-bit reverse-bits lookup table stores the bit-reversed value for every possible byte from 0 to 255. The table is useful because bit reversal then becomes an O(1) array lookup instead of repeated shifting and masking. The interesting part is how the table is generated efficiently: either by reversing each byte directly or by using a recurrence that builds each entry from a smaller one.

The Direct Generation Idea

The most straightforward algorithm loops through all 256 byte values and reverses each one bit by bit.

1def reverse_byte(value):
2    result = 0
3    for i in range(8):
4        result = (result << 1) | (value & 1)
5        value >>= 1
6    return result
7
8
9table = [reverse_byte(i) for i in range(256)]
10print(table[:8])

This is easy to understand:

• take the least significant bit
• append it to the reversed result
• shift the input right
• repeat eight times

For generating the table once, this is already perfectly practical.

The Recurrence Behind Faster Table Generation

There is also a neat recurrence for generating the table without reversing every byte from scratch.

If rev[i] is the reversed 8-bit value of i, then:

rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << 7)

Why it works:

• 'i >> 1 removes the least significant bit from i'
• the reversed version of that smaller value is already known
• shifting it right by one makes room for the bit that used to be the least significant bit
• '(i & 1) << 7 moves that original least significant bit to the most significant position in the reversed byte'

That means every new entry can be derived from a previously computed smaller entry.

A Full Table Generator Using The Recurrence

1def build_reverse_table():
2    rev = [0] * 256
3    for i in range(1, 256):
4        rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << 7)
5    return rev
6
7
8table = build_reverse_table()
9print(table[0])
10print(table[1])
11print(table[13])

This works because i >> 1 is always smaller than i, so its reversed value has already been computed earlier in the loop.

Why Lookup Tables Are Used

Suppose you need to reverse bits for a large stream of bytes. Doing eight bit operations every time may be fine, but a table lookup is usually faster and simpler in hot code.

1def reverse_bytes(data, table):
2    return bytes(table[b] for b in data)
3
4
5table = build_reverse_table()
6print(reverse_bytes(bytes([0b00001101]), table))

This pattern shows up in low-level graphics work, communication protocols, digital signal processing, and some specialized compression or encoding tasks.

Extending The Idea

The same idea can be extended beyond 8 bits, but the lookup table grows exponentially. A 16-bit table would need 65536 entries, which is often too large to justify when compared with simpler arithmetic methods.

That is why 8-bit tables are common: they are tiny, cheap to generate, and easy to combine for larger word sizes if needed.

For example, a 32-bit reversal can be built from four reversed bytes plus byte reordering.

Common Pitfalls

The most common mistake is forgetting that the lookup table is for fixed-width values. An 8-bit reversal assumes exactly eight bits, including leading zeros.

Another issue is using signed integers carelessly in languages where right shift on signed values can behave differently. Lookup-table generation is safest with unsigned byte logic.

It is also easy to overcomplicate the problem. If you reverse bits only a few times, generating a table may be unnecessary.

Finally, make sure you distinguish bit reversal from byte-order reversal. Reversing the order of bits inside each byte is not the same as reversing the order of bytes in a multi-byte integer.

Summary

• An 8-bit reverse-bits table stores the reversed value for all 256 possible bytes.
• The direct algorithm reverses each byte using shifts and masks.
• A useful recurrence is rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << 7).
• Lookup tables make repeated bit reversal fast and simple.
• The technique works best for small fixed widths such as 8-bit values.