Published on Oct 05, 2016

Recurse Center Day 8: Hamming Weight: I Am So Confused

I’m reimplementing my cryptopals solutions in Python, and today I worked a challenge that required me to calculate the Hamming distance between two equal-length strings. Hamming distance is a measure of difference - it indicates how many corresponding characters in the input strings are different. For example, the Hamming distance between TARA and TART is 1. Well, that’s not entirely true. More often it’s more relevant to determine the specific number of differing bits. As you can see below, in this case, the strings differ by 3 bits.

ASCII Binary
A 01000001
T 01010100

Calculating Hamming Distance

Given two bytes, we can calculate their Hamming distance in two steps:

  1. XOR (exclusive or) the bytes
  2. Determine the Hamming weight of the XORed bytes

XOR the Bytes

Remember that Hamming distance is a measure of how many bits differ between two equal-length strings. XOR is relevant because when XORing two bytes, by the nature of the exclusive or operation, the only time a 1 bit will appear in the resulting byte is when the two input bits are different, and that’s exactly what we’re trying to determine here!

So for the above example: 01000001 ⨁ 01010100 = 00010101

Now all we need to do is count the number of occurrences of 1 in the resulting bit string.

Enter Hamming Weight

Hamming weight is formally defined as the count of the symbols in a string different than the zero-symbol of the given alphabet, Hamming weight is especially useful for evaluating bit strings, which are represented by an alphabet with two symbols, 0 and 1. So to find the Hamming weight of a bit string is to simply count the number of 1s in the string.

I Am So Confused

While I was able to implement a function for calculating the Hamming weight of a given byte , despite several attempts to understand how and why it works, I still don’t and I am very confused. I think the main issue is that I don’t entirely understand bitwise operations so I need to focus on addressing that before I delve into understanding Hamming weight implementations. If you know of any good resources for learning and practicing bitwise operations, will you tweet them at me please and thank you?