[ellis-bell]

XOR is used a ton in the theoretical underpinnings of cryptography. It's used in the one time pad which is essentially the "smallest" cryptographic scheme that is perfectly secure (perfectly secure has a mathematical definition in this context, it's not saying there can never be any attacks).

In general the reason why is that if you have two random variables x and y, where x has any distribution (so for example x could even be "attack normandy on june 6" with certainty) and y is uniformly distributed across all n-bit strings (so it could be any string of n zeros and ones with equal probability), then you can show that x ^ y appears as if it is also uniformly distributed across all n-bit strings as well.

Because of this property it's used frequently in many higher order methods as well.