The linear algebra of bit twiddling

The previous post looked at the tempering step of the Mersenne Twister, formulating a sequence of bit operations as multiplication by a matrix mod 2. This post will look at the components more closely. The theorems of linear algebra generally hold independent of the field of scalars. Typically the field is ℝ or ℂ, but most of basic linear algebra works the same over every field [1]. In particular, we can do linear algebra over a finite field, and we’re interested in the most finite of finite fields GF(2), the field with just two elements, 0 and 1. In GF(2), addition corresponds to XOR. We will denote this by ⊕ to remind us that although it’s addition, it’s not the usual addition, i.e. 1 ⊕ 1 = 0. Similarly, multiplication corresponds to AND. We’ll work with 8-bit numbers to make the visuals easier to see. Shifting a number left one bit corresponds to multiplication by a matrix with 1’s below the diagonal main. Shifting left by k bits is the same as shifting left by 1 bit k times, so…