Notation $x \in X$ means $x$ is an element of the set $X$. $\mathbb{F}$ a finite field. $\mathbb{F}^N = {(x_1,\dots,x_N): x_1,\dots,x_n \in \mathbb{F}}$. $\mathbb{F}[X]$ is the set of polynomials in one variable with coefficients in $\mathbb{F}$. $\mathbb{F}[X, Y]$ is the set of polynomials in two variables with coefficients in $\mathbb{F}$. STARKs Over the past years many different STARKs implementations appeared. So far one thing they all have in common is that the order of the base field is a smooth prime $p$. That means, it is a prime such that $p-1$ is divisible by a high power of two. Say, $p-1 = 2^nk$ with $n$ larger than $20$. These choices are efficiency-driven. Such primes allow for fast implementations of algorithms needed in the generation of the proof, like the Fast Fourier Transform.