# Definition of the multiplication matrix of element of finite field.

I've been reading this article XOR-counts and lightweight multiplication with fixed elements in binary finite fields and I came across the term Multiplication matrix and I've wondered if that's some terminology from finite fields theory I perhaps don't know (English is not my native language and I don't study math in English). It seems really odd that the author mentions the term in an abstract and doesn't define it properly (but then maybe it's just a widely used term).
From what I've understood multiplication matrix is matrix such that: $$\forall \alpha \in \mathbb{F}_{2^n} \exists A \in \mathbb{F}_2$$ such that $$\forall x \in \mathbb{F}_{2^n}: \alpha \cdot x = A \cdot \hat{x}$$, where $$\hat{x}$$ is element $$x$$ written as an n-dimensional vector over $$\mathbb{F}_2$$. Thank you for any help.

Multiplication by an element of $$\mathbb F_{2^n}$$ on elements of $$\mathbb F_{2^n}$$ produces a $$\mathbb F_2$$ linear transformation on $$\mathbb F_{2^n}$$. As such, you can pick a basis of $$\mathbb F_{2^n}$$ (necessarily having $$n$$ elements) and find a transformation matrix (an element of $$M_n(\mathbb F_2)$$) for each element.
Our goal in this work is to explore some connections and properties of the direct and sequential XOR-count metrics and then to apply these to get some theoretical results regarding optimal implementations of matrices that represent multiplication with a ﬁxed ﬁeld element $$α ∈ \mathbb F_{2^k}$$. Optimal choices of these matrices (called multiplication matrices) can then be used for local optimizations of matrices over $$\mathbb F_{2^k}$$.
• Could you please describe a way to finding such a transformation matrix? I realize that when I pick a basis $B$, the element $\alpha x$ has only one way of being written in this basis. I just can't think of a way to get matrix $A$ from $x$ and $\alpha x$. I can even think of a general example where multiple matrices that fulfill these criteria exist. Does $B^{-1}AB$ change with basis? By my geometrical intuition, it shouldn't, since linear transformation just a rotation of vector space. But I guess that might be the goal of this article: to find in which basis has $[A]_B$ the lowest XOR-count. – kytka Sep 24 at 21:49