should note that the term “binary” has two different meanings here, which will become clear from the context. A binary operation on a set $G$ is a map from the cartesian product $G\times G$ into $G$. Referring to the representation of real numbers in base $b=2$, the elements of the set $\mathcal{A}_2^m$ are called binary vectors, and for $m=1$ one speaks of binary digits or bits.

Let us consider the case $b=2$ first. There are two well known examples for addition of digit vectors, which will be discussed below.

For $n\in {\ensuremath{\mathbb{N}}}$, $n\ge 2$, let ${\ensuremath{\mathbb{Z}}}/n{\ensuremath{\mathbb{Z}}}$ denote the additive group of residue classes modulo $n$. We identify this cyclic group with the set of integers $\{0,1, \ldots, n-1\}$ equipped with addition modulo $n$.

We identify $\mathcal{A}_2$ with ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$. For ${\ensuremath{\mathbf x}}, {\ensuremath{\mathbf y}}\in \mathcal{A}_2^m$, ${\ensuremath{\mathbf x}}= (x_0, \ldots, x_{m-1})$ and ${\ensuremath{\mathbf y}}= (y_0, \ldots, y_{m-1})$, we define $${\ensuremath{\mathbf x}}+ {\ensuremath{\mathbf y}}= (x_0\oplus y_0, \ldots, x_{m-1}\oplus y_{m-1}),$$ where ‘$\oplus$’ denotes addition on ${\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}}$, $0\oplus 0=1\oplus 1 =0$, and $0\oplus 1=1\oplus 0 =1$. The pair $(\mathcal{A}_2^m, +)$ is an abelian group. In fact, it is isomorphic to the product group $({\ensuremath{\mathbb{Z}}}/2{\ensuremath{\mathbb{Z}}})^m$. We call this binary operation [*addition without carry*]{}, or [*${\ensuremath{\texttt{XOR}}}$-addition*]{} of 