Complex Multiplication

Two Complex Numbers $x=a+ib$ and $y=c+id$ are multiplied as follows:

$$xy = (a+ib)(c+id)= ac+ibc+iad-bd$$

$$= (ac-bd)+i(ad+bc).$$

However, the multiplication can be carried out using only three Real multiplications, $ac$, $bd$, and $(a+b)(c+d)$ as

$$\begin{eqnarray*} \Re[(a+ib)(c+id)] &=& ac-bd\\ \Im[(a+ib)(c+id)] &=& (a+b)(c+d)-ac-bd. \end{eqnarray*}$$

Complex multiplication has a special meaning for Elliptic Curves.

See also Complex Number, Elliptic Curve, Imaginary Part, Multiplication, Real Part

References

Cox, D. A. Primes of the Form $x^2 + ny^2$: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.