Complex Multiplication

Two complex numbers  and  are multiplied as follows:

			(1)
			(2)
			(3)

In component form,

(4)

(Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar  is then given by

(5)

Surprisingly, complex multiplication can be carried out using only three real multiplications, , , and  as

			(6)
			(7)

Complex multiplication has a special meaning for elliptic curves.

REFERENCES:

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.

Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 1, 1999.