Multiplying two complex numbers using only three multiplications of real numbers

I have problem given below.

Show how two complex numbers $(a+ib)$ and $(c+id)$ may be multiplied using only three multiplications of real numbers, where $i=\sqrt{-1}$. You may use any number of additions and subtractions.

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Please help me. How to solve this?

You are interested in two numbers : $\alpha_1 = ac-bd \text{ and } \alpha_2 = ad+bc$. You can compute 3 products, viz. $P_1 = ac, P_2 = bd, \text{ and } P_3 = (a+b)(c+d)$. Then $\alpha_1 = P_1 - P_2, \text{ and } \alpha_2 = P_3 - P_2 - P_1$.

This process is often called Karatsuba multiplication, and is used in algorithm design quite frequently.

Prahlad Vaidyanathan has already answered on how performing the product between two complex numbers with only three real multiplications. Now the question is: can we perform the product between two complex numbers with less than three real multiplications?

The answer is NO and is provided by Winograd's theorem in

S. Winograd, "On the number of multiplications required to compute certain functions", Commun. Pure Appl. Math. 23 (1970), 165-179.

The minimum number of multiplications required in the computation of the product between two complex numbers is three.