On the circuit complexity of the standard and the Karatsuba methods of multiplying integers

The goal of the present paper is to obtain accurate estimates for the complexity of two practical multiplication methods: standard (school) and Karatsuba [1]. Here, complexity is the minimal possible number of gates in a logic circuit implementing the required function over the basis {AND,OR,XOR,NAND,NOR,XNOR}. One can find upper estimates for the said methods e.g. in [2]. The standard method has complexity M(n) ≤ 6n − 8n. In the case n = 2, the complexity K(n) of the Karatsuba method can be deduced from the recursion K(2n) ≤ 3K(n) + 49n− 8 as K(2) ≤ 26 9 · 3 − 49 · 2 + 4. We intend to show that the above estimates may be improved with the help of the result [3] stating that a sum of n bits may be computed via 4.5n operations instead of 5n as in the naive approach. The resulting bounds are M(n) ≤ 5.5n − 6.5n− 1 + (n mod 2) and