∴ T (β)∈ O (β 2)
The time complexity can be reduced from the following observation:
X1 Y2+ X2 Y1= (X1 + X2).(Y1 + Y2) - X1 Y1- X2Y2
Thus instead of 4 we actually require 3 multiplications each of two
β/2- bit long numbers.
Thus T(β)=3T(β/2)+C.β
⇒ T(β)= O(βlog23)
Fastest multiplication known to date for β bit numbers takes O (βlogβ loglogβ) time.
Division
Simple division of two β bit numbers will take O(β2) time through simple approach. In the numbers theoretic problems applied to cryptography over the group Zn*.we will consider division by a number to be equivalent to multiplication by its inverse. Thus time complexity of division is same as that of multiplication.
Reference:
1. Introduction to Algorithms , Second Edition, T. H. Cormen, C. E. Leiserson, R. Rivest and C. Stein, Prentice Hall India .
2. Computers and Intractability: A Guide to the Theory of NP-Completeness , M. Garey and D. S. Johnson, W. H. Freeman .
3. Introduction to AutomataTheory, Languages and Computation , J. E. Hopcroft and J. D. Ullman, Addison-Wesley