The key operation in encryption and decryption process is the exponentstion. Given an integer x it is easy to write a linear algo O((n)) to find x^n for any integer n. Describe the maths of a fast exponentation algo that can compute x^n in o(log2n) time. Justify

Following are the solution to this question:

Explanation:

could be approximated by getting a small n loop as well as multiplying x for each incarnation in a linear time. It's very simple. Its key concept is to do it in log2n time.  

Therefore, could be determined and divided by itself instead of calculation. It must be done frequently to ensure which half the research is done out with each stage.

Runtime:

Its repetition relationship of the above function is:

This can be resolved by master theorem, so it's obvious. The running time of this repeating ration, by master theorem, is .

Pseudocode:

int exponent( int x_1, int y_1 )//defining a method exponent

{

if(y_1==1) //use if to check n value equal to 1  

  {return x_1;} //return x value

  Int t= exponent(x_1,y_1/2);//call method    

return t*t;//calculate  square

}

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