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Example for exponentiation:

Consider the value of b is 15

The repeated squaring algorithm calculates the value ${\mathrm{b}}^{15}$ by following method:

${\mathrm{a}}^{15}=\mathrm{a}*{\mathrm{a}}^2*{\mathrm{a}}^4*{\mathrm{a}}^8$

In the above calculation,

鈥�${\mathrm{a}}^2$鈥� is having one multiplication (i.e.${\mathrm{a}}^2=\mathrm{a}脳\mathrm{a}$ ).

鈥�${\mathrm{a}}^4$鈥� is having one multiplication (i.e.${\mathrm{a}}^4={\mathrm{a}}^2脳{\mathrm{a}}^2$).

鈥� ${\mathrm{a}}^8$鈥� is having one multiplication (i.e. ${\mathrm{a}}^8={\mathrm{a}}^4脳{\mathrm{a}}^4$).

The expression 鈥�$\mathrm{a}脳{\mathrm{a}}^2脳{\mathrm{a}}^4脳{\mathrm{a}}^8$鈥� has three multiplications.

Therefore, the above method takes totally 6 multiplications.

Other method for exponentiation:

Consider the value 鈥渂 鈥� is 鈥�15 鈥�.

Split the value 鈥� 15 鈥� into 鈥� 3 6 , and 12 鈥�.

Initially, find 鈥�${\mathrm{a}}^3=\mathrm{a}*\mathrm{a}*\mathrm{a}$ 鈥�.

This step contains two multiplications process.

Find 鈥�${\mathrm{a}}^6={\mathrm{a}}^3*{\mathrm{a}}^3$ 鈥�.

This step contains one multiplication process.

And find 鈥�${\mathrm{a}}^{12}={\mathrm{a}}^6*{\mathrm{a}}^6$ 鈥�.

This step contains one multiplication process.

Then finally, 鈥�${\mathrm{a}}^{15}$鈥� can be calculated by the following:

${\mathrm{a}}^{15}={\mathrm{a}}^{12}*{\mathrm{a}}^3$

The above step contains one multiplication process.

Thus, the above method takes a totally5 multiplications.

Therefore, the new method is performed by using fewer multiplications than the given algorithm.