COMP 2711 Ex 16

1

Encrypt $M=100$, where $p=11$, $q=29$, $e=3$

By repeated squaring

$$3=2+1$$ $$\begin{array}{rl}& 100^1\mathrm{mod}(11\cdot 29)=100\\ & 100^2\mathrm{mod}(11\cdot 29)=111\end{array}$$

The encrypted message $y$ is

$$\begin{array}{rl}y& =M^e\mathrm{mod}(p\cdot q)\\ & =100^3\mathrm{mod}(11\cdot 29)\\ & =(100^1\cdot 100^2)\mathrm{mod}(11\cdot 29)\\ & =(100^1\mathrm{mod}(11\cdot 29)\cdot 100^2\mathrm{mod}(11\cdot 29))\mathrm{mod}(11\cdot 29)\\ & =(100\cdot 111)\mathrm{mod}(11\cdot 29)\\ & =254\end{array}$$