Question

What sequence of pseudorandom numbers is generated using the

pure multiplicative generator $\mathbf{xn}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{3}\mathbf{xn}\mathbf{mod}\mathbf{11}$with seed $\mathit{x}\mathbf{0}\mathbf{=}\mathbf{2}$?

Short answer

sequence of pseudorandom numbers is generated using the pure multiplicative generator is

$\mathbf{6}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{10}\mathbf{,}\mathbf{8}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{10}\mathbf{,}\mathbf{8}\mathbf{,}\mathbf{2}$

Step-by-step solution

Step 1

${\mathit{x}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}\left(3x_n\right)\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{,}{\mathit{x}}_{\mathbf{0}}\mathbf{=}\mathbf{2}$

The given recursive function and the initial value

Step 2

$$\begin{gathered} {\mathit{x}}_{\mathbf{0}}\mathbf{=}\mathbf{2} \\ \mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{6} \\ \mathit{n}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{\left(}\mathbf{18}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{7} \\ \mathit{n}\mathbf{=}\mathbf{2}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{=}\mathbf{\left(}\mathbf{21}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{10} \\ \mathbf{n}\mathbf{=}\mathbf{3}\mathbf{,}{\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{\left(}\mathbf{30}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{8} \\ \mathit{n}\mathbf{=}\mathbf{4}\mathbf{,}{\mathit{x}}_{\mathbf{5}}\mathbf{=}\mathbf{\left(}\mathbf{24}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{2} \\ \mathit{n}\mathbf{=}\mathbf{5}\mathbf{,}{\mathit{x}}_{\mathbf{6}}\mathbf{=}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{6} \\ \mathbf{n}\mathbf{=}\mathbf{6}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{=}\mathbf{\left(}\mathbf{18}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{7} \\ \mathbf{n}\mathbf{=}\mathbf{7}\mathbf{,}{\mathbf{x}}_{\mathbf{8}}\mathbf{=}\mathbf{\left(}\mathbf{21}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{10} \\ \mathit{n}\mathbf{=}\mathbf{8}\mathbf{,}{\mathit{x}}_{\mathbf{9}}\mathbf{=}\mathbf{\left(}\mathbf{30}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{8} \\ \mathit{n}\mathbf{=}\mathbf{9}\mathbf{,}{\mathit{x}}_{\mathbf{10}}\mathbf{=}\mathbf{\left(}\mathbf{24}\mathbf{\right)}\mathbf{mod}\mathbf{\left(}\mathbf{11}\mathbf{\right)}\mathbf{=}\mathbf{2} \end{gathered}$$

It is easily observed that this pure multiplicative generator is inherently pseudo-random as it only produces the numbers$\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{10}$and$\mathbf{8}$.