User:Ejc.cryptography

$a_{r}={\frac {{\sqrt {2\pi }}x_{w}}{L}}[exp(-{\frac {x_{w}^{2}}{2}}(k_{w}+k_{r})^{2})+exp(-{\frac {x_{w}^{2}}{2}}(k_{w}-k_{r})^{2})]$

${\frac {GMm}{r^{2}}}={\frac {mv^{2}}{r}}\Rightarrow v={\sqrt {\frac {GM}{r}}}$

$z_{n}=-{\frac {x\times x_{n}+y\times y_{n}}{z}}$

${\begin{bmatrix}x\\y\\z\\\end{bmatrix}}\cdot {\begin{bmatrix}x_{n}\\y_{n}\\z_{n}\\\end{bmatrix}}=0$

${\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\\\end{bmatrix}}={\frac {v}{\sqrt {x_{n}^{2}+y_{n}^{2}+z_{n}^{2}}}}{\begin{bmatrix}x_{n}\\y_{n}\\z_{n}\\\end{bmatrix}}$

$x\times x_{n}+y\times y_{n}+z\times z_{n}=0$

$y_{1}={g}^{x_{1}}{\pmod {p}}$

$y_{2}={g}^{x_{2}}{\pmod {p}}$

$s={y_{1}}^{x_{2}}{\pmod {p}}$

$s={y_{2}}^{x_{1}}{\pmod {p}}$

$s=(g^{x_{1}})^{x_{2}}{\pmod {p}}$

$s=(g^{x_{2}})^{x_{1}}{\pmod {p}}$

$s=g^{x_{1}x_{2}}{\pmod {p}}$

$p=53\,$

$g=18\,$

$x_{1}=8\,$

$y_{1}=18^{8}{\pmod {53}}=24\,$

$x_{2}=11\,$

$y_{2}=18^{11}{\pmod {53}}=48\,$

$s=24^{11}{\pmod {53}}=15\,$

$s=48^{8}{\pmod {53}}=15\,$

$24=18^{x_{1}}{\pmod {53}}\,$