Let $E_{{?}_{?}}$ be the elliptic curve $y^2=x^3+41x+83$ defined over $Z_{179}.$ It can be shown that #$E=$

$182$ and $P=(9,110)$ is an element of order $182$ in E$.$ The Simple Elliptic Curve$-$

Based Cryptosystem defined on $E$ has $Z_1{?}_{79}$ as a plaintext space. Suppose the private

key is $m=16.$

$($a$)$

$($b$)$ Decrypt the following string of ciphertext:

$((91,0),148),((106,0),153),((78,0),111),((137,0),21),((29,1),81),((151,1),5),$

$((115,1),149)$

$($c$)$ Assuming that each plaintext represents one alphabetic character, convert

the plaintext into an English word. $($Note: here we will use the correspondence $A$

harr$1,$dotsZharr$26,$ because $0$ is not allowed in the plaintext$).$

Note: The Simple Elliptic Curve$-$Based Cryptosystem is defined in the

ImprovementsToTheBasicModel.pdf file.