please help me with this exercise as soon as possible I have only $2$ days please I am waiting for your response.

INSTRUCTIONS: You must write good mathematics; clear, meaningful, consistent. Show each step of your work to receive full credit. This time, you are not allowed to use any programming languages. However you can use scientific calculators. Upload a SINGLE PDF file. It must be readable. Please name your PDF file "studentID$\_$name$\_$surname" e$.$g$."210210210\_$faith$\_$temiz". If you use a text editor, you shouldn't write expressions like x $2$ instead of $x^2$ or etc. No late homework is accepted.

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$(25$ points$)$ Find your private keys $p$ and $q$ in the table below. Choose an appropriate encryption exponent $e$ and send me an RSA$-$encrypted message of length at least $5$ via converting letters to ASCII codes $(0-255).$ Also find the decryption exponent $d$ for me$.$

$(25$ points$)$ Write an equation for an elliptic curve over $F_p.$ Find two points on the curve which are not $($additive$)$ inverse of each other. Show that the points are indeed on the curve. Find the sum of these points.

Hint:

https:$//$graui$.$de$/$code$/$elliptic$2/$

$(25$ points$)$ Let $p$ be a prime number of length $k$ bits. Let $H(x)=x^2(modp)$ be a hash function which maps any message to a $k-$bit hash value.

$($a$)$ Is this function pre$-$image resistant? Why?

$($b$)$ Is this function second pre$-$image resistant? Why?

$($c$)$ Is this function collision resistant? Why?

$(25$ points$)$ Find a primitive root in $F_4,$ which is not $6,$ say $a.$ Show that $a$ is indeed a primitive root. You are given the discrete logarithm problem

$a^x-=6,(modq)$

Solve the discrete logarithm problem by using the babystep$-$giantstep algorithm.

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