CONSTRUCTION $12\mathrm{.}16$

Let $9$ be as in the text. Define a public$-$key encryption scheme as follows:

Gen:

on input $1"$ run G$(1")$ to obtain $($G$,$q$,$g$).$ Then choose a uniform a $$ Zq and compute h :$=$ g$*.$ The public key is $($G$,$ q$,$ g$,$ h$)$ and the private key is $($G$,$ q$,$ g$,$ x$).$ The message space is G$.$

Enc: on input a public key ph $=($G$,$ g$,$ g$,$ h$)$ and a message m $$ G$,$ choose a uniform y $$ Zq and output the ciphertext

$$g$$h$$m$.$

$$ Dec: on input a private key sk $=($G$,$ q$,$g$,$x$)$ and a ciphertext

$($C$1,$ C$2),$ output

m:$=$ c$2/$C$].$

The El Gamal encryption scheme.

To see that decryption succeeds, let $($C$1,$ C$2)=($g$",$ hy$.$m$)$ with h$=$ g$*.$ Then

C$2$

m $=$

hY$.$ M

$(94)2$

$($g$2)$y$.$m

gxy

$=$

$=$ m$.$

Question

Consider the El Gamal Public Key Encryption scheme construction $12\mathrm{.}16$ in slide $13$ of module $12\mathrm{.}3.$ Suppose the same message m is encrypted twice using this scheme. What is the probability that the ciphertext will be the same in both cases? Explain.

$[$Let y$1$ and y$2$ denote the choice of y in the two encryptions of m$.$ Observe that since g is a generator gy$1$ can equal gy$2$ if and only if y$1=$y$]$

Plz do not use chat gpt

give me a little explanation of $10-12$ line not very long