Show me the steps to solve

Plain RSA Encryption $[26$ Marks$].$

$($a$)(8$ marks$)$ In the Plain RSA encryption scheme, the two primes were chosen to be p$=23$ and q$=17,$ and

the public parameter e $=3$ was chosen. The sender wants to send the message "HELLO" character by

character. Since $$H$$ is $72$ in ASCII, the sender first wants to send the message m $=72.$ Provide a descrip$-$

tion of the $($Gen$,$ Enc, Dec$)$ algorithms in the plain RSA encryption scheme for the above parameters.

$($b$)$ In the Plain RSA encryption scheme, the Gen algorithm runs GenRSA$(1$n$)$ to obtain N$,$ e$,$ d where N denotes

the modulus that is a product of two large primes p and q$,$ the public exponent e satisfies gcd $($e$,$ $($N$))=1$

and d denotes the modular inverse of e with respect to Euler$$s totient function $($N$).$ The public key is

$$N$,$ e$$and the private key is d$.$

i$.(5$ marks$)$ Suppose the adversary Eve gets to know the Euler totient $($N$).$ Describe how Eve can use

this information to factor N $($to derive the primes p and q$).$ Illustrate this for the modulus N$=18923$

and $($N$)=18648.$

ii$.(5$ marks$)$ Suppose the adversary Eve gets to know the private key d$.$ Describe how Eve can use this

information to factor N $($to derive the primes p and q$).$

$($c$)$ Alice has decided to use the Plain RSA scheme for encryption and has generated two large primes p and q

and computed N$=$ p$$q$.$ She has also chosen a public encryption key eA $=3$ satisfying gcd$($eA$,$ $($N$))=1$

and computed the corresponding private key dA$,$ where gcd denotes the greatest common divisor.

Her friend Bridget hears about this, and also wants to try the RSA scheme. So Alice assists Bridget by

sharing the modulus N and choosing a new public encryption key eB $=5$ satisfying gcd$($eB$,$ $($N$))=1,$

and computing the corresponding private key dB$.$ Alice gives Bridget the public key $($N$,$ eB$)$ and the

private key dB$.$