Task $2$: Cryptanalysis of RSA with Shared Modulus

words$)$

Suppose two users Alice and Bob share the same RSA modulus $N$ and their public

exponents $e_A$ and $e_B$ are relatively prime $($i$.$e$.,s**e_A+t**e_B=1$ for some

integers $s$ and $t).$ Charles wants to send the message $m$ to both Alice and Bob, so

he encrypts to produce ciphertexts $C_A-=m^{e_A}$modN and $C_B-=m^{e_B}$modN.

Discuss and show if an attacker Eve can deduce $m$ by employing "Ciphertext

only attack" after intercepting $C_A$ and $C_B.$ Provide a clear explanation of the attack

using mathematical notations $($hint: You may utilize extended Euclidean

algorithm in the equation $s**e_A+t**e_B=1).$ Furthermore, research and

summarise real$-$life incidents or theoretical attacks where sharing an RSA

modulus $N$ or similar cryptographic setups led to security breaches. Ensure to

include references to credible sources in your discussion. Your explanation

should be detailed and adhere to an academic style.