I was studying various cryptographic encryption schemes and why modular arithmetic is being used so much while encrypting messages. Why don't we use simple multiplication with a large known prime with the message, as factorizing large numbers takes exponential time complexity, so it would be hard for an intruder to decrypt the message?
So, simply using an encryption scheme that multiplies a large prime on the message to be encrypted is also hard to decrypt for an intruder without knowledge of the prime number multiplied.
Why was there a need felt for modular arithmetic-based encryption schemes?
wanted just a motivation why we should study modular arithmetic...
Factorising large numbers is only difficult if the number has at least two distinct very large prime factors. Multiplication by a large prime could therefore only be used to securely encrypt messages whose plaintexts had at least one very large prime factor. And if a stickybeak were to obtain just two messages encrypted with the same prime number $\ p\ $, he or she would nearly always be able to recover $\ p\ $ by taking the greatest common divisor of the two encrypted messages, since it's unlikely that both plaintexts would be divisible by the same sufficiently large prime to prevent recovery of $\ p\ $ from the gcd of the encrypted texts. Even if this were to happen, it's virtually certain that interception of a third message encrypted with $\ p\ $ would enable the interceptor to recover it.