Why is RSA Encryption/Decryption secure?

Let’s analyze security from different perspectives:

What Does an Outsider See?

An outsider observing an RSA-encrypted message has access to:

• The public key: $(e,n)$

• The ciphertext: $c = m^e\ mod\ n$

To decrypt the message and retrieve m, the outsider would need the private key $d$. However:

Why can't the outsider calculate $d$?

To find $d$, the outsider must compute the modular inverse of $e\ mod\ ϕ(n)$, where $ϕ(n)=(p−1)(q−1)$. This requires knowledge of $ϕ(n)$, which can only be calculated if it can factorize $n$ into $pq$

However, due to Integer Factorization Problem, it is very hard to factorize $n$! hence, cannot calculate private key $d$

Can Alice Learn Bob’s Private Value?

Suppose Alice is an authorized sender and knows:

• Bob’s public key: (e,n).

• Her own plaintext message: m.

• The ciphertext she sent: $c=m^e\ mod\ n$

Even with this information:

Similarly, Alice cannot derive Bob’s private key $d$, as she would need to factorize $n$ to compute $ϕ(n)$, which is very hard due to Integer Factorization Problem.