If Alice were to choose n to be a prime, instead of the product of two primes, what exactly would go wrong in the RSA cryptosystem? How would Bob decrypt her message?
Asked
Active
Viewed 7,556 times
2 Answers
25
How would Bob decrypt her message?
That'd be easy, all we would need to assume that Bob has the public key. If $n$ is a prime, he is able to compute $d = e^{-1} \bmod{\lambda(n)}$, because $\lambda(n) = n - 1$. He can then raise the ciphertext to the $d$th power modulo $n$, and that's the padded plaintext.
Since anyone with the public key can do this, RSA based on a prime modulus is insecure.
poncho
- 154,064
- 12
- 239
- 382
0
That is impossible; it cannot happen. $n$ is not just chosen, instead we choose two primes $p$ and $q$ and set $n=p\cdot q$. As you can see, by definition $n$ is composite.
mikeazo
- 39,117
- 9
- 118
- 183