# Why does RSA give better security on longer messages?

I am trying to understand the notion of RSA security.

Choosing a public exponent where $e = 3$ facilitates the calculations, considering that it is secure if the plaintext or message is long.

If the message is short, it affects security but why?

Does that relates to the $2^n$ possibilities, if the message is short the probability of knowing the message is high?

• Typical RSA messages consist of a random symmetric key, so this concept doesn't really apply in practice. We also use special padding modes instead of plain RSA, which prevents many attacks (AFAIK including yours). – CodesInChaos Mar 17 '12 at 19:08

If you have a $n$-bit modulus, and use a message $x$ shorter than $n/3$ bits, the modular part of modular exponentiation doesn't come to play when calculating the ciphertext as $c = x^3 \bmod n$. The effect is that you can simply calculate the (integer) cubic root of the ciphertext $c$ to extract the plaintext, and don't have to deal with the key at all.
I think similar attacks are possible if the message is only slightly longer than $n/3$ bits.