# Relationship between exponent and modulus in RSA (as RSA properties as listed in X.509)

In one of my assignments, I had the following question (Please read on. Not a homework assignment :) ):

X.509 (1998 version) lists properties that RSA keys must satisfy to be secure. One such requirement is that the public exponent $e > \log_2(n)$ where n is the modulus. Consider the following justifications for this:

1. We require $e > \log_2(n)$ in order to prevent an attack to disclose the plaintext by taking the eth root modulus n.
2. We require $e > \log_2(n)$ in order to prevent an attack to disclose the plaintext by taking the eth integer root.

Which of these two is the correct justification of the above requirement? For the incorrect justification, explain why is it incorrect.

My question here is: Where is the X.509 certificate listed mentioning the above requirement? I tried googling a lot but could not find any such requirement listed in any online available material unfortunately. Also, if any such requirement does exist, what is the justification for the same?

• X.509 is a standard for certificates. The original specification appears to have had a "security considerations" section discussing these requirements. An X.509 certificate is any certificate that follows the X.509 specification. As for the justification, imagine what happens if you encrypt $2$ using $e$ if $e<\log_2(n)$ using textbook RSA. – SEJPM Nov 15 '15 at 17:28

This advice was removed in the 2000 edition of the standard. It was arguably misguided, and at the very least the justification given was very incorrect (even when using RSA for encryption without padding, what's stated in the justification works only for a very small portion of plaintexts, including for $e=3$; it thus does not work at all for random plaintext, or when using RSA with proper padding).