# RSA - Compare two values for encryption exponent, both are relatively prime and $e < λ(n)$

So we have $q = 311$, $p = 163$, $e = 101$, I found $d = 39281$ using Euclidean Algorithm and checked by encrypting and decrypting a message.

Next it is asking if $e = 101$ was a good choice or would the value $e = 9131$ be better.

What I know:

• $101$ = prime

• $39281$ = not prime ( $23 x\times 397$)

Both options for $e$ are relatively prime to $(p - 1) \cdot (q - 1)$ and both options for $e$ satisfy $𝑒 < λ(n)$.

Are there any other things I need to consider for $e$ to answer the question? I have looked around the forum and web but I'm stuck right now.

Hints preferred not a full answer!

• @fgrieu As hints are preferred over a full answer you might as well turn the hints into an answer :P Welcome to crypto, Mr.Mushy99... May 9 '18 at 15:05

Hint: compute $d$ for the alternate value of $e$ considered. Isn't something rather remarkable? What is the probability that it occurs by chance?
While that is totally immaterial given that $p$ and $q$ are billions-n'-billions times too small to prevent factorization (gross understatement), what if we scaled things? Why would it matter (caution: there are several levels of answer to that later one)?