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Are there any other commonly used exponents? Why are they selected?

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We want a number co-prime with p-1 and q-1. We also want modular exponentiation tp be efficient. For this purpose we want it to be a small number with few set bits. To meet the co-prime requirement we can pick a prime number and verify and we are reasonably likely to succeed.

For all these together we are looking at small primes of the form: $2^n+1$ known as Fermat primes. These numbers leads to efficient public key operations.

Using 3 is an obvious choice but some don't like it, partly due to real attacks when not padding and partly irrational fear. slightly larger such primes became popular.

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Here is the beginning of that answer from info security stackexchange for the reader’s convenience:

There is no known weakness for any short or long public exponent for RSA, as long as the public exponent is "correct" (i.e. relatively prime to p-1 for all primes p which divide the modulus).

If you use a small exponent and you do not use any padding for encryption and you encrypt the exact same message with several distinct public keys, then your message is at risk: if $e = 3,$ and you encrypt message $m$ with public keys $n_1,n_2,n_3$ then you have $c_i= m^3 \pmod{n_i}$ and use the chinese remainder theorem to recover the message by a non modular cube root extraction.

The weakness, here, is not the small exponent; rather, it is the use of an improper padding (namely, no padding at all) for encryption.

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