How does openssh actually choose the public exponent 'e' when generating a RSA key?

Question

What I've read:

• https://crypto.stackexchange.com/a/12256
• https://crypto.stackexchange.com/a/20328

Seems in practice, will first choose an e, then choose the 2 big primes p1 and p2.

What I've learned so far:

• e is usually between (2^16, 2^32), e.g 2^16 + 1 = 65537. // TODO .. not sure
• e should not be too large, otherwise the public-key operation is slow.

I've also try to read openssh source code, but can't figure out it yet.

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Questions

1. How openssh actually choose e, when creating a RSA key?
2. If possible, can u point out relevant file/functions in the source code.
3. If a software choose a fixed e (e.g 65537) instead of a random value, will that make the RSA key less secure?

Answer 1

1. $e=65537$.
2. See here and here.
3. No, using fixed $e$ is the industry standard, and $e=65537$ is a particularly popular choice.