I am a novice programmer and am just finishing up an RSA encryption program that I am writing for practice. Currently I have the program generate a relatively small random value for the public key e. When adding the finishing touches, I realized that there was no point for e to be random. Is this thinking correct?
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Yes, this thinking is correct; there is no requirement that the public exponent $e$ to be random. After all, it doesn't matter whether $e$ can be guessed by an attacker; we'll be including that value in the public key anyways.
Common practice is currently to use the fixed value $65537 =2^{16} +1$ for $e$. Any odd value of $e > 1$ will work; however, smaller values of $e$ will tend to make the system brittle against errors in performing the RSA padding.
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1$\begingroup$ some combinations of $e$ and $n$ don't work. $e$ and φ(n) must be coprime. $\endgroup$ Commented Dec 4, 2012 at 22:23
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1$\begingroup$ You could also add that the reason we use values of the form $2^n + 1$ for $e$ is that they have low Hamming weight, which makes modular exponentiation (and thus, encryption) very efficient. $\endgroup$– ThomasCommented Dec 5, 2012 at 2:52