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Does deriving the public key from an RSA private key always yield the same result? And if so, does this generally apply to all asymmetric cryptosystems?

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Does deriving the public key from an RSA private key always yield the same result?

Well, if $e$ is the correct public exponent, then any value of the form $e + k \cdot \text{lcm}(p-1, q-1)$ (for any integer $k$) is also a correct public exponent (in the sense that it would function equivalently when used with the same modulus).

Of course, we usually select $e$ to be small; the alternative public exponents are huge (and hence much slower). In addition, publishing two different public exponents of that form would allow someone to efficiently factor the modulus, and hence taking advantage of it might not be that great of an idea...

And if so, does this generally apply to all asymmetric cryptosystems?

Not in general. Another example where it doesn't apply would be NTRU; the private key is a polynomial $F$ (that meets certain properties); the public key is another polynomial $pF^{-1}G$ (where $G$ is a sparse polynomial); we can select a different $G$ and obtain a different public key that works equivalently.

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