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1

I'll first assume $\phi$ is known, which would be the situation in RSA at key selection. A common method could be to select (odd) $\mu$ at random in some appropriate interval until $\gcd(\mu,\phi)=1$ is satisfied. In the context of RSA, it is fine (if we ignore performance an interoperability issues) to choose an RSA exponent at random in $[3\dots\phi-3]$ ...

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many thanks to everyone. When I was reading originally the FIPS 197 document I made one big mistake: I assumed that the appendix C had only the cipher portion, similar to the appendix B, and missed the uncipher portions. Answering my own question, yes, translation of the variable temp to the one I proposed initially was correct. However, my error come ...

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You're right in that there's little chance you can break the logarithm in a well-chosen 512 bit group (using a home computer, in reasonable time — as pointed out by SEJPM, it is possible investing some time and a good amount of money). However, in your case, the parameters are bad: The order of $(\mathbb Z/p\mathbb Z)^\ast$, that is $p-1$, is a smooth ...

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