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I am implementing a protocol for authentication where parameters need to be selected with $gcd(\mu,\phi)=1$ where $\phi=(p−1)(q−1)$. What is the best way to select the parameters while implementing

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In the context, do you know $\phi$? If yes, one way could be to select (odd) $\mu$ at random until $\gcd(\mu,\phi)=1$ is satisfied. Or if that fits, just $\mu=1$. –  fgrieu Dec 28 '13 at 16:45
Random is the accepted method? Whether RSA uses random method? –  user5507 Dec 28 '13 at 16:49
When there is nothing else specified, random can be an accepted method. In particular, it is fine to choose an RSA exponent at random that is coprime with $\phi=(p−1)(q−1)$. If that needs to be done for unknown $\phi$, one option is to select primes $p$ and $q$ of same bit size such that $(p-1)/2$ and $(q-1)/2$ are primes, making an overwhelming majority of odd $\mu$ with $3\le\mu<N-3\sqrt N$ suitable. –  fgrieu Dec 28 '13 at 17:00

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