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I understand that if $e\in{\mathbf{Z^*_{\phi(N)}}}$ then $\gcd(e,\phi(N))=1$ and if $e\not\in{\mathbf{Z^*_{\phi(N)}}}$ than $\gcd(e,\phi(N))\neq{}1$.

But I couldn't figure out why this implies bijection of $f(x)=x^e$.

I also tried to see examples but that didn't help me to explain the phenomenon.

A. Maman
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1 Answers1

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The correct form is: The integer $a \mod Z_n$ has multiplicative inverse iff $gcd(n,a)=1$

Here, you are working on exponents, so you must consider the modulo as $\phi(n)$ not $n$.

Therefore, here you can find the inverse of $e$ iff $gcd(e, \phi(e))=1$. This guarantees that you be able to find the inverse of $e$ using extended Euclidean algorithm.

m123
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