I remember my professor mentioning that RSA encryption fails when $e$ is odd, but cannot seem to figure out why it is so, and can't find a proof in a textbook/online. I tried verifying it by proof-by-contradiction by considering the even case, i.e. $a^{2t}\pmod n$ (where n=pq), but don't see where the contradiction would be coming from. Do I need to bring in Euler's totient $\varphi(n)=\varphi(pq)=\varphi(p-1)(q-1)$ somehow?
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3In brief, you need (or at least, really want) $\gcd(e, \varphi(n))=1$ and $\varphi(n)$ is always even (well, for $n>2$ anyway). See this question – lulu Oct 22 '23 at 13:12