I am somewhat familiar with the RSA algorithm, and I know that the security comes from the fact that the totient $\phi(m)$ is difficult to compute when $m$ is large and of unknown factorisation. I have more or less assumed that something like this lies at the base for most modern public-key encryption, but how widespread in the real world are public key encryption schemes (including RSA) for which "finding $\phi(m)$ from $m$" is the main bottleneck for a brute-force attack?
I'm not asking how many such schemes there are out there that people have invented (although I actually only know of RSA, so if someone were to, say, drop the names of one or three others, that would be cool), but rather, in a sense, what qualification can I use when I say "_____ of modern public key encryption hinges on the fact that $\phi(m)$ is hard to compute" to laypeople, for instance in a high school math class where we take a day off from the usual curriculum. Can I say "Almost all of modern encryption", or "Most" or do I have to moderate it to "A lot", or even just "Some"?