Public key encryption is the basis for secure communication over the internet and thus the basis for our internet economy. (If my students buy a song using iTunes then they are using public key encryption. See the Wikipedia article on TSL and SSL protocols.)
A fundamental form of public key encryption (widely used now and the also the first example of public key encryption) is the RSA algorithm. It is based on Euler's theorem that $a^{\phi(n)}\equiv 1 \mod n$ for all $a$ relatively prime to $n$.
Without Euler's theorem we would not have RSA; without RSA we would not have IDEA, SSL, TSL and our internet economy.
