Much of the research in number theory, like mathematics as a whole, has been inspired by hard problems which are easy to state. A famous example is “Fermat’s Last Theorem”. Starting in the 1970’s number theoretic problems have been suggested as the basis for cryptosystems, such as RSA and Diffie-Hellman. In 1985 Koblitz and Miller independently suggested that the discrete logarithm problem on elliptic curves might be more secure than the “conventional” discrete logarithm on multiplicative groups of finite fields. Since then it has inspired a great deal of research in number theory and geometry in an attempt to understand its security.
I’ll give a brief historical tour concerning the elliptic curve discrete logarithm problem, and the closely connected Weil Pairing algorithm.