Arithmetic aspects of elliptic curve cryptography

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The study of encryption schemes based on elliptic curves has attracted considerable interestfrom the research community ever since their discovery 20 years ago by Miller and(independently) Koblitz. Elliptic curve systems are unique among all public key systems inthat they have successfully maintained the highest known security levels for the longestperiod of time among all such systems available today. My research aims to improve ourunderstanding of elliptic curve cryptosystems by investigating the mathematical foundationsof elliptic curve arithmetic.  My methods combine both rigorous mathematics and cryptologyin order to achieve results that are beyond the reach of either approach in isolation. Inearlier work, I show for the first time that the security of elliptic curve cryptosystemsdepends in almost all cases purely on the key size of the elliptic curve, assuming thegeneralized Riemann hypothesis. I intend to build upon this work and extend these results tohyperelliptic curve cryptosystems, while at the same time exploring new constructions ofexpander graphs and public key cryptosystem designs arising from the techniques that I used.