2009 Number Theory Seminar - 08

I'll discuss some connections between algorithms for finding rational points on curves and some cryptosystems. Akiyama and Goto have proposed a cryptosystem based on rational points on curves over function fields (stated in the equivalent form of sections of fibrations on surfaces). It is easy to construct a curve passing through a few given points, but finding the points, given only the curve, is hard. In this talk I will show how to break their original cryptosystem by using algebraic points instead of rational points and discuss the possibility of changing their original system to create a secure one.