Applications of algebraic curves to cryptography

In this thesis we study two different applications of algebraic curves to cryptography.; First, we present an infinite family of hyperelliptic curves of genus two over a finite field of characteristic two and show that it satisfies all the conditions that are needed for the vector decomposition problem (VDP). We construct a signature scheme based on VDP and we generalize the ElGama1 signature scheme for cyclic groups to a signature scheme for n-dimensional vector spaces.; Secondly, we use algebraic functions with two poles to obtain efficient secret sharing schemes. We present a method to find the lower bounds for the minimum distance of geometric codes. We apply this to the two-point codes on a Hermitian function field. The lower bounds turn out to be sharp and they meet the formulas by Homma and Kim for the actual minimum distance of the Hermitian two-point codes with a shorter proof and fewer cases for the formulas. Moreover, our approach gives an efficient error correcting algorithm to decode up to half the actual minimum distance.