We discuss a number of applications of the theory of function fields to cryptography. By considering the function fields of irreducible hyperelliptic curves we can investigate the security of hyperelliptic curve cryptosystems with the help of number-theoretic ideas. One application is the computation of regulators and class numbers of hyperelliptic function fields with the help of truncated Euler products. Hereby, we provide sharp estimates for the divisor class number of a hyperelliptic function field, i.e. the cardinality of the Jacobian of the corresponding hyperelliptic curve. These estimates can be used to develop an effective method of computing the divisor class number. Furthermore, we provide heuristics on the distribution of the divisor class number within the bounds on the divisor class number. These heuristics suggest that, although the bounds are sharp, the approximation is in general far better. We also explain these heuristics based on recent results of Katz and Sarnak. Finally, we show how these methods can be extended to any algebraic function field given a way to compute the group operation in the corresponding Jacobian.