One common way of constructing public-key cryptographic systems is to utilize the problem of finding a discrete logarithm in some abelian group. Groups that have been utilized for this purpose include $\F_{q}^{*}$ in the Massey-Omura and ElGamal cryptosystems and the group of points on an elliptic curve in elliptic curve cryptosystems. (See~\cite{Koblitz} for background, for example.) Another possibility which has been proposed by Buchmann and Williams (see, e.g., \cite{BW90}) is to use the group of ideal classes in a number field. In order to make sure that the discrete logarithm problem is computationally hard, one needs to know something about the structure of the group involved, e.g. that it is divisible by a large prime. One situation in which it is easy to show this is in the case of the class group of $\Q(\zeta_{p})$; according to a well-known theorem of Kummer, $p$ divides the order of the class group of $\Q(\zeta_{p})$ if and only if $p$ divides the numerator of a Bernoulli number $B_{m}$ for some even $m$ such that $2 \leq m \leq p-3$. Such primes are called irregular; the others are called regular. We extend the concept of regular and irregular primes to the setting of arbitrary totally real number fields $k_{0}$, using the values of the zeta function $\zeta_{k_{0}}$ at negative integers as our ``higher Bernoulli numbers''. In this setting the author proved in his thesis~(\cite{dissert}), building on work of Greenberg and Kudo, that under a certain technical condition Kummer's criterion can be extended to give information about whether $p$ divides the class group of $k_{0}(\zeta_{p})$. Thus we are interested in the feasibility of finding these analogues of irregular primes, since their associated class groups may be especially suitable for cryptography. In the case where $k_{0}$ is a real quadratic field, Siegel, in~\cite{Siegel68}, presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (See also~\cite{Zagier} and~\cite{Cohen76} for a discussion of these formulas. The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We will briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of $k_{0}$-irregularity of a prime number. (The author has already discussed one of these algorithms in~\cite{Holden98}.)