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}.)