Math @ Duke

Publications [#9681] of Joshua B. Holden
Papers Accepted
 Joshua Holden, Comparison of algorithms to calculate quadratic irregularity of prime numbers,
Mathematics of Computation
, accepted 2000 [math.NT/0010286], [available here]
(last updated on 2000/11/09)
Abstract: In previous work, the author has extended 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''.
We are
interested in the feasibility of finding these analogues of
irregular
primes, both as pure theory and also because they are
associated with class
groups which may be especially suitable
for cryptography.
In the case where $k_{0}$ is a real quadratic field, Siegel
presented
two formulas for calculating these zetavalues: one using
entirely
elementary methods and one which is derived from the theory
of modular
forms. (The author would like to thank Henri Cohen for
suggesting an
analysis of the second formula.) We 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
(more generally, ``quadratic irregularity'') of a prime
number.


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