  In this paper we generalize the quantum algorithm for computing short
discrete logarithms previously introduced by Eker{\aa} so as to allow for
various tradeoffs between the number of times that the algorithm need be
executed on the one hand, and the complexity of the algorithm and the
requirements it imposes on the quantum computer on the other hand.
  Furthermore, we describe applications of algorithms for computing short
discrete logarithms. In particular, we show how other important problems such
as those of factoring RSA integers and of finding the order of groups under
side information may be recast as short discrete logarithm problems. This
immediately gives rise to an algorithm for factoring RSA integers that is less
complex than Shor's general factoring algorithm in the sense that it imposes
smaller requirements on the quantum computer.
  In both our algorithm and Shor's algorithm, the main hurdle is to compute a
modular exponentiation in superposition. When factoring an n bit integer, the
exponent is of length 2n bits in Shor's algorithm, compared to slightly more
than n/2 bits in our algorithm.
