  Whilst lattice-based cryptosystems are believed to be resistant to quantum
attack, they are often forced to pay for that security with inefficiencies in
implementation. This problem is overcome by ring- and module-based schemes such
as Ring-LWE or Module-LWE, whose keysize can be reduced by exploiting its
algebraic structure, allowing for faster computations. Many rings may be chosen
to define such cryptoschemes, but cyclotomic rings, due to their cyclic nature
allowing for easy multiplication, are the community standard. However, there is
still much uncertainty as to whether this structure may be exploited to an
adversary's benefit. In this paper, we show that the decomposition group of a
cyclotomic ring of arbitrary conductor can be utilised to significantly
decrease the dimension of the ideal (or module) lattice required to solve a
given instance of SVP. Moreover, we show that there exist a large number of
rational primes for which, if the prime ideal factors of an ideal lie over
primes of this form, give rise to an "easy" instance of SVP. It is important to
note that the work on ideal SVP does not break Ring-LWE, since its security
reduction is from worst case ideal SVP to average case Ring-LWE, and is one
way.
