This work provides refined polynomial upper bounds for the condition number of the transformation between RLWE/PLWE for cyclotomic number fields with up to 6 primes dividing the conductor. We also provide exact expressions of the condition number for any cyclotomic field, but under what we call the twisted power basis. Finally, from a more practical perspective, we discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, for which we also study the RLWE/PLWE equivalence of a concrete non-cyclotomic family of number fields. We think this family could be of particular interest due to its arithmetic efficiency properties.

翻译：本文提供了循环数域RLWE/PLWE之间变换的条件数的精细多项式上界，其中的循环数域有至多6个质数作为导数。我们还对于任意一个循环数域提供了在扭动的幂基础下的条件数的精确表达式。最后，从实际应用的角度，我们讨论了循环数域在环同态加密中快速多项式算术的优势和局限性。为此，我们还研究了一个非循环域的RLWE/PLWE等价的具体系列数域的算术效率性质。我们认为这个系列数域由于其算术效率优势可能具有特别的价值。