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：本文针对导子至多被6个素数整除的分圆数域，给出了RLWE/PLWE变换条件数的精细多项式上界。同时，在所谓的扭曲幂基下，我们推导了任意分圆域条件数的精确表达式。最后，从实际应用角度出发，我们讨论了分圆数域在实现同态加密快速多项式算术时的优势与局限性，并为此研究了具体非分圆数域族的RLWE/PLWE等价性。我们认为该数域族因其算术效率特性而具有特殊研究价值。