Paper 2016/220

Algorithms on Ideal over Complex Multiplication order

Paul Kirchner


We show in this paper that the Gentry-Szydlo algorithm for cyclotomic orders, previously revisited by Lenstra-Silverberg, can be extended to complex-multiplication (CM) orders, and even to a more general structure. This algorithm allows to test equality over the polarized ideal class group, and finds a generator of the polarized ideal in polynomial time. Also, the algorithm allows to solve the norm equation over CM orders and the recent reduction of principal ideals to the real suborder can also be performed in polynomial time. Furthermore, we can also compute in polynomial time a unit of an order of any number field given a (not very precise) approximation of it. Our description of the Gentry-Szydlo algorithm is different from the original and Lenstra- Silverberg’s variant and we hope the simplifications made will allow a deeper understanding. Finally, we show that the well-known speed-up for enumeration and sieve algorithms for ideal lattices over power of two cyclotomics can be generalized to any number field with many roots of unity.

Note: One mistake was removed.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Contact author(s)
paul kirchner @ ens fr
2016-04-06: revised
2016-02-29: received
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Creative Commons Attribution


      author = {Paul Kirchner},
      title = {Algorithms on Ideal over Complex Multiplication order},
      howpublished = {Cryptology ePrint Archive, Paper 2016/220},
      year = {2016},
      note = {\url{}},
      url = {}
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