Algorithms on Ideal over Complex Multiplication order
This work addresses computational problems in algebraic number theory, particularly for cryptographers and mathematicians working with ideal lattices, by generalizing and simplifying existing algorithms, though it is incremental in nature.
The paper extends the Gentry-Szydlo algorithm from cyclotomic orders to complex-multiplication orders and more general structures, enabling polynomial-time solutions for testing equality in polarized ideal class groups, finding generators, solving norm equations, and computing units in number fields.
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.