Abstract
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.
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Communicated by John Cremona.
This material is based on research sponsored by DARPA under Agreement Number FA8750-13-2-0054 and by the Alfred P. Sloan Foundation. The US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA or the US Government.
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Lenstra, H.W., Silverberg, A. Roots of Unity in Orders. Found Comput Math 17, 851–877 (2017). https://doi.org/10.1007/s10208-016-9304-1
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DOI: https://doi.org/10.1007/s10208-016-9304-1