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Algorithms on Ideal over Complex Multiplication order [article]

Paul Kirchner
2016 arXiv   pre-print
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  ...  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.  ...  We then build the CM order O ⊗ Z[X]/(X e − 1), by concatenating the basis of I i = (ω i ). We now use [21, Theorem 1.2] to find the generators of the roots of unity of this order.  ... 
arXiv:1602.09037v1 fatcat:h4eyuz7sdrcu3nq7lb54g5pcya

On FGLM Algorithms with Tate Algebras

Xavier Caruso, Tristan Vaccon, Thibaut Verron
2021 Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation  
In the present article, we extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for fast change of ordering, this strategy has two other important bene ts.  ...  In [CVV19, CVV20] the formalism of Gröbner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed.  ...  Algorithm 6: GB( 1 , . . . , , , ≤) Input : 1 , . . . , the multiplication matrices over the unit ball of , the image of 1 ∈ {X; r} in , ≤ a monomial ordering Output :A Gröbner basis of the ideal ⊂ {X;  ... 
doi:10.1145/3452143.3465521 fatcat:gklflrd66vawjm7dbvrqgmtce4

On FGLM Algorithms with Tate Algebras [article]

Xavier Caruso
2021 arXiv   pre-print
In the present article, we extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for fast change of ordering, this strategy has two other important benefits.  ...  In [CVV19, CVV20] the formalism of Gröbner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed.  ...  Algorithm 6 :== 6 GB( 1 , . . . , , , ≤) Input : 1 , . . . , the multiplication matrices over the unit ball of , the image of 1 ∈ {X; r} in , ≤ a monomial ordering Output :A Gröbner basis of the ideal  ... 
arXiv:2102.05324v1 fatcat:ff4wfztbyvgnvnkb7oteadjbfi

Fast Construction of Secure Discrete Logarithm Problems over Jacobian Varieties [chapter]

Jinhui Chao, Kazuto Matsuo, Shigeo Tsujii
2000 IFIP Advances in Information and Communication Technology  
This paper presents efficient algorithms to calculate the CM type and ideal factorization of Frobenius endomorphisms of Jacobian varieties over finite fields F P in polynomial time of logp.  ...  However, lacking of efficient point-counting algorithms for such varieties over finite fields makes the design of secure cryptosystems very difficult.  ...  Thus one derives primal ideal q:J's in Op lying over p such that (p) = Nq:J = N ( w). 2 : 2 For all roots of unity { ( E K}, calculate the order #.:J(F p) -N(1-(7ro). 3 : If { #.  ... 
doi:10.1007/978-0-387-35515-3_25 fatcat:3w5cig26izdpvia32sxrk37rly

Fast interpolation in algebraic soft decision decoding of Reed-Solomon codes

Vera Miloslavskaya, Peter Trifonov
2010 2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON)  
A generalization of the binary interpolation algorithm to the case of variable root multiplicity is presented.  ...  For each ideal in this sequence one can derive a Gröbner basis, and obtain the required polynomial as the smallest element of the basis of the last ideal. 1) Multiplication of ideals: Multiplication  ...  Figure 7 illustrates the decoding time of (63, 48) code over GF (2 6 ) with the Kötter-Vardy method based on the iterative interpolation algorithm [11] and the proposed one.  ... 
doi:10.1109/sibircon.2010.5555311 fatcat:jel3wegvw5ewllfiu2ubvy3j7a

Complex multiplication and canonical lifts

David R. Kohel
2008 Algebraic Geometry and Its Applications  
Algorithms for p-adic canonical lifts give rise to very efficient means of constructing high-precision approximations to CM points on moduli spaces of abelian varieties.  ...  In particular, algorithms for 2-adic and 3-adic lifting of Frobenius give rise to CM constructions in dimension 2 (see [6] and [2]).  ...  Complex multiplication The Main Theorem of Complex Multiplication gives the relation between the ideal classes of a CM order O and abelian varieties with endomorphism ring O.  ... 
doi:10.1142/9789812793430_0003 fatcat:u7wucgrh4zebhazt6nuoyykxma

Page 2617 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
another kind of multiplication for ideals.  ...  The authors choose to omit all complexity analysis of the algorithms, because “complexity theory is too important an issue to be dealt with lightly”.  ... 

Isogeny graphs with maximal real multiplication [article]

Sorina Ionica
2019 arXiv   pre-print
Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one.  ...  Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal.  ...  We are indebted to John Boxall for sharing his ideas regarding the computation of isogenies preserving the real multiplication, and providing guidance for improving the writing of this paper.  ... 
arXiv:1407.6672v4 fatcat:syv4rplqdvhvfmw6o3vxwf7grq

Computing the support of local cohomology modules [article]

Josep Àlvarez Montaner, Anton Leykin
2006 arXiv   pre-print
Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules H^r_I(R) for any ideal I⊆ R using the Čech complex.  ...  These applications are illustrated by examples of computations using our implementation of the algorithm in Macaulay 2.  ...  In order to construct the algebraic characteristic cycle over Q we would need to find the absolute primary decomposition of the ideal C we obtain with Algorithm 3.1.  ... 
arXiv:math/0606142v2 fatcat:ownthnm42rgnlkzskefkf3p5bq

Computing endomorphism rings of abelian varieties of dimension two [article]

Gaetan Bisson
2013 arXiv   pre-print
Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.  ...  Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite  ...  Definition 2 . 3 . 23 For any order O in a complex multiplication field K, denote by I O the group consisting of all pairs (a, ρ) satisfying aa = ρO, where a is an invertible fractional ideal of O and  ... 
arXiv:1209.1189v2 fatcat:dgyyc3jaancpbhov2wo2ntgfxy

Computing endomorphism rings of abelian varieties of dimension two

Gaetan Bisson
2015 Mathematics of Computation  
Although its correctness and complexity bound rely on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.  ...  two over finite fields.  ...  For any order in a complex multiplication field K, denote by I the group consisting of all pairs (a, ρ) satisfying aa = ρ , where a is an invertible fractional ideal of and ρ is a totally positive element  ... 
doi:10.1090/s0025-5718-2015-02938-x fatcat:cqnh3tyjrbcuvcvpokastfaafy

Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals

Andrew Ferguson, Huu Phuoc Le
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals.  ...  For an 𝑛 × 𝑛 symmetric matrix with polynomial entries of degree 𝑑, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM  ...  We also thank the anonymous reviewers for their helpful comments on our paper.  ... 
doi:10.1145/3476446.3536182 fatcat:inolyr27jreahjca5367z3dhx4

Computing the support of local cohomology modules

Josep Àlvarez Montaner, Anton Leykin
2006 Journal of symbolic computation  
Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules H r I (R) for any ideal I ⊆ R using theČech complex.  ...  The algorithm, in particular, is useful for answering questions regarding vanishing of local cohomology modules and computing Lyubeznik numbers.  ...  In order to construct the algebraic characteristic cycle over Q we would need to find the absolute primary decomposition of the ideal C we obtain with Algorithm 3.1.  ... 
doi:10.1016/j.jsc.2006.09.001 fatcat:tayiwhftbrfpfknwrub7rsg5cy

Bounds for Bilinear Complexity of Noncommutative Group Algebras [article]

Alexey Pospelov
2010 arXiv   pre-print
We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication.  ...  These lower bounds are built on the top of Bl\"aser's results for semisimple algebras and algebras with large radical and the lower bound for arbitrary associative algebras due to Alder and Strassen.  ...  is called the length of the quadratic algorithm and the minimal length over all quadratic algorithms for ϕ is called the multiplicative complexity of ϕ and is denoted by C(ϕ).  ... 
arXiv:1003.4679v1 fatcat:ld2pgtta4nd5bn63gebrtabn74

Isogeny graphs with maximal real multiplication

Sorina Ionica, Emmanuel Thomé
2019 Journal of Number Theory  
We consider the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring has class number one and is locally maximal at , for a fixed prime.  ...  Background and notations It is well known that in the case of elliptic curves with complex multiplication by an imaginary quadratic field K, the lattice of orders of K has the structure of a tower.  ...  We are indebted to John Boxall for sharing his ideas regarding the computation of isogenies preserving the real multiplication, and providing guidance for improving the writing of this paper.  ... 
doi:10.1016/j.jnt.2019.06.019 fatcat:iakoufei7zc2zgzltwdvgwzp6q
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