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Algorithms on Ideal over Complex Multiplication order
[article]

2016
*
arXiv
*
pre-print

We show in this paper that the Gentry-Szydlo

arXiv:1602.09037v1
fatcat:h4eyuz7sdrcu3nq7lb54g5pcya
*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*. ...##
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On FGLM Algorithms with Tate Algebras

2021
*
Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation
*

In the present article, we extend the FGLM

doi:10.1145/3452143.3465521
fatcat:gklflrd66vawjm7dbvrqgmtce4
*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; ...##
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On FGLM Algorithms with Tate Algebras
[article]

2021
*
arXiv
*
pre-print

In the present article, we extend the FGLM

arXiv:2102.05324v1
fatcat:ff4wfztbyvgnvnkb7oteadjbfi
*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*...##
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Fast Construction of Secure Discrete Logarithm Problems over Jacobian Varieties
[chapter]

2000
*
IFIP Advances in Information and Communication Technology
*

This paper presents efficient

doi:10.1007/978-0-387-35515-3_25
fatcat:3w5cig26izdpvia32sxrk37rly
*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 { #. ...##
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Fast interpolation in algebraic soft decision decoding of Reed-Solomon codes

2010
*
2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON)
*

A generalization of the binary interpolation

doi:10.1109/sibircon.2010.5555311
fatcat:jel3wegvw5ewllfiu2ubvy3j7a
*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*. ...##
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Complex multiplication and canonical lifts

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. ...

##
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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”. ...##
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Isogeny graphs with maximal real multiplication
[article]

2019
*
arXiv
*
pre-print

Our setting considers genus 2 jacobians with

arXiv:1407.6672v4
fatcat:syv4rplqdvhvfmw6o3vxwf7grq
*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. ...##
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Computing the support of local cohomology modules
[article]

2006
*
arXiv
*
pre-print

Based

arXiv:math/0606142v2
fatcat:ownthnm42rgnlkzskefkf3p5bq
*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. ...##
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Computing endomorphism rings of abelian varieties of dimension two
[article]

2013
*
arXiv
*
pre-print

Although its correctness and

arXiv:1209.1189v2
fatcat:dgyyc3jaancpbhov2wo2ntgfxy
*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 ...##
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Computing endomorphism rings of abelian varieties of dimension two

2015
*
Mathematics of Computation
*

Although its correctness and

doi:10.1090/s0025-5718-2015-02938-x
fatcat:cqnh3tyjrbcuvcvpokastfaafy
*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 ...##
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Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals

2022
*
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
*

In this paper, we focus

doi:10.1145/3476446.3536182
fatcat:inolyr27jreahjca5367z3dhx4
*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. ...##
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Computing the support of local cohomology modules

2006
*
Journal of symbolic computation
*

Based

doi:10.1016/j.jsc.2006.09.001
fatcat:tayiwhftbrfpfknwrub7rsg5cy
*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. ...##
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Bounds for Bilinear Complexity of Noncommutative Group Algebras
[article]

2010
*
arXiv
*
pre-print

We study the

arXiv:1003.4679v1
fatcat:ld2pgtta4nd5bn63gebrtabn74
*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). ...##
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Isogeny graphs with maximal real multiplication

2019
*
Journal of Number Theory
*

We consider the case of genus-2 Jacobians with

doi:10.1016/j.jnt.2019.06.019
fatcat:iakoufei7zc2zgzltwdvgwzp6q
*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. ...
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