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Modular Composition Modulo Triangular Sets and Applications

2013
*
Computational Complexity
*

We generalize Kedlaya

doi:10.1007/s00037-013-0063-y
fatcat:v52lbh5mtbbmdgympfgfjv6hby
*and*Umans'*modular**composition*algorithm to the multivariate case. ... As a main*application*, we give fast algorithms for many operations involving*triangular**sets*(over a finite field), such as*modular*multiplication, inversion, or change of order. ... This requires m + 1 ≤ 3*applications*of Φ P or Ψ P ,*and*a*modular**composition**modulo*P . ...##
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On the complexity of computing with zero-dimensional triangular sets

2013
*
Journal of symbolic computation
*

We generalize Kedlaya

doi:10.1016/j.jsc.2012.05.008
fatcat:cd3rc7zokvf6jdmwf7s3jhppha
*and*Umans'*modular**composition*algorithm to the multivariate case. ... As a main*application*, we give fast algorithms for many operations involving*triangular**sets*(over a finite field), such as*modular*multiplication, inversion, or change of order. ... This requires m + 1 ≤ 3*applications*of Φ P or Ψ P ,*and*a*modular**composition**modulo*P . ...##
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On the complexity of computing with zero-dimensional triangular sets
[article]

2011
*
arXiv
*
pre-print

Conversely, we also show how to reduce the problem of

arXiv:1109.4323v1
fatcat:xeu6hjh3avfvdj7oxk2i4id3di
*modular**composition*to change of order for*triangular**sets*, so that all these problems are essentially equivalent. ... Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya*and*Umans' algorithm for*modular**composition*. ... is supported by NSERC*and*the Canada Research Chair program. ...##
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Modular composition via factorization

2018
*
Journal of Complexity
*

*Modular*

*composition*is the problem to compute the

*composition*of two univariate polynomials

*modulo*a third one. ... In this article, we explore particular cases of moduli over finite fields for which

*modular*

*composition*turns out to be cheaper than in the general case. ... INTRODUCTION Let be an effective field,

*and*let f , g, h be polynomials in [x] . The problem of

*modular*

*composition*is to compute f ∘ g

*modulo*h. ...

##
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Phaseshaping Oscillator Algorithms For Musical Sound Synthesis

2010
*
Proceedings of the SMC Conferences
*

The authors believe that nested phaseshaping is a flexible tool that has many practical uses in the design

doi:10.5281/zenodo.849811
fatcat:clbhakwffzddxd7txxpixg4vga
*and*implementation of*modular*sound synthesis*applications*. ... Figure 13 shows also that nested phaseshaping is a practical tool that provides a*modular*approach to sound synthesis*and*is therefore instantly*applicable*in systems such as Max, Pure Data,*and*Reaktor ...##
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Analysis of Modular Arithmetic
[chapter]

2005
*
Lecture Notes in Computer Science
*

Moreover, they run in time linear in the program size

doi:10.1007/978-3-540-31987-0_5
fatcat:45dwqtfg7zht3nbhg7hho5ssvy
*and*polynomial in the number of program variables*and*can be implemented by using the same*modular*integer arithmetic as the target language to be ... The difficulty here is that the ring Zm of integers*modulo*m = 2 w , w > 1, has zero divisors*and*thus cannot be embedded into a field. ... We thank Martin Hofmann for pointing us to the topic of analyzing*modular*arithmetic*and*the anonymous referees for valuable comments. ...##
###
Analysis of modular arithmetic

2007
*
ACM Transactions on Programming Languages and Systems
*

Moreover, they run in time linear in the program size

doi:10.1145/1275497.1275504
fatcat:e4tv4aufdvcj5ar2a65ywmiwg4
*and*polynomial in the number of program variables*and*can be implemented by using the same*modular*integer arithmetic as the target language to be ... The difficulty here is that the ring Zm of integers*modulo*m = 2 w , w > 1, has zero divisors*and*thus cannot be embedded into a field. ... Acknowledgments We thank Martin Hofmann for pointing us to the topic of analyzing*modular*arithmetic*and*the anonymous referees for valuable comments. ...##
###
Faster Modular Composition
[article]

2021
*
arXiv
*
pre-print

A new Las Vegas algorithm is presented for the

arXiv:2110.08354v1
fatcat:r4abnf5vundljczjbfdfh75axu
*composition*of two polynomials*modulo*a third one, over an arbitrary field. ... The previous fastest algebraic algorithms, due to Brent*and*Kung in 1978, require O(n^1.63) field operations in general,*and*n^3/2+o(1) field operations in the particular case of power series over a field ... The second*and*third steps are*modular**compositions*. ...##
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Page 2794 of Mathematical Reviews Vol. , Issue 97E
[page]

1997
*
Mathematical Reviews
*

*modulo*some prime,

*and*hence not a square. ... The construction of the corresponding

*set*B uses a

*modular*additive exact completion for certain moduli tending to infinity. ...

##
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Inversion Modulo Zero-Dimensional Regular Chains
[chapter]

2012
*
Lecture Notes in Computer Science
*

We consider the questions of inversion

doi:10.1007/978-3-642-32973-9_19
fatcat:2fvtfqsqbvfyjbhvmnd7ounx6u
*modulo*a regular chain in dimension zero*and*of matrix inversion*modulo*such a regular chain. ... We acknowledge the support of the Canada Research Chairs Program*and*of NSERC. ... Multiplication*modulo**triangular**sets*. In the context of*triangular**sets*, the first non-trivial algorithmic question is*modular*multiplication. ...##
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Homotopy techniques for multiplication modulo triangular sets

2011
*
Journal of symbolic computation
*

We study the cost of multiplication

doi:10.1016/j.jsc.2011.08.015
fatcat:fjdd6sfwhzffbmkac6lmbd2a7e
*modulo**triangular*families of polynomials. ...*Applications*are given to notably addition of algebraic numbers in small characteristic. ...*set*, or further tasks such as*modular*inversion or*modular**composition*. ...##
###
Fast algorithms, modular methods, parallel approaches and software engineering for solving polynomial systems symbolically

2007
*
ACM Communications in Computer Algebra
*

In this thesis, we study

doi:10.1145/1358190.1358195
fatcat:qck42ui4rzcgti62umwt2njmje
*and*apply fast algorithms,*modular*methods, parallel approaches*and*software engineering techniques to improve the efficiency of symbolic solvers for computing*triangular*decomposition ... Then, by introducing the notion of equiprojectable decomposition, a sharp*modular*method for*triangular*decompositions based on Hensel lifting techniques is obtained. ... W r } is a non-critical*triangular*de-*composition*of T*and*, for all 1 ≤ ≤ r, F is a GCD-free basis of A ∪ B*modulo*W . mergeTwoGcdFreeBasesModT((a 1 , . . . , a e ), (b 1 , . . . , b s ), T ) == 1: bases ...##
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A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers
[chapter]

2005
*
Lecture Notes in Computer Science
*

Most interestingly, we reduce the case of solving univariate polynomials f (x)

doi:10.1007/11426639_15
fatcat:uapgrebvpbctxay7x54f7bqtfm
*modulo*some*composite*number N of unknown factorization to the case of solving bivariate polynomials over the integers. ... Our method has several*applications*. ... Upper*Triangular*Shape In this subsection, we analyze a variant of Coppersmith's univariate*modular*approach, where one solves polynomial equations*modulo*a divisor of N . ...##
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On finding small solutions of modular multivariate polynomial equations
[chapter]

1998
*
Lecture Notes in Computer Science
*

Let P(x) --0 (rood N) be a

doi:10.1007/bfb0054124
fatcat:brktftrufnhebd2gxwy6n4ev2i
*modular*multivariate polynomial equation, in m variables,*and*total degree k with a small root x0. ... However, the*modular*equation*modulo*a large*composite*N is not known to be solvable in polynomial time (unless the factorization of N is known). ...*Application*to Attacks on KMOV Let N be a large*composite*integer, product of two secret prime numbers p*and*q. ...##
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Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries

2012
*
Mathematics in Computer Science
*

These results are based on multi-

doi:10.1007/s11786-012-0112-y
fatcat:ue4xju344je6padm65xbvbmkqi
*modular**and*evaluation-interpolation techniques. We show how to use intrinsic symmetries in order to increase the efficiency of these techniques. ...*Applications*to splitting fields of univariate polynomials are presented. (2000) . Primary 12Y05; Secondary 12F10, 12-04. Mathematics Subject Classification ... Arithmetic in extension fields with*modular*technique In this section we present arithmetic in the number field K in general*setting*by using our multi-*modular**and**modular*evaluation-interpolation framework ...
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