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### Modular Composition Modulo Triangular Sets and Applications

Adrien Poteaux, Éric Schost
2013 Computational Complexity
We generalize Kedlaya 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 .  ...

### On the complexity of computing with zero-dimensional triangular sets

Adrien Poteaux, Éric Schost
2013 Journal of symbolic computation
We generalize Kedlaya 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 .  ...

### On the complexity of computing with zero-dimensional triangular sets [article]

Adrien Poteaux, Éric Schost
2011 arXiv   pre-print
Conversely, we also show how to reduce the problem of 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.  ...

### Modular composition via factorization

Joris van der Hoeven, Grégoire Lecerf
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.  ...

### Phaseshaping Oscillator Algorithms For Musical Sound Synthesis

Jari Kleimola, Victor Lazzarini, Joseph Timoney, Vesa Välimäki
2010 Proceedings of the SMC Conferences
The authors believe that nested phaseshaping is a flexible tool that has many practical uses in the design 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  ...

### Analysis of Modular Arithmetic [chapter]

Markus Müller-Olm, Helmut Seidl
2005 Lecture Notes in Computer Science
Moreover, they run in time linear in the program size 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

Markus Müller-Olm, Helmut Seidl
2007 ACM Transactions on Programming Languages and Systems
Moreover, they run in time linear in the program size 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]

Vincent Neiger, Bruno Salvy, Éric Schost, Gilles Villard
2021 arXiv   pre-print
A new Las Vegas algorithm is presented for the 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.  ...

### 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.  ...

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

Marc Moreno Maza, Éric Schost, Paul Vrbik
2012 Lecture Notes in Computer Science
We consider the questions of inversion 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.  ...

### Homotopy techniques for multiplication modulo triangular sets

Alin Bostan, Muhammad F.I. Chowdhury, Joris van der Hoeven, Éric Schost
2011 Journal of symbolic computation
We study the cost of multiplication 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

Yuzhen Xie
2007 ACM Communications in Computer Algebra
In this thesis, we study 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  ...

### A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers [chapter]

Johannes Blömer, Alexander May
2005 Lecture Notes in Computer Science
Most interestingly, we reduce the case of solving univariate polynomials f (x) 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 .  ...

### On finding small solutions of modular multivariate polynomial equations [chapter]

Charanjit S. Jutla
1998 Lecture Notes in Computer Science
Let P(x) --0 (rood N) be a 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.  ...

### Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries

Sébastien Orange, Guénaël Renault, Kazuhiro Yokoyama
2012 Mathematics in Computer Science
These results are based on multi-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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