research-article

Decomposition of generic multivariate polynomials

Authors:

Jean-Charles Faugère,

Joachim von zur Gathen,

Ludovic PerretAuthors Info & Claims

ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

Pages 131 - 137

https://doi.org/10.1145/1837934.1837963

Published: 25 July 2010 Publication History

Abstract

We consider the composition f =g o h of two systems g= (g₀, ..., g_t) and h=(h₀, ..., h_s) of homogeneous multivariate polynomials over a field K, where each g_j ∈ K[y₀, ..., y_s] has degree ℓ each h_k ∈ K[x₀, ..., x_r] has degree m, and f_i = g_i(h₀, ..., h_s) ∈ K[x₀, ..., x_r] has degree n = ℓ · m, for 0 ≤ i ≤ t. The motivation of this paper is to investigate the behavior of the decomposition algorithm Multi-ComPoly proposed at ISSAC'09 [18]. We prove that the algorithm works correctly for generic decomposable instances -- in the special cases where ℓ is 2 or 3, and m is 2 -- and investigate the issue of uniqueness of a generic decomposable instance. The uniqueness is defined w.r.t. the "normal form" of a multivariate decomposition, a new notion introduced in this paper, which is of independent interest.

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Cited By

Duff TKorotynskiy VPajdla TRegan M(2023)Using monodromy to recover symmetries of polynomial systemsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597106(251-259)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597106
Minaud BDerbez PFouque PKarpman P(2018)Key-Recovery Attacks on ASASAJournal of Cryptology10.1007/s00145-017-9272-x31:3(845-884)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s00145-017-9272-x
Biryukov ABouillaguet CKhovratovich D(2014)Cryptographic Schemes Based on the ASASA Structure: Black-Box, White-Box, and Public-Key (Extended Abstract)Advances in Cryptology – ASIACRYPT 201410.1007/978-3-662-45611-8_4(63-84)Online publication date: 2014
https://doi.org/10.1007/978-3-662-45611-8_4
Show More Cited By

Index Terms

Decomposition of generic multivariate polynomials
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

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cover image ACM Other conferences

ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

July 2010

366 pages

ISBN:9781450301503

DOI:10.1145/1837934

Program Chair:
Wolfram Koepf
London, Ontario, Canada

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 25 July 2010

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ISSAC '10: International Symposium on Symbolic and Algebraic Computation

July 25 - 28, 2010

Munich, Germany

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ISSAC '10 Paper Acceptance Rate 45 of 110 submissions, 41%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Duff TKorotynskiy VPajdla TRegan M(2023)Using monodromy to recover symmetries of polynomial systemsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597106(251-259)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597106
Minaud BDerbez PFouque PKarpman P(2018)Key-Recovery Attacks on ASASAJournal of Cryptology10.1007/s00145-017-9272-x31:3(845-884)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s00145-017-9272-x
Biryukov ABouillaguet CKhovratovich D(2014)Cryptographic Schemes Based on the ASASA Structure: Black-Box, White-Box, and Public-Key (Extended Abstract)Advances in Cryptology – ASIACRYPT 201410.1007/978-3-662-45611-8_4(63-84)Online publication date: 2014
https://doi.org/10.1007/978-3-662-45611-8_4
Zhao SFeng RGao X(2012)On functional decomposition of multivariate polynomials with differentiation and homogenizationJournal of Systems Science and Complexity10.1007/s11424-012-1144-825:2(329-347)Online publication date: 11-Apr-2012
https://doi.org/10.1007/s11424-012-1144-8

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