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42nd ACM International Symposium on Symbolic and Algebraic Computation (ISSAC)

作者： Vaccon, Tristan Yokoyama, Kazuhiro Univ Limoges CNRS UMR 7252 XLIM F-87060 Limoges France Rikkyo Univ Dept Math Tokyo Japan

ISBN: (纸本)9781450350648

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Grobner bases taking into account the valuation of K. While generalizing the classical theory of Grobner bases, it is not clear how modern algorithms for computing Grobner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated f5 algorithm of faugere. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Grobner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical f5 algorithm.

关键词： algorithms Tropical geometry Grobner bases f5 algorithm p-adic precision

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Enhancement in Time Efficiency and Expected Capacity of f5 algorithm 7

Enhancement in Time Efficiency and Expected Capacity of F5 A...

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7th International Conference on Computational Intelligence and Communication Networks (CICN)

作者： Rawat, Pooja Pandey, Amit Kumar Kushwah, Rajendra Singh Inst Technol & Management Comp Sci & Engn Gwalior India

ISBN: (纸本)9781509000760

Most steganography algorithms are weak against attacks like visual and statistical but among all steganography algorithms f5 algorithm is more robust and secure. It can be more efficient if we reduce the encryption and decryption time and improve the embedding capacity. In this paper we present a better f5 algorithm with less encryption and decryption time and higher embedding capacity with the help of quantization table.

关键词： Image Steganography f5 algorithm Quantization table Luminance Chrominance

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On affine tropical f5 algorithms

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JOURNAL Of SYMBOLIC COMPUTATION 2021年 102卷 132-152页

作者： Vaccon, Tristan Verron, Thibaut Yokoyama, Kazuhiro Univ Limoges XLIM Limoges France JKU Inst Algebra Linz Austria Rikkyo Univ Dept Math Tokyo Japan

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Grobner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Grobner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Grobner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the f5 strategy to a new definition of tropical Grobner bases in an affine setting. We also provide a competitor with an adaptation of the f4 strategy to tropical Grobner bases computations. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical f5 algorithm. We also illustrate its merits as a first step before an fGLM algorithm to compute (classical) lex bases over p-adics. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： algorithms Tropical geometry Grobner bases f5 algorithm p-adic precision

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Enhancement in Time Efficiency and Expected Capacity of f5 algorithm

Enhancement in Time Efficiency and Expected Capacity of F5 A...

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International Conference on Computational Intelligence and Communication Networks

作者： Pooja Rawat Amit Kumar Pandey Rajendra Singh Kushwah Computer Science and Engineering Institute of Technology and Management

ISBN: (纸本)9781509000777

关键词： Image Steganography f5 algorithm Quantization table Luminance Chrominance

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Axioms for a theory of signature bases

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JOURNAL Of SYMBOLIC COMPUTATION 2024年 123卷

作者： Lairez, Pierre Univ Paris Saclay Inria F-91120 Palaiseau France

Twenty years after the discovery of the f5 algorithm, Grobner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Grobner bases with signatures with the purpose of uncoupling the theory and the algorithms, giving general results applicable in many different settings (e.g. Grobner for submodules, f4-style reduction, noncommutative rings, non-Noetherian settings, etc.), and extending the reach of signature algorithms. (c) 2023 Published by Elsevier Ltd.

关键词： Grobner basis f5 algorithm Signature basis

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Matrix-f5 algorithms and tropical Grobner bases computation

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JOURNAL Of SYMBOLIC COMPUTATION 2018年第Nov./Dec.期89卷 227-254页

作者： Vaccon, Tristan Univ Limoges CNRS XLIM UMR 7252 F-87060 Limoges France

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of GrObner bases taking into account the valuation of K. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a p-adic field. We design a strategy to compute such tropical GrObner bases by adapting the Matrix-f5 algorithm. Two variants of the Matrix-f5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster. Our study is performed both over any exact field with valuation and some inexact fields like Q(p) or f-q((t)). In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Grobner bases when the valuation is non-trivial. Numerical examples are provided. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： GrObner bases Tropical geometry f5 algorithm p-Adic precision p-Adic algorithm

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On Affine Tropical f5 algorithms 18

On Affine Tropical F5 Algorithms

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43rd ACM International Symposium on Symbolic and Algebraic Computation (ISSAC)

作者： Vaccon, Tristan Verron, Thibaut Yokoyama, Kazuhiro Univ Limoges Limoges France CNRS XLIM Limoges France Johannes Kepler Univ Linz Inst Algebra Linz Austria Rikkyo Univ Dept Math Tokyo Japan

ISBN: (纸本)9781450355506

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Grobner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Grobner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Grobner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the f5 strategy to a new definition of tropical Grobner bases in an affine setting. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical f5 algorithm. We also illustrate its merits as a first step before an fGLM algorithm to compute (classical) lex bases over p-adics.

关键词： algorithms Tropical geometry Grobner bases f5 algorithm p-adic precision

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Signature-based algorithms for Grobner Bases over Tate Algebras 20

Signature-based Algorithms for Grobner Bases over Tate Algeb...

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45th International Symposium on Symbolic and Algebraic Computation

作者： Caruso, Xavier Vaccon, Tristan Verron, Thibaut Univ Bordeaux CNRS INRIA Bordeaux France Univ Limoges Limoges France CNRS XLIM UMR 7252 Limoges France Johannes Kepler Univ Linz Inst Algebra Linz Austria

ISBN: (纸本)9781450371001

Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Grobner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Grobner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SAGEMATH. We discuss their superiority based on numerical evidence.

关键词： algorithms Power series Tate algebra Grobner bases f5 algorithm p-adic precision

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An Enhanced Steganographic Code and Its Application in Voice-Over-IP Steganography

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IEEE ACCESS 2019年 7卷 97187-97195页

作者： Zhang, Lejun Hu, Xiaoyan Rasheed, Waqas Huang, Tianwen Zhao, Chunhui Yangzhou Univ Coll Informat Engn Yangzhou 225127 Jiangsu Peoples R China Harbin Engn Univ Coll Informat & Commun Engn Harbin 150001 Heilongjiang Peoples R China

Voice-over IP (VoIP) technology is a kind of digital transmission technology based on IP network. It is one of the important methods to use voice service in VoIP as steganographic carrier to ensure secure transmission. However, the traditional steganographic code has some problems, such as low embedding efficiency and weak concealment, which cannot meet the requirements of VoIP streaming media information hiding for the security of secret information. Therefore, a steganographic algorithm combining f5 and simplified wet paper code (SWPC) algorithm is proposed. The main idea is to embed secret messages in each row of the carrier matrix using the f5 algorithm, and then the SWPC algorithm is used to embed the columns according to the wet and dry characteristics of the wet paper code without affecting the results before row embedding. We use the VoIP streams encoded by the ITU-T G.729a codec as a carrier to verify the proposed scheme. The experimental results demonstrate that the proposed scheme can achieve relatively better IP speech data steganographic transparency and that it can outperform f5-WPC and SWPC approaches.

关键词： f5 algorithm steganography streaming media voice over IP wet paper code

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Matrix-f5 algorithms over finite-precision complete discrete valuation fields

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JOURNAL Of SYMBOLIC COMPUTATION 2017年第Part2期80卷 329-350页

作者： Vaccon, Tristan Univ Rennes 1 F-35014 Rennes France JSPS Rikkyo Univ Tokyo Japan

Let (f(1,) . . . ,f(s)) is an element of Q(p) [X-1,X- . . . ,X-n](s) be a sequence of homogeneous polynomials with p-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since Q(p) is not an effective field, classical algorithm does not apply. We provide a definition for an approximate Grobner basis with respect to a monomial order w. We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals < f(1), . . . , f(i)> are weakly-w-ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic. Two variants of that strategy are available, depending on whether one leans more on precision or time-complexity. for the analysis of these algorithms, we study the loss of precision of the Gauss row echelon algorithm, and apply it to an adapted Matrix-f5 algorithm. Numerical examples are provided. Moreover, the fact that under such hypotheses, Grobner bases can be computed stably has many applications. firstly, the mapping sending (f(1,) . . . , f(s)) to the reduced Grobner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allow to perform lifting on the Grobner bases, from Z/p(k)Z to Z/p(k+k')Z or Z. finally, asking for the same hypotheses on the highest-degree homogeneous components of the entry polynomials allows to extend our strategy to the affine case. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： f5 algorithm Grobner bases Moreno-Socias conjecture p-Adic algorithm p-Adic precision Differential precision

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