Equations for the projective closure and effective Nullstellensatz

Caniglia, Leandro; Galligo, André; Heintz, Joos

doi:10.1016/0166-218x(91)90105-6

Cited by 23 publications

(14 citation statements)

References 10 publications

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“…The reader is referred to [6,10,11,40] for general theory on the topic, although the speed estimates there can be considered superseded. The most advanced implementations are detailed in [29,30,31].…”

Section: Gröbner Bases Algorithms F -Fmentioning

confidence: 99%

All in the XL Family: Theory and Practice

Yang

Chen²

2005

Lecture Notes in Computer Science

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Abstract. The XL (eXtended Linearization) equation-solving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XL and its relatives may be used as direct attacks against multivariate Public-Key Cryptosystems and as final stages for many "algebraic cryptanalysis" used today. We analyze the applicability and performance of XL and its relatives, particularly for generic systems of equations over medium-sized finite fields.In examining the extended family of Gröbner Bases and XL from theoretical, empirical and practical viewpoints, we add to the general understanding of equation-solving. Moreover, we give rigorous conditions for the successful termination of XL, Gröbner Bases methods and relatives. Thus we have a better grasp of how such algebraic attacks should be applied. We also compute revised security estimates for multivariate cryptosystems. For example, the schemes SFLASH v2 and HFE Challenge 2 are shown to be unbroken by XL variants.

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Section: Gröbner Bases Algorithms F -Fmentioning

confidence: 99%

All in the XL Family: Theory and Practice

Yang

Chen²

2005

Lecture Notes in Computer Science

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“…We shall propose some changes and explain why. 5 usually by LU decomposition, even though it does not yield all nonsingular matrices.…”

Section: The Old: Tts/2 a Previously Proposed Variantmentioning

confidence: 99%

A More Secure and Efficacious TTS Signature Scheme

Chen

Yang

2004

Information Security and Cryptology - ICISC 2003

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-derived SFLASH family of schemes, which uses monomials in a large field for the central portion, previously usually acknowledged as fastest. We show a small flaw in TTS/2 and present an improved TTS implementation which we call TTS/4. We will examine in some detail how well TTS/4 performs, how it stands up to previously known attacks, and why it represents an advance over TTS/2. Based on this topical assessment, we consider TTS in general and TTS/4 in particular to be competitive or superior in several aspects to other schemes, partly because the theoretical roots of TTS induce many good traits. One specific area in which TTS/4 should excel is in low-cost smartcards. It seems that the genre has great potential for practical deployment and deserves further attention by the cryptological community.

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“…When the ideal is zero-dimensional, Caniglia et al (1991) showed that we can even lower the degree bound to d O(n) .…”

Section: A Degree Boundmentioning

confidence: 96%

A Fast Algorithm for Gröbner Basis Conversion and its Applications

Tran

2000

Journal of Symbolic Computation

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The Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zerodimension as well as positive dimension. Typically, the target point of the walking path lies on the intersection of very many cones, which ends up with initial forms of a considerable number of terms. Therefore, it is crucial to the performance of the conversion to change the target point since we have to compute a Gröbner basis with respect to the elimination term order of such large initial forms.In contrast to heuristic methods found in the literature, in this paper the author presents a deterministic method to vary the target point in order to ensure the generality of the position, i.e. we always have just a few terms in the initial forms. It turns out that this theoretical result brings a dramatic speed-up in practice. We have implemented the Gröbner walk method together with the deterministic method for varying the target point in the kernel of Mathematica. Our experiments show the superlative performance of our improved Gröbner walk method in comparison with other known methods. Our best performance is 3 × 10 4 times faster than the direct computation of the reduced Gröbner basis with respect to pure lexicographic term order (using the Buchberger algorithm and the sugar cube strategy). We also discuss the complexity of the conversion algorithm and prove a degree bound for polynomials in the target Gröbner basis.In the second part of the paper, we present some applications of the conversion method for implicitization and geometric reasoning. We compare the efficiency of the improved Gröbner walk method with other methods for elimination such as multivariate resultant methods.

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Equations for the projective closure and effective Nullstellensatz

Cited by 23 publications

References 10 publications

All in the XL Family: Theory and Practice

All in the XL Family: Theory and Practice

A More Secure and Efficacious TTS Signature Scheme

A Fast Algorithm for Gröbner Basis Conversion and its Applications

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