Computing Gröbner Bases for Vanishing Ideals of Finite Sets of Points

Farr, Jeffrey B.; Gao, Shuhong

doi:10.1007/11617983_11

Cited by 16 publications

(10 citation statements)

References 14 publications

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“…Reconstruction of ( k , n )‐SSIS schemes is accomplished by computing Lagrange interpolation polynomial. Actually, we can adopt Neville's approach such as explained in to obtain this polynomial. In this algorithm, we first calculate the constant interpolants, then all the interpolants continuing higher and higher order interpolants.…”

Section: Experiments and Comparisonmentioning

confidence: 99%

Reducing shadow size in smooth scalable secret image sharing

Liu

Yang

Yeh

2014

Security Comm Networks

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A (k, n)-scalable secret image sharing (SSIS) scheme shares a secret image into n independent shadow images. This (k, n)-SSIS scheme can simultaneously provide the properties of threshold (k or more shadow images are required in reconstruction; but k -1 or less shadow images reveal no information on the secret image) and the scalability (when k or more than k shadow images participate in reconstruction, more shadow images can reveal more information amount of secret image). In this paper, we deal with (k, n)-SSIS scheme with the smooth scalability such that the information amount of revealed image is "smoothly" proportional to the number of involved shadows, the whole secret image can be reconstructed only when all the n shadow images participate in reconstruction. Comparing with the existing smooth (k, n)-SSIS schemes, the proposed scheme has the less size of shadow image.

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Section: Experiments and Comparisonmentioning

confidence: 99%

Reducing shadow size in smooth scalable secret image sharing

Liu

Yang

Yeh

2014

Security Comm Networks

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“…There are still more approaches to the problem of finding a Gröbner basis of the vanishing ideal I (A) (e.g., the algorithm in [8] is a higher-dimensional analogue of Newton interpolation, in particular, based on induction over #A, or [9], whose approach is based on coding theory) and on the related topic of multivariate interpolation. The interested reader may consult [10] for the survey of the literature.…”

Section: Further Comments On the Literaturementioning

confidence: 99%

The vanishing ideal of a finite set of closed points in affine space

Lederer¹

2008

Journal of Pure and Applied Algebra

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Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.

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“…Such vanishing ideals over finite sets of points have been studied over arbitrary fields [30]. This paper presents solutions to such problems in Z 2 m , within a CAD-based verification framework.…”

Section: Symbolic Algebra and Number Theorymentioning

confidence: 99%

Equivalence Verification of Polynomial Datapaths Using Ideal Membership Testing

Shekhar

Kalla

Enescu³

2007

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Abstract-This paper addresses the problem of equivalence verification of RTL descriptions that implement arithmetic computations (such as ADD, MULT) over bit-vectors with finite widths. A bit-vector of size m represents integer values from 0 to 2 m −1; implying that the corresponding integer values are reduced modulo 2 m (%2 m ). This suggests that bit-vector arithmetic can be efficiently modeled as algebra over finite integer rings, where the bit-vector size (m) dictates the cardinality of the ring (Z2m ). This paper models the arithmetic datapath verification problem as equivalence testing of polynomial functions from Z 2 n 1 × Z 2 n 2 × · · · × Z 2 n d → Z2m . We formulate the equivalence problem f ≡ g into that of proving whether f − g ≡ 0%2 m . Fundamental concepts and results from number, ring and ideal theory are subsequently employed to develop systematic, complete algorithmic procedures to solve the problem. We demonstrate application of the proposed theoretical concepts to high-level (behavioral/RTL) verification of bit-vector arithmetic within practical CAD settings. Using our approach, we verify a set of arithmetic datapaths at RTL where contemporary verification approaches prove to be infeasible.

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Computing Gröbner Bases for Vanishing Ideals of Finite Sets of Points

Cited by 16 publications

References 14 publications

Reducing shadow size in smooth scalable secret image sharing

Reducing shadow size in smooth scalable secret image sharing

The vanishing ideal of a finite set of closed points in affine space

Equivalence Verification of Polynomial Datapaths Using Ideal Membership Testing

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