Hilbert Series for Ideals Generated by Generic Forms

Fröberg, Ralf; Hollman, Joachim

doi:10.1006/jsco.1994.1008

Cited by 25 publications

(24 citation statements)

References 4 publications

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“…We prove that such algebraic independence at low degree holds with m ≤ n + n−2 2 in Theorem 10. This improves on a result of [29,Theorem 2.2] where Fröberg and Hollman proved that the squares of m generic linear forms are semigeneric as long as m ≤ n + 15 and n ≤ 6. Finally, we consider algebraic independence at higher degree and prove in Theorem 11 that BinaryError-LWE samples give rise to a semi-regular sequence with high probability for m = n + 1 and for a sufficiently large field.…”

Section: Binaryerror-lwesupporting

confidence: 73%

“…The main difficulty in proving Fröberg's conjecture is to prove that the polynomial h is not identically zero or that Z O is not empty (see [29]). To prove Fröberg's conjecture it is sufficient to find one explicit family of polynomials which can be proven semi-regular for any m and n. Proving Assumptions 1 or 2 would provide such family and hence prove Fröberg's conjecture.…”

Section: Binaryerror-lwementioning

confidence: 99%

“…Furthermore, any non-trivial partial results on our assumptions would lead to progress on the general Fröberg's conjecture. over a sufficiently big finite field, sequences with the same number of polynomials (m) and same number of variables (n), m = n+1 in characteristic 0, m polynomials of degree 2 with n ≤ 11, and m polynomials of degree 3 with n ≤ 8 [20, 28,29]. Indeed, Fröberg and Hollman [29] already investigated semi-regularity for powers of generic linear forms.…”

Section: Binaryerror-lwementioning

confidence: 99%

“…over a sufficiently big finite field, sequences with the same number of polynomials (m) and same number of variables (n), m = n+1 in characteristic 0, m polynomials of degree 2 with n ≤ 11, and m polynomials of degree 3 with n ≤ 8 [20, 28,29]. Indeed, Fröberg and Hollman [29] already investigated semi-regularity for powers of generic linear forms. In characteristic 0, [29, Lemma 2.1] proves that a sequence of n + 1 squares of generic linear forms in n variables is generically semi-regular.…”

Section: Binaryerror-lwementioning

confidence: 99%

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Algebraic algorithms for LWE problems

Albrecht

Cid

Faugère

et al. 2015

ACM Commun. Comput. Algebra

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Abstract. The Learning with Errors (LWE) problem, proposed by Regev in 2005, has become an ever-popular cryptographic primitive, due mainly to its simplicity, flexibility and convincing theoretical arguments regarding its hardness. Among the main proposed approaches to solving LWE instancesnamely, lattice algorithms, combinatorial algorithms, and algebraic algorithms -the last is the one that has received the least attention in the literature, and is the focus of this paper. We present a detailed and refined complexity analysis of the original Arora-Ge algorithm, which reduced LWE to solving a system of high-degree, error-free polynomials. Moreover, we generalise their method and establish the complexity of applying Gröbner basis techniques from computational commutative algebra to solving LWE. As a result, we show that the use of Gröbner basis algorithms yields an exponential speed-up over the basic Arora-Ge algorithm. On the other hand, our results show that such techniques do not yield a subexponential algorithm for the LWE problem. We also apply our algebraic algorithm to the BinaryError-LWE problem, which was recently introduced by Micciancio and Peikert. We show that BinaryError-LWE in dimension n can be solved in subexponential time given access to a quasi-linear number of samples m under a regularity assumption.We also give precise complexity bounds for BinaryError-LWE given access to linearly many samples. Our approach outperforms the best currently-known generic heuristic exact CVP solver as soon as m/n ≥ 6.6. The results in this work depend crucially on the assumption that the encountered systems have no special structure. We give experimental evidence that this assumption holds and also prove the assumption in some special cases. Therewith, we also make progress towards proving Fröberg's long-standing conjecture from algebraic geometry.

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Section: Binaryerror-lwesupporting

confidence: 73%

Section: Binaryerror-lwementioning

confidence: 99%

Section: Binaryerror-lwementioning

confidence: 99%

Section: Binaryerror-lwementioning

confidence: 99%

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Algebraic algorithms for LWE problems

Albrecht

Cid

Faugère

et al. 2015

ACM Commun. Comput. Algebra

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“…It should be mentioned that the Hilbert series of the non-commutative graded algebra defined by the squares of s linear forms with generic coefficients is not the same as the one of the non-commutative graded algebras defined by s quadratic forms with generic coefficients, whereas they may be the same in the commutative case [6]. In fact, using the same technique we can prove the following result: …”

Section: Koszul Property For Generic Pointsmentioning

confidence: 86%

Koszul property for points in projective spaces

Conca¹,

Trung²,

Valla³

2001

MATH. SCAND.

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A graded K-algebra R is said to be Koszul if the minimal R-free graded resolution of K is linear. In this paper we study the Koszul property of the homogeneous coordinate ring R of a set of s points in the complex projective space P n . Kempf proved that R is Koszul if s ≤ 2n and the points are in general linear position. If the coordinates of the points are algebraically independent over Q, then we prove that R is Koszul if and only if s ≤ 1 + n + n 2 /4. If s ≤ 2n and the points are in linear general position, then we show that there exists a system of coordinates x 0 , . . . , x n of P n such that all the ideals (x 0 , x 1 , . . . , x i ) with 0 ≤ i ≤ n have a linear R-free resolution.

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Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables

Snellman

1998

Journal of Symbolic Computation

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If K is a field, let the ring R consist of finite sums of homogeneous elements inThen, R contains M, the free semi-group on the countable set of variables {x 1 , x 2 , x 3 , . . .}. In this paper, we generalize the notion of admissible order from finitely generated sub-monoids of M to M itself; assume that > is such an admissible order on M. We show that we can define leading power products, with respect to >, of elements in R , and thus the initial ideal gr(I) of an arbitrary ideal I ⊂ R . If I is what we call a locally finitely generated ideal, then we show that gr(I) is also locally finitely generated; this implies that I has a finite truncated Gröbner basis up to any total degree. We give an example of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic initial order > lex on M.

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Hilbert Series for Ideals Generated by Generic Forms

Cited by 25 publications

References 4 publications

Algebraic algorithms for LWE problems

Algebraic algorithms for LWE problems

Koszul property for points in projective spaces

Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables

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