Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

Faugère, Jean-Charles; Rahmany, Sajjad

doi:10.1145/1576702.1576725

Cited by 37 publications

(33 citation statements)

References 7 publications

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“…Moreover, computing a Hironaka decomposition can be a difficult task. In the case where the invariant ring is not a polynomial algebra one can use also SAGBI Gröbner bases, see for instance [27]; we will not need this strategy in this work.…”

Section: Theorem 32 (Hilbert's Finiteness Theorem) the Invariant Rimentioning

confidence: 99%

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

et al. 2013

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ABSTRACT. In 2004, an algorithm is introduced to solve the DLP for elliptic curves defined over a non prime finite field F q n . One of the main steps of this algorithm requires decomposing points of the curve E(F q n ) with respect to a factor base, this problem is denoted PDP. In this paper, we will apply this algorithm to the case of Edwards curves, the well-known family of elliptic curves that allow faster arithmetic as shown by Bernstein and Lange. More precisely, we show how to take advantage of some symmetries of twisted Edwards and twisted Jacobi intersections curves to gain an exponential factor 2 ω(n−1) to solve the corresponding PDP where ω is the exponent in the complexity of multiplying two dense matrices. Practical experiments supporting the theoretical result are also given. For instance, the complexity of solving the ECDLP for twisted Edwards curves defined over F q 5 , with q ≈ 2 64 , is supposed to be ∼ 2 160 operations in E(F q 5 ) using generic algorithms compared to 2 130 operations (multiplications of two 32-bits words) with our method. For these parameters the PDP is intractable with the original algorithm.The main tool to achieve these results relies on the use of the symmetries and the quasi-homogeneous structure induced by these symmetries during the polynomial system solving step. Also, we use a recent work on a new algorithm for the change of ordering of Gröbner basis which provides a better heuristic complexity of the total solving process.

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Section: Theorem 32 (Hilbert's Finiteness Theorem) the Invariant Rimentioning

confidence: 99%

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

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“…All computation of the reduced Gröbner basis of I D would implicitly use the grading R = g∈Ĝ R g since it computes S-polynomials. There exist several versions of the F 5 -algorithm (see [7,5]), we present here a variant of the matrix version. The [7,5]).…”

Section: Abelian-f 5 Algorithmmentioning

confidence: 99%

“…There exist several versions of the F 5 -algorithm (see [7,5]), we present here a variant of the matrix version. The [7,5]). The key of the Abelian-F 5 algorithm is the following : the polynomials f i are G D -homogeneous, and also the polynomials m μ f i .…”

Section: Abelian-f 5 Algorithmmentioning

confidence: 99%

“…In [2] Colin proposes to use invariants [19] to solve the system. This method is very efficient if the Hironaka Decomposition of the ring of invariants is simple, but for the Cyclic-n problem [12] for example, it seems better to use a second method based on SAGBI Gröbner Basis techniques [7]. However, it remains an open issue to solve efficiently the system in the general case.…”

Section: Introductionmentioning

confidence: 99%

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Gröbner bases of ideals invariant under a commutative group

Faugère

Svartz

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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We propose efficient algorithms to compute the Gröbner basis of an ideal I ⊂ k[x 1 ,...,x n ] globally invariant under the action of a commutative matrix group G, in the non-modular case (where char(k) doesn't divide |G|). The idea is to simultaneously diagonalize the matrices in G, and apply a linear change of variables on I corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on I are diagonal. This action induces a grading on the ring R = k[x 1 ,...,x n ], compatible with the degree, indexed by a group related to G, that we call G-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into |G| submatrices of roughly the same size. In the same way, we are able to split the canonical basis of R/I (the staircase) if I is a zero-dimensional ideal. Therefore, we derive abelian versions of the classical algorithms F 4 , F 5 or FGLM. Moreover, this new variant of F 4 /F 5 allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of F 4 . Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.

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“…An important feature of sparse GBs is that their definition depends only on the ambient semigroup algebra and not on an embedding in a polynomial algebra. In this sense, they differ conceptually from SAGBI bases, even though the sparse-FGLM algorithm has similarities with the SAGBI-FGLM algorithm proposed in [16]. In the special case S M = N n , sparse Gröbner bases in k[S M ] are classical Gröbner bases, and sparse-FGLM is the usual FGLM algorithm.…”

Section: Introductionmentioning

confidence: 99%

Sparse Gröbner bases

Faugère

Spaenlehauer

Svartz³

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a semigroup algebra, i.e. an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce sparse Gröbner bases, an analog of classical Gröbner bases for semigroup algebras, and we propose sparse variants of the F 5 and FGLM algorithms to compute them. Our prototype "proof-of-concept" implementation shows large speed-ups (more than 100 for some examples) compared to optimized (classical) Gröbner bases software. Moreover, in the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope P ⊂ R n and under regularity assumptions, we prove complexity bounds which depend on the combinatorial properties of P. These bounds yield new estimates on the complexity of solving 0-dim systems where all polynomials share the same Newton polytope (unmixed case). For instance, we generalize the bound min(n 1 , n 2 ) + 1 on the maximal degree in a Gröbner basis of a 0-dim. bilinear system with blocks of variables of sizes (n 1 , n 2 ) to the multilinear case: n i − max(n i ) + 1. We also propose a variant of Fröberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems.

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Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

Abstract: International audienc

Cited by 37 publications

References 7 publications

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

Gröbner bases of ideals invariant under a commutative group

Sparse Gröbner bases

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