Computer algebra symbolic and algebraic computation

Buchberger, Bruno; Collins, George E.; Loos, Rüdiger; Albrecht, R.

doi:10.1145/1089310.1089312

Cited by 108 publications

(91 citation statements)

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“…They provide an algorithmic solution for solving several problems related to polynomial systems (some of them can be found in [1]). The historical method for computing Gröbner bases is Buchberger's algorithm [8,7]. Recently, more efficient algorithms have been proposed.…”

Section: Definition 2 For Any N-uplementioning

confidence: 99%

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Cryptanalysis of 2R− Schemes

Faugère

Perret

2006

Lecture Notes in Computer Science

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Abstract. In this paper, we study the security of 2R − schemes [17,18], which are the "minus variant" of two-round schemes. This variant consists in removing some of the n polynomials of the public key, and permits to thwart an attack described at Crypto'99 [25] against two-round schemes. Usually, the "minus variant" leads to a real strengthening of the considered schemes. We show here that this is actually not true for 2R − schemes. We indeed propose an efficient algorithm for decomposing 2R − schemes. For instance, we can remove up to n 2 equations and still be able to recover a decomposition in O(n 12 ). We provide experimental results illustrating the efficiency of our approach. In practice, we have been able to decompose 2R − schemes in less than a handful of hours for most of the challenges proposed by the designers [18]. We believe that this result makes the principle of two-round schemes, including 2R − schemes, useless.

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Section: Definition 2 For Any N-uplementioning

confidence: 99%

“…The historical method for computing Gröbner bases is Buchberger's algorithm [8,7]. Recently, more efficient algorithms have been proposed.…”

Section: Appendix Amentioning

confidence: 99%

Cryptanalysis of 2R− Schemes

Faugère

Perret

2006

Lecture Notes in Computer Science

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“…A wellestablished approach to solve PoSSo is to compute a Gröbner basis [7,11,12]. The cost of solving a (zero-dimensional, i.e., finite number of solutions) system of m non-linear equations in n variables with the F 5 algorithm [4,16] is O n+Dreg Dreg ω , where D reg is the maximum degree reached during the Gröbner basis computation, and ω is the matrix multiplication exponent (or the linear-algebra constant) as defined in [47,Chapter 12].…”

Section: Theorem 2 ([25]mentioning

confidence: 99%

Practical Cryptanalysis of a Public-Key Encryption Scheme Based on New Multivariate Quadratic Assumptions

Albrecht

Faugère

Fitzpatrick

et al. 2014

Public-Key Cryptography – PKC 2014

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Abstract. In this paper, we investigate the security of a public-key encryption scheme introduced by Huang, Liu and Yang (HLY) at PKC'12. This new scheme can be provably reduced to the hardness of solving a set of quadratic equations whose coefficients of highest degree are chosen according to a discrete Gaussian distributions. The other terms being chosen uniformly at random. Such a problem is a variant of the classical problem of solving a system of non-linear equations (PoSSo), which is known to be hard for random systems. The main hypothesis of Huang, Liu and Yang is that their variant is not easier than solving PoSSo for random instances. In this paper, we disprove this hypothesis. To this end, we exploit the fact that the new problem proposed by Huang, Liu and Yang reduces to an easy instance of the Learning With Errors (LWE) problem. The main contribution of this paper is to show that security and efficiency are essentially incompatible for the HLY proposal. That is, one cannot find parameters which yield a secure and a practical scheme. For instance, we estimate that a public-key of at least 1.03 GB is required to achieve 80-bit security against known attacks. As a proof of concept, we present practical attacks against all the parameters proposed Huang, Liu and Yang. We have been able to recover the private-key in roughly one day for the first challenge i.e. Case 1 proposed by HLY and in roughly three days for the second challenge i.e. Case 2 .

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“…In order to apply RifSimp to equations (1,2) with M and F given by (3) -(4), we first convert the trigonometric functions to polynomials using the Weierstrass transformation, which is cos η = (1− u(t) 2 )/(1 + u(t) 2 ), sin η = 2u(t)/(1 + u(t) 2 ). This yields a rational polynomial differential system.…”

Section: Application To Spinning Topmentioning

confidence: 99%

Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems

Zhou

Jeffrey

Reid

et al. 2005

Computer Algebra and Geometric Algebra With Applications

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Abstract. The RifSimp package in Maple transforms a set of differential equations to Reduced Involutive Form. This paper describes the application of RifSimp to challenging real-world problems found in engineering design and modelling. RifSimp was applied to sets of equations arising in the dynamical studies of multibody systems. The equations were generated by the Maple package Dynaflex, which takes as input a graph-like description of a multibody mechanical system and generates a set of differential equations with algebraic constraints. Application of the standard RifSimp procedure to such Differential Algebraic Equations can require large amounts of computer memory and time, and can fail to finish its computations on larger problems.We discuss the origin of these difficulties and propose an Implicit Reduced Involutive Form to assist in alleviating such problems. This form is related to RifSimp form by the symbolic inversion of a matrix. For many applications such as numerically integrating the multibody dynamical equations, the extra cost of symbolically inverting the matrix to obtain explicit RifSimp form can be impractical while Implicit Reduced Involutive Form is sufficient.An approach to alleviating expression swell involving a hybrid analytic polynomial computation is discussed. This can avoid the excessive expression swell due to the usual method of transforming the entire input analytic differential system to polynomial form, by only applying this method in intermediate computations when it is required.

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Computer algebra symbolic and algebraic computation

Cited by 108 publications

References 0 publications

Cryptanalysis of 2R− Schemes

Cryptanalysis of 2R− Schemes

Practical Cryptanalysis of a Public-Key Encryption Scheme Based on New Multivariate Quadratic Assumptions

Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems

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