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Relevant bibliographies by topics / Algorithme de lenstra

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Academic literature on the topic 'Algorithme de lenstra'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Algorithme de lenstra"

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Sur, Samarendra Nath, Rabindranath Bera, Akash Kumar Bhoi, Mahaboob Shaik, and Gonçalo Marques. "Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems." Information 11, no. 6 (2020): 301. http://dx.doi.org/10.3390/info11060301.

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Massive multi-input-multi-output (MIMO) systems are the future of the communication system. The proper design of the MIMO system needs an appropriate choice of detection algorithms. At the same time, Lattice reduction (LR)-aided equalizers have been well investigated for MIMO systems. Many studies have been carried out over the Korkine–Zolotareff (KZ) and Lenstra–Lenstra–Lovász (LLL) algorithms. This paper presents an analysis of the channel capacity of the massive MIMO system. The mathematical calculations included in this paper correspond to the channel correlation effect on the channel capa

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Futa, Yuichi, and Yasunari Shidama. "Isomorphism Theorem on Vector Spaces over a Ring." Formalized Mathematics 25, no. 3 (2017): 171–78. http://dx.doi.org/10.1515/forma-2017-0016.

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Summary In this article, we formalize in the Mizar system [1, 4] some properties of vector spaces over a ring. We formally prove the first isomorphism theorem of vector spaces over a ring. We also formalize the product space of vector spaces. ℤ-modules are useful for lattice problems such as LLL (Lenstra, Lenstra and Lovász) [5] base reduction algorithm and cryptographic systems [6, 2].

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Futa, Yuichi, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. "Torsion Z-module and Torsion-free Z-module." Formalized Mathematics 22, no. 4 (2014): 277–89. http://dx.doi.org/10.2478/forma-2014-0028.

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Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

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Futa, Yuichi, and Yasunari Shidama. "Embedded Lattice and Properties of Gram Matrix." Formalized Mathematics 25, no. 1 (2017): 73–86. http://dx.doi.org/10.1515/forma-2017-0007.

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Summary In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].

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Futa, Yuichi, Hiroyuki Okazaki та Yasunari Shidama. "Torsion Part of ℤ-module". Formalized Mathematics 23, № 4 (2015): 297–307. http://dx.doi.org/10.1515/forma-2015-0024.

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Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

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Futa, Yuichi, та Yasunari Shidama. "Divisible ℤ-modules". Formalized Mathematics 24, № 1 (2016): 37–47. http://dx.doi.org/10.1515/forma-2016-0004.

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Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

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Futa, Yuichi, та Yasunari Shidama. "Lattice of ℤ-module". Formalized Mathematics 24, № 1 (2016): 49–68. http://dx.doi.org/10.1515/forma-2016-0005.

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Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

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OSWALD, Nicola M. R. "The Doeblin-Lenstra Conjecture for a Complex Continued Fraction Algorithm." Tokyo Journal of Mathematics 40, no. 1 (2017): 45–52. http://dx.doi.org/10.3836/tjm/1502179214.

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Futa, Yuichi, Hiroyuki Okazaki, and Yasunari Shidama. "Submodule of free Z-module." Formalized Mathematics 21, no. 4 (2013): 273–82. http://dx.doi.org/10.2478/forma-2013-0029.

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Abstract In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.

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Lapidus, Michel L., Machiel van Frankenhuijsen, and Edward K. Voskanian. "Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm." Mathematics 9, no. 6 (2021): 591. http://dx.doi.org/10.3390/math9060591.

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The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of t

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Dissertations / Theses on the topic "Algorithme de lenstra"

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Viry, Guy. "Algorithmes de factorisation des polynômes à coefficients entiers." Nancy 1, 1989. http://www.theses.fr/1989NAN10489.

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Cette thèse présente les algorithmes classiques de factorisation des polynômes à une ou plusieurs variables. Dans le cas des polynômes à une variable, deux nouvelles méthodes sont proposées. Dans la première, on calcule un facteur linéaire modulo un nombre premier P, puis on définit un multiple de ce facteur qui divise P sur ZX. Dans la seconde méthode, le calcul des produits des facteurs de P modulo P#N est remplacé par le calcul des sommes des images de ces facteurs de P. Dans le cas des polynômes à plusieurs variables, on donne deux méthodes pour diminuer les calculs de la dernière étape de

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McKee, James. "Some elliptic curve algorithms." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319564.

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Kapfunde, Goodwell. "Near-capacity sphere decoder based detection schemes for MIMO wireless communication systems." Thesis, University of Hertfordshire, 2013. http://hdl.handle.net/2299/11350.

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The search for the closest lattice point arises in many communication problems, and is known to be NP-hard. The Maximum Likelihood (ML) Detector is the optimal detector which yields an optimal solution to this problem, but at the expense of high computational complexity. Existing near-optimal methods used to solve the problem are based on the Sphere Decoder (SD), which searches for lattice points confined in a hyper-sphere around the received point. The SD has emerged as a powerful means of finding the solution to the ML detection problem for MIMO systems. However the bottleneck lies in the de

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Gao, Liyan. "Ellipsoidal approximation to polytopes and computational study of Lenstra's algorithm." Thesis, 2002. http://hdl.handle.net/1911/18081.

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General integer programming is an important mathematical approach for many decision-making problems. In this field, a major theoretical breakthrough came in 1983 when H. W. Lenstra, Jr. proposed a polynomial-time algorithm for general integer programs while the number of variables is fixed. Two key ingredients of Lenstra's algorithm are ellipsoidal approximation of polytopes and lattice basis reduction. However, the lack of practically efficient algorithms and software for the ellipsoidal approximation of polytopes has made it difficult to study the computational properties of Lenstra's algori

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Book chapters on the topic "Algorithme de lenstra"

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Hlawka, Edmund, Rudolf Taschner, and Johannes Schoißengeier. "The Algorithm of Lenstra, Lenstra and Lovász." In Geometric and Analytic Number Theory. Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/978-3-642-75306-0_7.

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Abbott, JA, and JH Davenport. "Polynomial factorisation: an exploration of Lenstra's algorithm." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/3-540-51517-8_144.

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Conference papers on the topic "Algorithme de lenstra"

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Menon, U. Vivek, Naga Raju Challa, and Kalapraveen Bagadi. "Lenstra Lenstra Lovász (LLL) Assisted Likelihood Ascent Search (LAS) Algorithm for Signal Detection in Massive MIMO." In 2019 International Conference on Vision Towards Emerging Trends in Communication and Networking (ViTECoN). IEEE, 2019. http://dx.doi.org/10.1109/vitecon.2019.8899594.

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Zapotecas-Martinez, Saul, Cuauhtemoc Mancillas-Lopez, Francisco Rodriguez-Henriquez, and Nareli Cruz-Cortes. "Reconfigurable Hardware Implementation of the Lenstra Factorization Algorithm." In 2006 3rd International Conference on Electrical and Electronics Engineering. IEEE, 2006. http://dx.doi.org/10.1109/iceee.2006.251939.

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