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Algebraic and Euclidean Lattices: Optimal Lattice Reduction and Beyond [article]

Thomas Espitau and Paul Kirchner and Pierre-Alain Fouque
2019 arXiv   pre-print
The recursive strategy over the rank can also be applied to the reduction of Euclidean lattices, and we can perform a reduction in asymptotically almost the same time as matrix multiplication.  ...  We introduce a framework generalizing lattice reduction algorithms to module lattices in order to practically and efficiently solve the γ-Hermite Module-SVP problem over arbitrary cyclotomic fields.  ...  : δ i,j = n h L 2 − αn h − 2 −Ω(p) ALGEBRAIC AND EUCLIDEAN LATTICES: OPTIMAL LATTICE REDUCTION AND BEYOND 33 by heuristic 1 and theorem 11.  ... 
arXiv:1912.04586v1 fatcat:v3a6uo45dncodenbrs24u3zr5q

Lattice Reduction over Imaginary Quadratic Fields [article]

Shanxiang Lyu, Christian Porter, Cong Ling
2020 arXiv   pre-print
Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms.  ...  Lastly, we numerically show the time-advantage of using algebraic LLL by considering lattice bases generated from wireless communications and cryptography.  ...  In Section II, backgrounds about quadratic fields and algebraic lattices are reviewed, and the concept of algebraic lattice reduction is induced.  ... 
arXiv:1806.03113v6 fatcat:6i2qafg6mvewhmzko3b3d4cafe

Lattice Polly Cracker cryptosystems

Massimo Caboara, Fabrizio Caruso, Carlo Traverso
2011 Journal of symbolic computation  
As a consequence, we concentrate on binomial ideals that correspond to Euclidean lattices.  ...  This paper develops the details and proofs related to computer algebra, the cryptographic details related to security, the comparison with other lattice cryptosystems and discusses the implementation.  ...  Future work will be concentrated on tying the hardness of finding a ''good'' (i.e. manageable and sufficiently fat) Gröbner basis, or a change of coordinates and a good Gröbner basis, to hard lattice problems  ... 
doi:10.1016/j.jsc.2010.10.004 fatcat:cb7pm2rmifcslk6tb3tzi7yfsy

Counting lattice points

Alexander Gorodnik, Amos Nevo
2012 Journal für die Reine und Angewandte Mathematik  
, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.  ...  We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function  ...  Counting lattice points in S-algebraic groups: height balls.  ... 
doi:10.1515/crelle.2011.096 fatcat:73hxxboukffu7a2isvijajhijy

Counting lattice points [article]

Alexander Gorodnik, Amos Nevo
2009 arXiv   pre-print
Applications include counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular  ...  (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.  ...  Counting lattice points in S-algebraic groups: height balls.  ... 
arXiv:0903.1515v1 fatcat:lvthr34srjd37ab5sl4x7ywzfa

Adaptive Compute-and-Forward with Lattice Codes Over Algebraic Integers [article]

Yu-Chih Huang, Krishna R. Narayanan, Ping-Chung Wang
2016 arXiv   pre-print
Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal  ...  This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for  ...  In [10] , instead of being confined in Euclidean domains or PIDs, we go beyond PIDs and construct lattices over rings of imaginary quadratic integers for compute-and-forward.  ... 
arXiv:1501.07740v2 fatcat:lqccugdp3vgx5dcnp3wfxhddy4

Algorithms for the Densest Sub-Lattice Problem [chapter]

Daniel Dadush, Daniele Micciancio
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
In computer science and cryptography, Rankin constants and the associated k-DSP problem in the Euclidean norm has been suggested as a building block for novel basis reduction algorithms [GHGKN06], as well  ...  , and has applications to the design of lattice reduction algorithms), and the shortest vector problem for arbitrary norms (which corresponds to setting k = 1) and its dual (k = n − 1).  ...  theory and algebraic number theory.  ... 
doi:10.1137/1.9781611973105.79 dblp:conf/soda/DadushM13 fatcat:gepxhjknzng4thpxjp3ylbbqoy

Lattice reduction using a Euclidean algorithm

A. Mujica
2017 Acta Crystallographica Section A: Foundations and Advances  
This is done in successive steps, each time solving a Diophantine linear equation by means of a Euclidean reduction algorithm in order to provide the generators of the reduced lattice.  ...  The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in theab  ...  MAT2010-21270-C04-03 and CSD2007-00045.  ... 
doi:10.1107/s2053273316015539 pmid:28042805 fatcat:awlq5frkhbfahftdii2df43dm4

Deconstruction, 2d lattice Yang-Mills, and the dynamical lattice spacing [article]

Joel Giedt
2003 arXiv   pre-print
That is, we compute observables in the fully quenched ensemble, equivalent to non-supersymmetric 2d lattice Yang-Mills with a dynamical lattice spacing and adjoint scalars.  ...  In an effort to understand these results, we examine the zero action configurations, with and without the deformation.  ...  This work was supported by the National Science and Engineering Research Council of Canada and the Ontario Premier's Research Excellence Award.  ... 
arXiv:hep-lat/0312020v3 fatcat:dp7h6riisvagndsp3hjdtlnvly

An Algebraic Family of Complex Lattices for Fading Channels With Application to Space–Time Codes

P. Dayal, M.K. Varanasi
2005 IEEE Transactions on Information Theory  
The two strategies proposed for the Frobenius norm reduction are based on the concepts of successive minima (SM) and basis reduction of an equivalent real lattice.  ...  The proposed construction is general and works for any dimension as long as a list of number fields of the same degree is available.  ...  The family of threaded algebraic space-time (TAST) codes presented in [10] are designed to achieve the full transmit diversity and also rates beyond one information symbol per channel use for a quasi-static  ... 
doi:10.1109/tit.2005.858923 fatcat:tmcvvbu3pnbnpm7igaavc7craq

Signal Codes: Convolutional Lattice Codes

Ofir Shalvi, Naftali Sommer, Meir Feder
2011 IEEE Transactions on Information Theory  
Error analysis and simulation results indicate that for the additive white Gaussian noise (AWGN) channel, convolutional lattice codes with computationally reasonable decoders can achieve low error rate  ...  We naturally term these codes "convolutional lattice codes" or alternatively "signal codes" due to the signal processing interpretation of the code construction.  ...  This technique is very effective for complexity reduction, and will be referred to as "x-range testing."  ... 
doi:10.1109/tit.2011.2158876 fatcat:gpma4tjbcvcqxgqz7vibcry4bq

Quantum stabilizer codes, lattices, and CFTs

Anatoly Dymarsky, Alfred Shapere
2021 Journal of High Energy Physics  
There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories.  ...  Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs.  ...  Acknowledgments We would like to thank Noam Elkies, Anton Gerasimov, Nikita Nekrasov, Vasily Pestun, and Eric Rains for discussions. AD is supported by the National Science Foundation under Grant No.  ... 
doi:10.1007/jhep03(2021)160 fatcat:xa33nkemh5bv7dtdtzamskkj4q

Quantum stabilizer codes, lattices, and CFTs [article]

Anatoly Dymarsky, Alfred Shapere
2020 arXiv   pre-print
There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories.  ...  Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs.  ...  Acknowledgments We would like to thank Noam Elkies, Anton Gerasimov, Nikita Nekrasov, Vasily Pestun, and Eric Rains for discussions. AD is supported by the National Science Foundation under Grant No.  ... 
arXiv:2009.01244v1 fatcat:cmie6lzgw5cqjpidpudldzhnqu

THE QCD ABACUS: A New Formulation for Lattice Gauge Theories [article]

Richard C. Brower
1997 arXiv   pre-print
This talk will emphasize the gauge and algebraic structure of the rishon or link Fermions and the special properties that may lead to fast discrete dynamics for numerical simulations and new theoretical  ...  A quantum Hamiltonian is constructed for SU(3) lattice QCD entirely from color triplet Fermions --- the standard quarks and a new Fermionic "constituent" of the gluon we call "rishons".  ...  Acknowledgments I wish to gratefully acknowledge Shailesh Chandrasekharan and Uwe-Jens Wiese for all aspects of this collaboration.  ... 
arXiv:hep-lat/9711027v1 fatcat:ga2qdj45kffn3flsnno5onmt3u

Improved Zero-Knowledge Identification with Lattices [chapter]

Pierre-Louis Cayrel, Richard Lindner, Markus Rückert, Rosemberg Silva
2010 Lecture Notes in Computer Science  
Döring, and R. Lindner. Efficiency improvement for NTRU. In A. Alkassar and J. H. Siekmann, editors, Sicherheit 2008: Sicherheit, Schutz und Zuverlässigkeit. Konferenzband der 4.  ...  ring Z[ζ], where ζ is some algebraic integer that is fixed for the reduction.  ...  Lattice Basis Reduction Experiments For our experiments we chose the sublattice dimension where lattice basis reduction algorithms like LLL/BKZ behave optimal in practice (see Section 4.4).  ... 
doi:10.1007/978-3-642-16280-0_1 fatcat:qphlhyww45avzim6wyo7bvnira
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