2 results
A sieve algorithm based on overlattices
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- Anja Becker, Nicolas Gama, Antoine Joux
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- Journal:
- LMS Journal of Computation and Mathematics / Volume 17 / Issue A / 2014
- Published online by Cambridge University Press:
- 01 August 2014, pp. 49-70
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In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.
This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$ in time $2^{0.3774\, n}$ using memory $2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.
Use of vitamin D supplements during infancy in an international feeding trial
- Eveliina Lehtonen, Anne Ormisson, Anita Nucci, David Cuthbertson, Susa Sorkio, Mila Hyytinen, Kirsi Alahuhta, Carol Berseth, Marja Salonen, Shayne Taback, Margaret Franciscus, Teba González-Frutos, Tuuli E Korhonen, Margaret L Lawson, Dorothy J Becker, Jeffrey P Krischer, Mikael Knip, Suvi M Virtanen, , Thomas Mandrup-Poulsen, Elias Arjas, Åke Lernmark, Barbara Schmidt, Jeffrey P. Krischer, Hans K. Åkerblom, Mila Hyytinen, Mikael Knip, Katriina Koski, Matti Koski, Eeva Pajakkala, Marja Salonen, David Cuthbertson, Jeffrey P. Krischer, Linda Shanker, Brenda Bradley, Hans-Michael Dosch, John Dupré, William Fraser, Margaret Lawson, Jeffrey L. Mahon, Mathew Sermer, Shayne P. Taback, Dorothy Becker, Margaret Franciscus, Anita Nucci, Jerry Palmer, Minna Pekkala, Suvi M. Virtanen, Jacki Catteau, Neville Howard, Patricia Crock, Maria Craig, Cheril L. Clarson, Lynda Bere, David Thompson, Daniel Metzger, Colleen Marshall, Jennifer Kwan, David K. Stephure, Daniele Pacaud, Wendy Schwarz, Rose Girgis, Marilyn Thompson, Shayne P. Taback, Daniel Catte, Margaret L. Lawson, Brenda Bradley, Denis Daneman, Mathew Sermer, Mary-Jean Martin, Valérie Morin, Lyne Frenette, Suzanne Ferland, Susan Sanderson, Kathy Heath, Céline Huot, Monique Gonthier, Maryse Thibeault, Laurent Legault, Diane Laforte, Elizabeth A. Cummings, Karen Scott, Tracey Bridger, Cheryl Crummell, Robyn Houlden, Adriana Breen, George Carson, Sheila Kelly, Koravangattu Sankaran, Marie Penner, Richard A. White, Nancy King, James Popkin, Laurie Robson, Eva Al Taji, Irena Aldhoon, Pavla Mendlova, Jan Vavrinec, Jan Vosahlo, Ludmila Brazdova, Jitrenka Venhacova, Petra Venhacova, Adam Cipra, Zdenka Tomsikova, Petra Krckova, Pavla Gogelova, Ülle Einberg, Mall-Anne Riikjärv, Anne Ormisson, Vallo Tillmann, Päivi Kleemola, Anna Parkkola, Heli Suomalainen, Anna-Liisa Järvenpää, Anu-Maaria Hämälainen, Hannu Haavisto, Sirpa Tenhola, Pentti Lautala, Pia Salonen, Susanna Aspholm, Heli Siljander, Carita Holm, Samuli Ylitalo, Raisa Lounamaa, Anja Nuuja, Timo Talvitie, Kaija Lindström, Hanna Huopio, Jouni Pesola, Riitta Veijola, Päivi Tapanainen, Abram Alar, Paavo Korpela, Marja-Liisa Käär, Taina Mustila, Ritva Virransalo, Päivi Nykänen, Bärbel Aschemeier, Thomas Danne, Olga Kordonouri, Dóra Krikovszky, László Madácsy, Yeganeh Manon Khazrai, Ernesto Maddaloni, Paolo Pozzilli, Carla Mannu, Marco Songini, Carine de Beaufort, Ulrike Schierloh, Jan Bruining, Margriet Bisschoff, Aleksander Basiak, Renata Wasikowa, Marta Ciechanowska, Grazyna Deja, Przemyslawa Jarosz-Chobot, Agnieszka Szadkowska, Katarzyna Cypryk, Malgorzata Zawodniak-Szalapska, Luis Castano, Teba Gonzalez Frutos, Mirentxu Oyarzabal, Manuel Serrano-Ríos, María Teresa Martínez-Larrad, Federico Gustavo Hawkins, Dolores Rodriguez Arnau, Johnny Ludvigsson, Malgorzata Smolinska Konefal, Ragnar Hanas, Bengt Lindblad, Nils-Osten Nilsson, Hans Fors, Maria Nordwall, Agne Lindh, Hans Edenwall, Jan Aman, Calle Johansson, Margrit Gadient, Eugen Schoenle, Dorothy Becker, Ashi Daftary, Margaret Franciscus, Carol Gilmour, Jerry Palmer, Rachel Taculad, Marilyn Tanner-Blasiar, Neil White, Uday Devaskar, Heather Horowitz, Lisa Rogers, Roxana Colon, Teresa Frazer, Jose Torres, Robin Goland, Ellen Greenberg, Maudene Nelson, Holly Schachner, Barney Softness, Jorma Ilonen, Massimo Trucco, Lynn Nichol, Erkki Savilahti, Taina Härkönen, Mikael Knip, Outi Vaarala, Kristiina Luopajärvi, Hans-Michael Dosch
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- Journal:
- Public Health Nutrition / Volume 17 / Issue 4 / April 2014
- Published online by Cambridge University Press:
- 24 June 2013, pp. 810-822
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Objective
To examine the use of vitamin D supplements during infancy among the participants in an international infant feeding trial.
DesignLongitudinal study.
SettingInformation about vitamin D supplementation was collected through a validated FFQ at the age of 2 weeks and monthly between the ages of 1 month and 6 months.
SubjectsInfants (n 2159) with a biological family member affected by type 1 diabetes and with increased human leucocyte antigen-conferred susceptibility to type 1 diabetes from twelve European countries, the USA, Canada and Australia.
ResultsDaily use of vitamin D supplements was common during the first 6 months of life in Northern and Central Europe (>80 % of the infants), with somewhat lower rates observed in Southern Europe (>60 %). In Canada, vitamin D supplementation was more common among exclusively breast-fed than other infants (e.g. 71 % v. 44 % at 6 months of age). Less than 2 % of infants in the USA and Australia received any vitamin D supplementation. Higher gestational age, older maternal age and longer maternal education were study-wide associated with greater use of vitamin D supplements.
ConclusionsMost of the infants received vitamin D supplements during the first 6 months of life in the European countries, whereas in Canada only half and in the USA and Australia very few were given supplementation.