David Thomson - Research
Research interests
I'm interested in:
- all aspects of finite fields and their applications;
- efficient arithmetic over finite fields;
- combinatorial algorithms, particularly over finite fields;
- cryptography, coding theory;
- various combinatorial objects, such as Latin squares and hypercubes;
- parallel implementations of combinatorial algorithms,
I am involved in the
Ottawa-Carleton Discrete Math group.
Thesis
A copy of my PhD Thesis, entitled ``On difference maps and their cryptographic applications'', can be found here.
Book Chapters
Refereed Journal Publications (click for preprints)
-
Artin-Schreier extensions of normal bases (with C. Weir), under revision.
- Generalized Sudoku arrays with strong regularity conditions (with M. Huggan, G. L. Mullen and B. Stevens), Designs, Codes and Cryptography, (available Online First).
- Fast and simple modular interpolation using factorial representations (with G. L. Mullen and D. Panario), American Mathematical Monthly to appear.
- Ambiguity and deficiency of permutations over finite fields with linearized difference map (with D. Panario, A. Sakzad, B. Stevens and Q. Wang), IEEE Transactions on Information Theory, 59, 5616-5626, 2013.
- Existence and properties of k-normal elements over finite fields (with S. Huczynska, G. L. Mullen and D. Panario), Finite Fields and their Applications, 24, 170-183, 2013.
- Subfield value sets of polynomials over finite fields, (with W.-S. Chou, J. Gomez-Calderon, G. L. Mullen and D. Panario) Functiones et Approximatio, Commentarii Mathematici, 48, 147-165, 2012.
- Sets of orthogonal hypercubes of class r (with J. Ethier, G. L. Mullen, D. Panario and B. Stevens), Journal of Combinatorial Theory, Series A, 119, 430-439, 2012.
- Gauss periods as low complexity normal bases (with M. Christopoulou, T. Garefalakis and D. Panario), Designs, Codes and Cryptography, 62, 43-62, 2012.
- Swan-like results for binomials and trinomials over F_q, q odd (with B. Hanson and D. Panario), Designs, Codes and Cryptography, 63, 273-283, 2011.
- Efficient pth root computations in finite fields of characteristic p (with D. Panario), Designs, Codes and Cryptography , 50, 351-358, 2009.
- The trace of an optimal normal element and low complexity normal bases (with M. Christopoulou, T. Garefalakis and D. Panario), Designs, Codes and Cryptography, 49, 199-215, 2008.
- Low complexity normal bases over finite fields of characteristic two (with A. Masuda, L. Moura and D. Panario),
IEEE Transactions on Computers, 57, 990-1001, 2008.
Refereed Conference Papers (click for preprints)
- Artin-Schreier extensions of normal bases (with C. Weir), extended abstract in Workshop on Coding and Cryptography 2015, 2015.
- On the characterization of a semi-multiplicative analogue of planar functions over finite fields (with A. Muratovic-Ribic, A. Pott and Q. Wang) Contemporary Mathematics: Proceedings of the 11th International Conference on Finite Fields and their Applications, 632, 8 pages, 2015.
- Ambiguity and deficiency of reversed Dickson permutations (with D. Panario and A. Sakzad) Contemporary Mathematics: Proceedings of the 11th International Conference on Finite Fields and their Applications, 632, 12 pages, 2015.
- On the Waring problem with multivariate Dickson polynomials'' (with A. Ostafe and A. Winterhof), Contemporary Mathematics: Proceedings of the 10th International Conference on Finite Fields and their Applications, 579, 153-161, 2012.
- The trace of an optimal normal element and low complexity normal bases (with M. Christopoulou, T. Garefalakis and D. Panario), extended abstract in Workshop on Coding and Cryptography 2007 (edited by D. Augot, N. Sendrier and J.-P. Tillich), 79 - 88, 2007.
Coming Conferences
Things I've Organized
- Fields-TIMC Workshop on Functions and Geometries over Finite Fields, July 8-11, 2015.
- CanaDAM 2015 two-part minisymposium on Finite Fields and Combinatorics, June 2013.
- Fields Undergraduate Network - Discrete Mathematics, July 22, 2011.
- Carleton Finite Fields Workshop, Carleton University, July 20-23, 2010.
- Canadian Undergraduate Mathematics Conference, Carleton Univeristy, July 8-11, 2009.
Education and Experience
- Since May 2015, I have been an Adjunct Research Professor in the School of Mathematics and Statistics at Carleton University.
- From August 2015-August 2016, I was Cyber Math Fellow at the Army Cyber Institute and Assistant Professor in the Department of Mathematical Sciences at the United States Military Academy in West Point, NY.
- Between January 2013 and May 2015, I was a Visiting Scholar in the School of Mathematics and Statistics at Carleton University, sponsored by Steven Wang.
- I obtained my PhD in November, 2012 from the School of Mathematics and Statistics
at Carleton University under the supervision of Daniel Panario.
- Between August and December, 2012, I was a Lecturer in the Department of Mathematics at The Pennsylvania State University in State College, PA, USA.
- Between January and April, 2011, I studied at the Claude Shannon Institute in Dublin (Ireland) under the supervision of Gary Mullen (Pennsylvania State University) and Gary McGuire (University College Dublin/Claude Shannon Institute).
- I did my B.Math (Honours project title: ``Binary irreducible polynomials'', 2006) and M.Sc (Pure Mathematics, 2007; thesis title: ``Low complexity normal bases'') at the
School of Mathematics and Statistics at Carleton University under the supervision of
Daniel Panario.
- I did a brief stint as a PhD student (September - December 2007) in the Department of Electrical and Computer Engineering at the University of Waterloo as a member of the Communications Security Laboratory under the supervision of Guang Gong.
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