Dr. Donald Mills

For additional information, read my (short!) biography.
 
I have many research interests (click here for my research statement). My primary interest is in the theory of finite fields and the applications of this theory to communications issues. I also have a strong interest in problems concerning Diophantine equations, algorithms for solving discrete math problems and their computational complexities, various aspects of algebraic graph theory, and just about anything having to do with combinatorial number theory.
 
While I was at West Point, I did work regarding the existence of primitive elements in cubic extensions of finite fields where the elements take on the form aC+b, C being a defining element of the cubic extension and a and b belonging to the underlying field. Click here to view a list of prime power values q for which an affirmative answer to the problem can be given (that is, for any such C there exist elements a and b in the ground field  GF(q) with a nonzero such that aC+b is a primitive element of GF(q^3)) via a sieving method due to S. Cohen of the University of Glasgow. The sieving method uses the factorization of q^3 - 1, and on the page listing said values of q I give the factors e and f with ef = q^3 - 1 that can be used to affirm the existence of primitive elements of the desired form. The paper corresponding to this is entitled "Primitive roots in cubic extensions of finite fields" (joint with G. McNay), and has recently appeared in the proceedings for the Sixth International Conference on Finite Fields and Applications.

I have a fascination with numbers -- one would hope so, given my line of work! In keeping with this, I am (slowly) building a page dedicated to sequences I find interesting. (I would welcome any input to this page!) For an exhaustive list of integer sequences, go to the On-Line Encyclopedia of Integer Sequences web page (link located in "Important Links" section).
 

Graduate Education: I received an M.S. in Mathematical Sciences in May 1995, and earned the Ph. D. in Mathematical Sciences in August 1999, both from Clemson University.
Publications
Ongoing Research
Presentations and Participations

Publications

1. Root-based polynomial compositions over finite fields. Ph.D. thesis, Clemson University, August 1999. abstract
2. “Computing composed products of polynomials,” Finite fields: theory, applications, and algorithms (Waterloo, ON, 1997), 1-15. Contemp. Math. 225, Amer. Math. Soc., Providence, RI, 1999. (with J.V. Brawley and S. Gao)
3. “Associative rational functions in two variables,” Finite Fields and Applications (D. Jungnickel and H. Niederreiter, eds.), 43-56. Proceedings of the Fifth International Conference on Finite Fields and Applications. Springer-Verlag, Berlin, 2001. (with J.V. Brawley and S. Gao)
4. "Factorizations of root-based polynomial compositions," Discrete Mathematics 240 (2001), no. 1-3, 161-173.
5. (*) “On the crosscorrelation of sequences with decimation factor d = (p^n + 1)/(p+1) - (p^n - 1)/2,” Applicable Algebra in Engineering, Communication, and Computing 12 (2001), no. 3, 255-263. (with Z. Hu, X. Li, E. Mueller, W. Sun, W. Willems, Y. Yang,  and Z. Zhang)
6. “On the evaluation of Weil sums of Dembowski-Ostrom polynomials,”  Journal of Number Theory 92 (2002), 87-98. pdf
7. “A bivariate analogue to the composed product of polynomials,” Algebra Colloq. 10 (2003), no. 4, 451-460.  (with K. Neuerburg) pdf
8. "Primitive roots in cubic extensions of finite fields," Finite Fields with Applications to Coding Theory, Cryptography and Related Areas (G.L. Mullen, H. Stichtenoth, H. Tapia-Recillas, eds.), 239-250. Proceedings of the Sixth International Conference on Finite Fields and Applications. Springer-Verlag, Berlin, 2002. (with G. McNay) pdf
9. “Two divisibility properties of Smarandache sequences,” International Journal of Mathematics, Game Theory, and Algebra 12 (2002), 1-8.
10. “APN functions in odd characteristic,” Discrete Mathematics 267 (2003), no. 1-3, 95-112. (with H. Dobbertin, E. Mueller, A. Pott, W. Willems)
11. “Primitive polynomials with first and second coefficients prescribed,” Finite Fields and Their Applications 9 (2003), no. 3, 334-350. (with S. Cohen) pdf
12. "Root-based compositions of quasiordinary polynomials and their generalizations: structure, geometric interpretations,  and decomposition results," Communications in Algebra 32 (2004), no. 1, 1-18. (with K. Neuerburg) preprint
13. "Existence of primitive polynomials with three coefficients prescribed," to appear in JP J. Algebra Number Theory Appl. 4 (2004), no. 1, 1-22. submitted version expanded version
14. "Counting the number of unavoidable sets of a given size in a finite set of integers," INTEGERS 5, no. 2, 16 pages (electronic). (with P. Mitchell) [link]
15. "Some observations on subset sum representations," submitted. [preprint]
16. "Polynomials generated by the Fibonacci sequence", [preprint].
17. “Correlation properties of sequences generated by polynomial compositions,” a technical report appearing in the 8^th Annual ARL/USMA Technical Symposium (AUTS), published November 2000 in the symposium proceedings (with C.T. Retter and A.B. Cooper; non-refereed).
18. "Correlation and linear span properties of nonlinear binary sequences generated by elements of large orders," a technical report appearing in the 9^th Annual AUTS, published November 2001 in the symposium proceedings (with C.T. Retter and A.B. Cooper; non-refereed).
19. "Existence and applications of primitive composed sums over finite fields," a technical report appearing in the 10^th Annual AUTS, published November 2001 in the symposium proceedings (with C.T. Retter; non-refereed).

Ongoing Research

(ranging from work in its initial stages to near-completed projects)

1. Subset sum representations of sets of (Gaussian) integers of size n with sets of (Gaussian) integers of size at most n
2. Roles played by various polynomials in forming geometries in PG(n,q)
3. Evaluations/estimations of various exponential sums over finite fields (including additive sums with prescribed-weight exponents and Brewer-like sums)
4. Ducci-type discrete dynamical systems over finite fields
5. Avoidable sets of positive integers and generalizations of this concept
6. "Good" algebraic Goppa codes over finite fields generated by certain polynomial compositions
7. Properties of certain classes of polynomials over finite fields, including value set and irreducibility determinations

---

 

Conference and Seminar Participations

2006

1. Gave a talk on January 12 on polynomial sequences whose coefficients are Fibonacci polynomials at the Joint Mathematics Meetings (January 12-15, San Antonio, TX).

2005

1. Gave a talk on January 6 on connections between subset sum representations and totally dependent sequences at the Joint Mathematics Meetings (January 5-8, Atlanta, GA).
2. Participated in the 2005 conference on Combinatorial and Additive Number Theory (CANT 2005), held at the CUNY Graduate Center in May 2005.
3. Delivered an invited talk on polynomial sequences whose coefficients are Fibonacci polynomials at the Integers Conference, University of West Georgia, October 27-30, 2005.

2004

1. Gave a talk on November 6 on subset sum representations of totally dependent sequences at the 2004 Southern Regional Weekend Algebra Conference, held on the SELU campus, Hammond, LA. Link
2. Delivered a talk on February 26 on possible connections between composed products of univariate polynomials and algebraic Goppa codes in the Algebra Seminar at SIUC. Transparencies
3. Co-conducted a special session on coding and design-theoretic applications of polynomials at the Joint Mathematics Meetings held January 7-10 in Phoenix, AZ. Link

2003

1. Delivered an invited talk on avoidable sets of positive integers at the Integers Conference, University of West Georgia, October 31-November 2. Transparencies Accompanying Graph
2. Gave a series of talks on multi-dimensional linear recurrence sequences over rings and modules in the Algebra Seminar at SIUC, October & November. Notes (PDF)
3. Gave a series of talks on properties of Dickson polynomials in the Speladion Seminar at SIUC, October & November.
4. Gave an invited talk on the number of unavoidable sets of positive integers of various sizes at the 2003 conference on Applications of Computer Algebra, held in Raleigh, NC July 28-31. (See above for preprint and abstract for the talk.)
5. Gave an invited talk on the existence of primitive polynomials over finite fields with certain coefficients prescribed in advance in Clemson University's Algebra/Discrete Mathematics Seminar on March 6.
6. Delivered a colloquium talk on avoidable sets of positive integers at Montclair St. University on March 4.
7. Gave an invited lecture on certain cryptographic and coding-theoretic applications of primitive elements at the U.S. Military Academy on March 3.
8. Delivered talks in February on avoidable families of sets of positive integers in the Combinatorics and Graph Theory Seminar at SIUC.

2002

1. Gave talks in November on exponential sums over Galois rings in the Algebra Seminar at SIUC.
2. Gave an invited talk on the existence and applications of primitive composed sums to communications issues at the Tenth Annual USMA/ARL Technical Symposium, held at Aberdeen Proving Ground in Aberdeen, MD on November 6.
3. Gave talks in September on primitive polynomials over finite fields in the Algebra Seminar at SIUC.

4. Conducted a seminar at USMA during the first half of the Spring 2002 semester on the theory and applications of elliptic curves over finite fields.
5. Gave a talk on a multivariate analogue to the composed product operation at the AMS Session on Field Theory and Linear Algebra at the Joint Mathematics Meetings held January 6-9 in San Diego, CA. At the same meeting, I moderated an MAA panel, sponsored by CRAFTY, on issues concerning the teaching of discrete mathematics to students in the first two years of their college experience.

2001
 

1. Gave an invited talk on linear span and correlation properties of certain multiple-trace sequences at the Ninth Annual USMA/ARL Technical Symposium, held at Aberdeen Proving Ground in Aberdeen, MD on November 5.
2. Gave an invited colloquium talk on properties of root-based polynomial compositions at Midwestern State University, Wichita Falls, TX on October 18.
3. Delivered a contributed talk on estimations of Weil sums of Dembowski-Ostrom polynomials at the American Mathematical Society's (AMS) Fall Eastern Sectional Meeting, held at Williams College on October 12-14.
4. Co-conducted a tutorial (with Patrick Mitchell of Midwestern State University) on cryptographic protocols and techniques at the Second Annual IEEE SMC Information Assurance Workshop, held June 5-6 at the United States Military Academy in West Point, NY.
5. Gave a contributed talk on estimations of Weil sums of Dembowski-Ostrom polynomials at the 6th International Conference on Finite Fields and Applications, held May 21-25 at the Hotel Mision de Los Angeles in Oaxaca, Mexico.
6. Gave a contributed talk on the existence of primitive polynomials with two coefficients prescribed in advance at the Spring 2001 AMS Sectional Meeting, held at the Stevens Institute of Technology, Hoboken, NJ on April 28 and 29.
7. Gave an invited talk on the existence of primitive polynomials with two coefficients prescribed in advance at the University of Delaware on April 13.
8. Gave an invited talk on the existence of primitive polynomials satisyfing certain properties at the University of Pennsylvania on February 20.
9. Gave a contributed talk on the existence of primitive sums in cubic extensions of finite fields in the AMS Session on Number Theory at the Joint Mathematics Meetings held January 10-13 in New Orleans, LA. At the same meeting, I participated in a focus group discussion sponsored by Elsevier Science on the development of an online service for research scientists.

2000
 

1. Delivered a contributed talk on bivariate root-based polynomial compositions at the AMS Eastern Sectional Meeting, held at Columbia University on November 4-5.
2. Gave an invited talk on correlation properties of subsequences of m-sequences at the Eighth Annual USMA/ARL Technical Symposium, held at Aberdeen Proving Ground in Aberdeen, MD on November 1.
3. Gave an invited talk on almost perfect nonlinear mappings at Wesleyan University on April 7.
4. Delivered an invited talk on the existence of primitive sums in cubic extensions of finite fields at the University of Delaware on March 24.
5. Delivered a contributed talk on crosscorrelations of sequences with a certain decimation factor in the AMS Session on Number Theory at the Joint Mathematics Meetings held January 19-22 in Washington, DC.

1997-1999
 

1. Gave an invited talk on the behavior of bivariate rational functions as group operations at an algebra conference held October 29-31, 1999 at the University of Louisiana at Lafayette.
2. Gave a contributed talk on almost perfect nonlinear mappings at the Missouri Algebra Conference, held October 16-17, 1999 at the University of Missouri at Columbia.
3. Delivered a contributed talk on the behavior of bivariate rational functions as group operations at the 5th International Conference on Finite Fields and Applications, held August 2-6, 1999 at the University of Augsburg in Augsburg, Germany.
4. Gave an invited talk on factorization patterns of root-based polynomial compositions at the Otto-von-Guericke University in Magdeburg, Germany on July 23, 1999.
5. Participated in a NATO-sponsored workshop entitled "Difference sets, sequences, and their correlation properties," held August 2-13, 1998 at the Kurhotel Bad Windsheim in Bad Windsheim, Germany.
6. Gave a contributed talk on efficient computation of root-based polynomial compositions at the 4th International Conference on Finite Fields and Applications, held August 11-15, 1997 at the University of Waterloo in Waterloo, ON.

Brief Biography

 

Updated by: Donald Mills

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