Publications

Publications

Ph.D. Dissertation

1. R.S. Coulter, “Planar functions and related topics in finite fields”, Ph.D. dissertation, Department of Computer Science and Electrical Engineering, The University of Queensland, Australia, 1998.

Articles (refereed journals and conference proceedings)

1. 1. R.S. Coulter and R.W. Matthews, Bent polynomials over finite fields, Bull. Austral. Math. Soc. 56 (1997), 429-437.
   2. R.S. Coulter and R.W. Matthews, Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr. 10 (1997), 167-184.
      (The original publication is available at www.springerlink.com)
   3. R.S. Coulter, Explicit evaluations of some Weil sums, Acta Arithmetica 83 (1998), 241-251.
   4. R.S. Coulter, Further evaluations of Weil sums,, Acta Arithmetica 86 (1998), 217-226.
   5. R.S. Coulter, G. Havas and M. Henderson, Functional decomposition of a class of wild polynomials, J. Combin. Math. Combin. Comp. 28 (1998), 87-94.
   6. R.S. Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math. 28 (1999), 171-184.
   7. R.S. Coulter and M. Henderson, A class of functions and their application in constructing semi-biplanes and association schemes, Discrete Math. 202 (1999), 21-31.
   8. A. Blokhuis, R.S. Coulter, M. Henderson and C.M. O’Keefe, Permutations amongst the Dembowski-Ostrom polynomials, Finite Fields and Applications: Proceedings of the Fifth International Conference on Finite Fields and Applications (D. Jungnickel and H. Niederreiter, eds.), 2001, pp. 37-42.
   9. R.S. Coulter, G. Havas and M. Henderson, Giesbrecht’s algorithm, the HFE cryptosystem, and Ore’s p^s-polynomials, Computer Mathematics: Proceedings of the Fifth Asian Symposium (ASCM 2001) (K. Shirayanagi and K. Yokoyama, eds.), Lecture Notes Series on Computing, vol. 9, World Scientific, 2001, pp. 36-45.
   10. R.S. Coulter, The number of rational points of a class of Artin-Schreier curves, Finite Fields Appl. 8 (2002), 397-413.
   11. R.S. Coulter and M. Henderson, The compositional inverse of a class of permutation polynomials over a finite field, Bull. Austral. Math. Soc. 65 (2002), 521-526.
   12. R.S. Coulter and R.W. Matthews, On the permutation behaviour of Dickson polynomials of the second kind, Finite Fields Appl. 8 (2002), 519-530.
   13. M. Henderson, R.S. Coulter, E. Dawson and E. Okamoto, Modelling trust structures for public key, Information Security and Privacy, Proceedings of the 7th Australasian Conference, ACISP 2002 (L.M. Batten and J. Seberry, eds.), 2002, pp. 56-70.
   14. R.S. Coulter, G. Havas and M. Henderson, On decomposition of sub-linearised polynomials, J. Austral. Math. Soc. 76 (2004), 317-328.
   15. R.S. Coulter and M. Henderson, A note on the roots of trinomials over a finite field, Bull. Austral. Math. Soc. 69 (2004), 429-432.
   16. R.S. Coulter and M. Henderson, On the splitting case of a semi-biplane construction, Proceedings of the Fifteenth Australasian Workshop on Combinatorial Algorithms, AWOCA 2004 (S-H. Hong, ed.), 2004, pp. 192-198.
   17. R.S. Coulter, M. Henderson and F. Lazebnik, On certain combinatorial Diophantine equations and their connection to Pythagorean numbers, Acta Arithmetica 122 (2006), 395-406.
   18. R.S. Coulter, The classification of planar monomials over fields of prime square order , Proc. Amer. Math. Soc. 134 (2006), 3373–3378.
   19. L.M. Batten, R.S. Coulter and M. Henderson, Extending abelian groups to rings, J. Austral. Math. Soc., 82 (2007), 297–313.
   20. R.S. Coulter, M. Henderson and P. Kosick, Planar polynomials for commutative semifields with specified nuclei, Des. Codes Cryptogr. 44 (2007), 275-286.
       (The original publication is available at www.springerlink.com)
       Note: there are 2 errors in this paper (that I know of!): Firstly, the length of the summand in Theorem 4.3 is not correct. Secondly, the new example of a commutative semifield is incorrect; a correct example is
       L(X)=X^243 – X^81 + X^9 + X^3 – X and D(X) = X^246 – X^10 with t(X) = X^9 – X.
       This yields a commutative semifield of order 3^8 with Nl=GF(3) and Nm=GF(9).
   21. R.S. Coulter and M. Henderson, Commutative presemifields and semifields, Adv. Math. 217 (2008), 282-304.
   22. R.S. Coulter and T. Gutekunst, Special subsets of difference sets with particular emphasis on skew Hadamard difference sets, Des. Codes Cryptogr. 53 (2009), 1-12.
       (The original publication is available at www.springerlink.com)
   23. R.S. Coulter, M. Henderson and R.W. Matthews, A note on constructing permutation polynomials, Finite Fields Appl. 15 (2009), 553-557.
   24. R.S. Coulter and P. Kosick, Commutative semifields of order 243 and 3125, Finite Fields: Theory and Applications — Proceedings of the 9th International Conference on Finite Fields and Applications (G. McGuire, G.L. Mullen, D. Panario and I.E. Shparlinski, eds.), Contemporary Mathematics, vol. 518, American Mathematical Society, 2010, pp. 129-136.
   25. R.S. Coulter and R.W. Matthews, Dembowski-Ostrom polynomials from Dickson polynomials, Finite Fields Appl. 16 (2010), 369-379.
   26. R.S. Coulter and R.W. Matthews, On the number of distinct values of a class of functions over a finite field, Finite Fields Appl. 17 (2011), 220-224.
   27. R.S. Coulter and F. Lazebnik, On the classification of planar monomials over fields of square order, Finite Fields Appl. 18 (2012), 316-336.
   28. R.S. Coulter and T. Gutekunst, Subsets of finite groups exhibiting additive regularity, Discrete Math. 313 (2013), 236-248.
   29. R.S. Coulter and M. Henderson, On a conjecture on planar polynomials of the form X(Tr_n(X)-uX), Finite Fields Appl. 21 (2013), 30-34.
   30. R.S. Coulter and P. Kosick, On expressing elements as a sum of squares where one square is restricted to a subfield, Finite Fields Appl. 26 (2014), 116-122.
   31. R.S. Coulter and S. Senger, On the number of distinct values of a class of functions with finite domain, Ann. Comb. 18 (2014), 233-243.
   32. C. Castillo, R.S. Coulter and S. Smith, A note on interpolation of permutations of a subset of a finite field, Bull. Austral. Math. Soc. 90 (2014), 213-219.
   33. C. Castillo and R.S. Coulter, A general representation theory for constructing groups of permutation polynomials, Finite Fields Appl. 35 (2015), 172-203.
   34. R.S. Coulter and R.W. Matthews, Closure planes, J. Algebraic Combin. 43 (2016), 735-749.
   35. R.S. Coulter, R.W. Matthews and C. Timmons, Planar polynomials and an extremal problem of Fischer and Matousek, J. Combin. Theory Ser. B 128 (2018), 96-103.
   36. R.S. Coulter and S. Mesnager, Bent functions from involutions over $\ff{2^n}$, IEEE Trans. Info. Th. 64 (2018), 2979-2986.
   37. P. Cesarz and R.S. Coulter, A Wilbrink-like equation for neo difference sets, Ann. Comb. 22 (2018), 245-253.
   38. R.S Coulter, On coordinatising projective planes of prime power order using finite fields, J. Austral. Math. Soc. 106 (2019), 184-199.
   39. L. Budaghyan, M. Calderini, C. Carlet, R.S. Coulter and I. Villa, On isotopic construction of APN functions, Proceedings of the International Conference on Sequences and their Applications, SETA 2018, Hong Kong, 2018.
   40. R.S. Coulter, S. De Winter, A. Kodess and F. Lazebnik,  A result on polynomials derived via graph theory, Math. Mag. 92 (2019), 288-295.
   41. S. Cioaba, R.S. Coulter, E. Fiorini and Q. Xiang, Preface to the Special Issue of Discrete Mathematics: Dedicated to the Algebraic and Extremal Graph Theory Conference, Discrete Math. 342 (2019), 2759.
   42. L. Budaghyan, M. Calderini, C. Carlet, R.S. Coulter and I. Villa, Generalized isotopic shift of Gold functions, Proceedings of the International Workshop on Coding and Cryptography, WCC 2019, Saint-Jacut-de-la-Mer, France, 2019.
   43. P.G. Cesarz and R.S. Coulter,  Images sets with regularity of differences, Cryptogr. Commun. 11 (2019), 1307-1337.
   44. L. Budaghyan, M. Calderini, C. Carlet, R.S. Coulter and I. Villa, On isotopic shift construction for planar functions, in 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France, 2019, pp.2962-2966.
   45. L. Budaghyan, M. Calderini, C. Carlet, R.S. Coulter and I. Villa, Constructing APN functions through isotopic shift, IEEE Trans. Info. Th. 66 (2020), 5299-5309.
   46. L. Budaghyan, M. Calderini, C. Carlet, R.S. Coulter and I. Villa, Generalized isotopic shift construction for APN functions, Des. Codes Cryptogr. 89 (2021), 19-32.
   47. R.S. Coulter and N.S. Kaleyski, Further observations on the distance invariant, Proceedings of the BFA 2021.
   48. E. Bergman, R.S. Coulter and I. Villa, The classification of planar monomials over fields of order $p^3$, Proceedings of the BFA 2021.
   49. E. Bergman and R.S. Coulter, Constructing functions with low differential uniformity, Mediterranean J. Math. 19 (2022), Paper #94 (online).
   50. E. Bergman, R.S. Coulter and I. Villa, Classifying planar monomials over fields of order a prime cubed, Finite Fields Appl. 78 (2022), Paper No. 101959 (online).
   51. L-A. Chen and R.S. Coulter, On the differential uniformity of the Wan-Lidl permutation polynomials, Proceedings of the BFA 2022.
   52. L-A. Chen and R.S. Coulter, Bounds on the differential uniformity of the Wan-Lidl polynomials, Cryptogr. Commun. 15 (2023), 1069-1085
   53. L-A. Chen and R.S. Coulter, Permutation resemblance, IEEE Trans. Info. Th. 69 (2023), 6711-6718.
       

       Book Chapters & Sections (refereed)

   54. R.S. Coulter, The $\kappa$-polynomials and related algebraic objects, Handbook of Finite Fields (G. Mullen and D. Panario, eds.), 2013.
   55. R.S. Coulter, Planar functions and commutative semifields, Handbook of Finite Fields (G. Mullen and D. Panario, eds.), 2013.
       

       Submitted articles and those in Preparation

   56. P.G. Cesarz and R.S. Coulter, A divisibility condition regarding orders of Lenz-Barlotti I.4 projective planes, 6 page manuscript, in draft.
   57. R.S. Coulter, Algebraic substructures of planar ternary rings, 16 page manuscript, in draft.
   58. E. Bergman, R.S. Coulter and B. Fain, On the distribution of squares in a finite field under a linear map, 7 page manuscript, in draft.
   59. L-A. Chen and R.S. Coulter, Algorithms for determining the permutation resemblance of a function, 23 page manuscript, in draft.
   60. R.S. Coulter and B. Fain, A class of functions and their application in constructing semisymmetric designs, in preparation.
   61. R.S. Coulter and B. Fain, Some classification results on semiplanar functions, in preparation.
   62. R.S. Coulter and M. Henderson, Isomorphism classes of Dembowski-Ostrom polynomials over fields of prime squared order, in preparation.
   63. Solutions to everything not solved here…, in preparation.

Unpublished articles — Journal der mathematischen Ablehnungen

• Planar polynomials and commutative semifields two dimensional over their middle nucleus and four dimensional over their nucleus (with M. Henderson, L. Hu, P. Kosick, Q. Xiang and X. Zeng)