I may be reached at mullen@math.psu.edu, or by calling 814-865-2312.
My main mathematical interest is finite fields, an interest which led me to found the journal Finite Fields and Their Applications. Academic Press began publication of this journal in 1995; Elsevier now publishes the journal.
I also serve on the editorial boards of Designs, Codes and Cryptography, Journal of Algebra and Its Applications (JAA), The Fibonacci Quarterly, and Quasigroups and Related Systems.
C. F. Laywine and I have published our textbook about latin squares, a mathematical topic which has always interested me. The book, which is titled Discrete Mathematics Using Latin Squares, was published by J. Wiley and Sons in September, 1998. In the book we develop a wide variety of topics from discrete mathematics. All of these topics are motivated and developed using various ideas from the theory of latin squares. The book is available by e-mailing CUSTSERV@WILEY.COM
C.B. Mummert and I have published Finite Fields and Applications . This is an upper division undergraduate level text designed to introduce the student to various areas of the theory of finite fields and some of their applications in combinatorcs, algebraic coding theory, and cryptography. The book was published by the Amer. Math. Soc. in 2008.
I am a member of the Number Theory Group of the PSU Mathematics faculty.
2. Wun-Seng Chou, 1990, Research Fellow, Academia Sinica, Taipei.
3. Sophie Frisch, 1993, Associate Professor, Technical University, Graz, Austria.
4. Ilene Morgan, 1994, Associate Professor, University of Missouri, Rolla.
5. John Ethier, 2008, The Pennsylvania State University.
6. Serge Ballif, in progress, The Pennsylvania State University.
2. Equivalence classes of functions over a finite field, Acta Arith. 29 (1976), 353-358.
3. Permutation polynomials in several variables over finite fields, Acta Arith. 31 (1976), 107-111.
4. Equivalence classes of polynomials over finite fields, Acta Arith. 31 (1976), 113-123.
5. Left automorphisms of functions in permutation groups, Det Kong. Norske Vid. Selskab, Series Skrifter (1977), No. 3, 1-3.
6. How many i-j reduced latin squares are there? Amer. Math. Mon. 85 (1978), 751-752.
7. Weak equivalence of functions over a finite field, Acta Arith. 35 (1979), 260-272.
8. Equivalence classes of matrices over finite fields, Linear Algebra & Appls. 27 (1979), 61-68.
9. Equivalence classes of matrices over a finite field, Internat. J. Math. & Math. Sci. 2 (1979), 487-491.
10. Local permutation polynomials over Zp, Fibonacci Quarterly 18 (1980), 104-108.
11. Local permutation polynomials in three variables over Zp, Fibonacci Quarterly 18 (1980), 208-214.
12. How many i-j reduced latin rectangles are there? (with J.R. Hamilton), Amer. Math Mon. 87 (1980), 391-394.
13. Equivalence classes of sets of functions over a finite field, Acta Arith. 36 (1980), 323-329.
14. Latin cubes of order <5 (with R.E. Weber), Discrete Math. 32 (1980), 291-297.
15. Local permutation polynomials over a finite field, Det Kong. Norske Vid. Selskab, Series Skrifter (1981), No. 1, 1-4.
16. A note on equivalence classes of matrices over a finite field (with J.V. Brawley), Internat. J. Math. & Math. Sci. 4 (1981), 279-287.
17. Equivalence classes of sets of polynomials over a finite field, Bull. Calcutta Math. Soc. 73 (1981), 87-92.
18. Permutation matrices and matrix equivalence over a finite field,Internat. J. Math. & Math. Sci. 4 (1981), 503-512.
19. Matrix equivalence over finite fields, Acta Arith. 41 (1982), 133-139.
20. Polynomials over finite fields which commute with linear permutations, Proc. Amer. Math. Soc. 84 (1982), 315-317.
21. Reducing multinomial coefficients modulo a prime power (with R.J. Martin), Computers and Mathematics with Applications 10 (1984), 37-41.
22. Polynomial functions (mod m) (with H. Stevens), Acta Mathematica Hungarica 44 (1984), 237-241.
23. Latin cubes and hypercubes of prime order (with C. Laywine), Fibonacci Quarterly 23 (1985), 139-145.
24. The structure of a group of permutation polynomials (with H. Niederreiter), J. Austral. Math. Soc. (A) 38 (1985), 164-170.
25. A note on isomorphism classes of reduced latin squares of order 7 (with L J. Brant), Utilitas Mathematica 27 (1985), 261-263.
26. Commuting polynomial vectors over an integral domain (with R. Lidl), Acta Arith. 46 (1986), 145-157.
27. A polynomial representation for logarithms in GF(q) (with D. White), Acta Arith. 47 (1986), 255-261.
28. Some results on enumeration and isotopic classification of frequency squares (with L.J. Brant), Utilitas Mathematica 29 (1986), 231-244.
29. Dickson polynomials over finite fields and complete mappings (with H. Niederreiter), Canad. Math. Bull. 30 (1987), 19-27.
30. Optimal characteristic polynomials for digital multistep pseudorandom numbers (with H. Niederreiter), Computing 39 (1987), 155-163.
31. Cycles of linear permutations over a finite field (with T.P. Vaughan), Linear Algebra & Appls. 108 (1988), 63-82.
32. Association schemes and derived PBIB designs of prime power order (with R. Hultquist & H. Niederreiter), Ars Combinatoria 25 (1988), 65-82.
33. Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order, Discrete Mathematics 69 (1988), 79-84.
34. A study of frequency cubes (with L.J. Brant), Discrete Mathematics 69 (1988), 115-121.
35. Partitions with "d(a) copies of a" (with A.K. Agarwal), J. Combinatorial Theory (A) 48 (1988), 120-135.
36. When does a polynomial over a finite field permute the elements of the field? (with R. Lidl), Amer. Math. Mon. 95 (1988), 243-246.
37. Value sets of Dickson polynomials over finite fields (with W.-S. Chou & J. Gomez-Calderon), J. Number Theory 30 (1988), 334-344.
38. Permutation polynomials and nonsingular feedback shift registers over finite fields, IEEE Trans. Information Theory, Vol. IT-35 (1989), 900-902.
39. Connections between r-factors of labelled graphs and frequency squares (with J. Denes & A.D. Keedwell), Ars Combinatoria 28 (1989), 201-202.
40. Pairs of primitive elements modulo p1, (with W.-S. Chou, S. Qi, & P.J.-S. Shiue), J. Sichuan University, Natural Science Edition 26 (1989), 189-195.
41. Mutually orthogonal frequency squares with non-constant frequency vectors (with C.F. Laywine), Ars Combinatoria 29 (1990), 259-264.
42. Another generalized Golomb-Posner code (with J. Denes & S.J. Suchower), IEEE Trans on Information Theory, Vol. IT-36 (1990), 408-411.
43. Figures of merit for digital multistep pseudorandom numbers (with D.A. Andre & H. Niederreiter), Mathematics of Computation 54 (1990), 737-748.
44. Orthogonal frequency squares of type F(4t;t) (with C. F. Laywine & S.J. Suchower), Utilitas Mathematica 37 (1990), 207-214.
45. Dickson polynomials over finite fields, Proc First Internat. Sym. on Alg. Structures and Number Theory, Hong Kong, Aug. 1988, S.P. Lam & K.P. Shum eds., (1990 ), 190-207. World Scientific.
46. Infinite latin squares containing nested sets of mutually orthogonal finite latin squares (with J. V. Brawley), Publicationes Mathematicae Debrecen 39 (1991), 1-7.
47. A simple construction for orthogonal latin rectanges (with P.J.-S. Shiue), J. Combinatorial Math & Comb. Computing 9 (1991), 161-166.
48. Galois rings and algebraic cryptography (with J. Gomez-Calderon), Acta Arith. 59 (1991), 317-328.
49. Cycle structure of Dickson permutation polynomials (with R. Lidl), Math. J. Okayama Univ. 33 (1991), 1-11.
50. Primitive elements in finite fields and Costas arrays (with S.D. Cohen), Applicable Algebra in Eng., Comm. and Computing 2 (1991), 45-53; Erratum AAECC 2 (1992), 297-299.
51. Generalizations of Bose's equivalence between complete sets of mutually orthgonal latin squares and affine planes (with C. F. Laywine), J. Comb. Theory (A) 61 (1992), 13-35.
52. Primitive polynomials over finite fields (with T. Hansen),Mathematics of Computation 59 (1992), 639-643 Supplement S47-S50.
53. Functions and polynomials over Galois rings (with J. V. Brawley), J. Number Theory 41 (1992), 156-166.
54. Scalar polynomial functions on the nonsingular n x n matrices over a finite field (with J.V. Brawley), Lin. Alg. & Appls. 174 (1992), 1-12.
55. Generating linear spans over finite fields (with W.-S. Chou), Acta Arith. 61 (1992), 183-191.
56. Point sets with uniformity properties and orthogonal hypercubes (with G. Whittle), Monatsh. Math. 113 (1992), 265-273.
57. A note on discrete lorgarithms in finite fields (with G. Meletiou), Applicable Algebra in Eng., Comm. and Computing 3 (1992), 75-78.
58. A series of combinatorial constructions motivated by Raj Chandra Bose, Journal of Combinatorics, Information and System Sciences 17, Special Issue dedicated to the memory of R.C. Bose, D.K. Ray-Chaudhuri & N.M. Singh eds., (1992), 43-66.
59. Additional conjectures concerning magic squares (with J. Denes), Pure Math. & Appl. (A), 3 (1992), 155-160.
60. Permutation polynomials over finite fields in Finite Fields, Coding Theory with Advances in Comm. and Computing, Proc. of Las Vegas Conference, August, 1991, Lecture Notes in Pure and Applied Math. 141 (1993), 131-151. Marcel Dekker, Inc.
61. A note on a finite field pseudorandom vector generator of Niki, Mathematica Japonica 38 (1993), 59-60.
62. When does a polynomial over a finite field permute the elements of the field? (with R. Lidl), Amer. Math. Mon. 100 (1993), 71-74.
63. Mutually orthogonal frequency hypercubes and affine geometries (with C.F. Laywine) in Coding Theory, Design Theory, Group Theory, Proc. Marshall Hall Conf., D. Jungnickel & S.A. Vanstone eds. (1993), 183-193.
64. Enumeration formulas for latin and frequency squares (with J. Denes), Discrete Mathematics 111 (1993), 157-163.
65. Good parameters for a class of node sets in quasi-Monte Carlo integration (with T. Hansen & H. Niederreiter), Math. of Comp. 61 (1993), 225-234.
66. Some data concerning the number of latin rectangles (with D. Purdy), J. Comb. Math. & Comb. Computing 13 (1993), 161-165.
67. Values of linear recurring sequences of vectors over finite fields (with I. Shparlinski), Acta Arith. 65 (1993), 221-226.
68. Frequency hypercubes, error-correcting codes, and fractional replication plans (with S.J. Suchower), J. Comb. Inform. & System Sciences 18, special issue dedicated to the 71st birthday of C.R. Rao, K. Shah ed., (1993), 5-18.
69. A note on power sets of latin squares (with J. Denes & S.J. Suchower), J. Comb. Math. & Comb. Comp. 16 (1994), 27-31.
70. Primitive normal polynomials over finite fields (with I. Morgan), Math Computation 63, Supplement S19-S23, 759-765.
71. Dickson polynomials and irreducible polynomials over finite fields (with S. Gao), J. Number Theory 49 (1994), 118-132.
72. The number of permutation polynomials of the form f(x) + cx over a finite field (with P. J.-S. Shiue & D. Wan), Proc. Edinburgh Math. Soc. 38 (1995), 133-149.
73. D-dimensional hypercubes and the Euler and MacNeish conjectures (with C.F. Laywine & G. Whittle), Monatsh. Math. 119 (1995), 223-238.
74. Permutation polynomials: A matrix analogue of Schur's conjecture and a survey of recent results, Finite Fields Appl. 1 (1995), 242-258.
75. The difference between permutation polynomials over finite fields (with S.D. Cohen & P. Shiue), Proc. Amer. Math. Soc. 123 (1995), 2011-2015.
76. A candidate for the "next Fermat problem," The Mathematical Intelligencer 17 (1995), 18-22.
77. Rubik's clock and its solution (with J. Denes), Math. Mag. 68 (1995), 378-381.
78. Dickson polynomials over finite fields: a survey of recent results in Number-Theoretical & Algebraic Methods in Computer Science, H.G. Zimmer, A.J. van der Poorten & I. Shparlinski eds., (1995), 139-149. World Scientific.
79. Tables of (t, m, s)-net and (t,s)-sequence parameters (with A. Mahalanabis & H. Niederreiter) in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, H. Niederreiter & P. J.-S. Shiue eds., Lecture Notes in Statistics 106 (1995), 58-86. Springer, New York.
80. Open problems in finite fields, Congressus Numerantium, 111 (1995), 97-103.
81. Combinatorial methods in the construction of point sets with uniformity properties, Mathematical & Computer Modelling, 23 (1996), 1-8.
82. Construction of digital (t, m, s)-nets from linear codes (with K.M. Lawrence, A. Mahalanabis & W.C. Schmid) in Finite Fields & Applications, S.D. Cohen & H. Niederreiter eds., Lecture Note Series of the London Math. Soc. 233 (1996), 189-208. Cambridge University Press.
83. Open problems and conjectures in finite fields (with I. Shparlinski) in Finite Fields & Applications, S.D. Cohen & H. Niederreiter eds., Lecture Notes of the London Math. Soc. 233 (1996), 243-268.
84. Completely normal primitive basis generators of finite fields (with I. Morgan), Utilitas Mathematica 49 (1996), 21-43.
85. Bases and the distribution of irreducible and primitive polynomials over finite fields in Applications of Finite Fields, D. Gollmann ed., Institute of Math. & Its Appl. Conf. Ser. 59 (1996), 1-18. Oxford Univ. Press.
86. Dickson polynomial discriminators (with P. Moree), J. Number Theory 59 (1996), 88-105.
87. An equivalence between (t, m, s)-nets and strongly orthogonal hypercubes (with W.C. Schmid), J. Combinatorial Theory A 76 (1996), 164-174.
88. Almost weakly self-dual bases of finite fields (with I. Morgan & M. Zivkovic), Appl. Alg. Eng. Comm. Comp. 8 (1997), 25-31.
89. The distribution of power residues in finite fields (with M. Caragiu), Southeast Asian Math. Soc. Bull. 21 (1997), 149-158.
90. Dickson-Stirling numbers (with L.C. Hsu & P. J.-S. Shiue), Proc. Edinburgh Math. Soc. 40 (1997), 409-423.
91. Improved (t, m, s)-net parameters from the Gilbert-Varshamov bound (with A.T. Clayman), Appl. Alg. Eng. Comm. Comp. 8 (1997), 491-496.
92. Irreducible polynomials and linear recurring arrays (with M. Liu), Discrete Appl. Math. 74 (1997), 93-99.
93. Distribution of irreducible polynomials of small degrees over finite fields (with K.H. Ham), Math. of Computation 66 (1997), 1195-1212.
94. Correlation immune functions over finite fields (with M. Liu & P. Lu), IEEE Trans. Infor. Thy. 44(1998), 1273-1276.
95. Orthogonal hypercubes and related designs, J. Statist. Planning & Inference 73 (1998), 177-188.
96. Irreducible polynomials over finite fields (with A. Knopfmacher & J. N. Ridley), Appl. Alg. Eng. Comm. Comp., 9 (1998), 265-269.
97. Updated tables of parameters of (t, m, s)-nets (with A.T. Clayman, K.M. Lawrence, H. Niederreiter & N.J.A. Sloane), J. Combinatorial Designs, 7 (1999), 381-393.
98. A quasigroup-based public-key cryptosystem (with D. Koscielny), Appl. Math and Comp. Sci., 9 (1999), 101-109.
99. A note on BIB designs with nested rows and columns (with J. MacDougall), Australasian J. Combinatorics, 20 (1999), 257-259.
100. Dickson polynomials, Encyclopedia of Math, Supplement II, Kluwer Academic Publishers, 2000, 154-156.
101. Parallel computing of a quasi-Monte Carlo algorithm for valuing derivatives (with J. X. Li), Parallel Computing, 29 (2000), 641-653.
102. A table of lower bounds for the number of mutually orthogonal frequency squares (with C.F. Laywine), Ars Combinatoria, 59 (2001), 85-96.
103. The world's most interesting class of integral polynomials (with R. Lidl), J. Comb. Math. Comb. Comp., 37 (2001), 87-100.
104. Constructions for mutually orthogonal frequency hyperrectangles of prescribed type (with B. Cheng), Discrete Mathematics 242 (2002), 55-64.
105. A constructive finite field method for scattering points on the surface of d-dimensional spheres (with B. Bajnok, S. B. Damelin, and J. Li), Computing 68/2 (2002), 97-109.
106. A hierarchy of complete orthogonal structures (with C. F. Laywine), Ars Combinatoria, 63 (2002), 75-88.
107. Frequency magic squares (with K. Hicks and C. Laywine) Utilitas Mathematica 62 (2002), 3-12.
108. Distribution of irreducible polynomials over finite fields (with K. H. Hicks and I. Sato) Finite Fields with Applications in Coding Theory, Cryptography and Related Areas, (edited by G. L. Mullen, H. Stichtenoth and H. Tapia-Recillas), Springer, 2002, 177-186.
109. Value sets of polynomials over finite fields (with P. Das) Finite Fields with Applications in Coding Theory, Cryptography and Related Areas, (edited by G. L. Mullen, H. Stichtenoth and H. Tapia-Recillas), Springer, 2002, 80-85.
110. Twin irreduciable polynomials over finite fields (with G. W. Effinger and K. H. Hicks) Finite Fields with Applications in Coding Theory, Cryptography and Related Areas, (edited by G. L. Mullen, H. Stichtenoth and H. Tapia-Recillas), Springer, 2002, 94-111.
111. Properties of codes with one check symbol from a quasigroup point of view (with V. Shcherbacov) Buletinul Academiei de Stiinte a Republicii Moldova, Mathematica 3(40) (2002), 71-86.
112. Irreducible polynomials over GF(2) with prescribed coefficients (with J.L. Yucas) Discrete Math. 274(2004), 265-279.
113. The number of linearly independent binary vectors with applications to the construction of hypercubes and orthogonal arrays, pseudo (t,m,s)-nets and linear codes (with S.B. Damelin, G. Michalski, and D. Stone) Monatsh. Math. 141(2004), 277-288.
114. Self-reciprocal polynomials over finite fields (with J.L. Yucas) Designs, Codes and Cryptography 33(2004), 275-281.
115. Bijections and congruences for generalizations of partition identities of Euler and Guy (with J.A. Sellers and A.V. Sills) Electronic J. Comb. R43, June 25, 2004.
116. n-T-quasigroups with one check symbol and their error detection capabilities (with V. Shcherbacov) Commentationes Mathematicae Universitatis Caroline 45(2004), 321-340.
117. A decade of Finite Fields and Their Applications Finite Fields Appl. 11(2005), 321-325.
118. Integers and polynomials: Comparing the close cousins Z and Fq[x] (with G. Effinger and K.H. Hicks) Mathematical Intelligencer 27(2005), 26-34.
119. Sets of partially orthogonal latin squares and projective planes (with A.D. Keedwell) Discrete Math. 288(2005), 49-60.
120. Orthogonal hypercubes and many-placed operations (with G.B. Belyavskaya) Quasigroups and Related Systems 13(2005), 73-86.
121. Check character systems using quasigroups: I (with G.B. Belyavskaya and V.I. Izbash) Designs, Codes and Cryptography 37(2005), 215-227.
122. Check character systems using quasigroups: II (with G.B. Belyavskaya and V.I. Izbash) Designs, Codes and Cryptography 37(2005), 405-419.
123. Transpositions and representability (with Z. Chen) Amer. Math. Mon. 112(2005), 913-919.
124. Frequency permutation arrays (with S. Huczynska) J. Combinatorial Designs 14(2006), 463-478.
125. Products of mixed covering arrays of strength two (with C.J. Colbourn, S. Martirosyan, D. Shasha, G.B. Sherwood, and J.L. Yucas) J. Combinaotrial Designs 14(2006), 124-138.
126. Frequency squares and hypercubes (with C.F. Laywine) Handbook of Combinatorial Designs, Sec. Ed., Edited by C.J. Colbourn and J.H. Dinitz, Chapman and Hall/CRC, 465-472.
127. Permutation polynomials Handbook of Combinatorial Designs, Sec. Ed., Edited by C.J. Colbourn and J.H. Dinitz, Chapman and Hall/CRC, 572-574.
128. The geometry of sets of orthogonal frequency hypercubes (with V.C. Mavron and T.P. McDonough) J. Combinatorial Designs 15(2007), 449-459.
129. Strongly orthogonal and uniformly orthogonal many-placed operations (with G.B. Belyavskaya) Algebraic & Discrete Math. (Ukraine) 1(2007), 1-17.
130. The cardinality of sets of k-independent vectors over finite fields (with S.B. Damelin and G. Michalski) Monatsh. Math. 150(2007), 289-295.
131. A polynomial analogue of the 3n+1 problem (with K. Hicks, J.L. Yucas, and R. Zavislak) Amer. Math. Mon. 115(2008), 615-622.
132. Dickson polynomials that decompose in cycles of the same length
(with I.M. Rubio, C. Corrada, and F.N. Castro)
Finite Fields & Applications (ed. by G.L. Mullen, D. Panario, and I.E.
Shparlinski), Contemporary Mathematics (Amer. Math. Soc.), Vol. 461(2008),
229-239.
ACCEPTED PAPERS
The number of solutions of Dickson polynomials over finite fields (with W.-S. Chou and B. Wassermann), Taiwanese J. Math. .
The extent to which subsets are additively closed (with S. Huczynska and J.L. Yucas), J. Combinatorial Thy., Ser. A.
Number of irreducible polynomials and pairs of relatively prime
polynomials in several variables over finite fields
(with X.-D. Hou), Finite Fields Appl..
SUBMITTED PAPERS
Formulas for permutation polynomials over finite fields (with B. Vioreanu)
A generalized counting and factoring technique for polynomials over finite fields (with R.C. Mullin and J.L. Yucas)
RESEARCH MONOGRAPHS
Dickson Polynomials (with R. Lidl & G. Turnwald), Pitman Monographs and Surveys in Pure & Applied Math. 65 (1993). Longman, Essex, England.
Unsolved Problems in Finite Fields and Their Applications (with
S. Gao and I. E. Shparlinski), in progress.
TEXTBOOKS
Discrete Mathematics Using Latin Squares (with C. F. Laywine). This is an upper division undergraduate/beginning graduate level text designed to introduce the student to various areas of discrete mathematics with all of the topics motivated and developed through the use of latin squares. The book was published by Wiley in 1998.
Finite Fields and Applications (with C.B. Mummert). This is an upper division undergraduate level text designed to introduce the student to various areas of the theory of finite fields and some of their applications in combinatorcs, algebraic coding theory, and cryptography. The book was published by the Amer. Math. Soc. in 2008.
CONFERENCE PROCEEDINGS EDITED
Finite Fields, Coding Theory and Advances in Communications and Computing (with P. J.-S. Shiue), Lecture Notes in Pure & Applied Math. 141 (1993). Marcel Dekker.
Finite Fields: Theory, Applications & Algorithms (with P. J.-S. Shiue), Contemporary Mathematics 168 (1994). American Math. Society.
Finite Fields: Theory, Applications & Algorithms (with R. C. Mullin), Contemporary Mathematics 225. American Math. Society.
Number Theory and Discrete Mathematics, A. Agarwal, G. Mullen et al. eds., Hindustan Book Agency, 2002.
Finite Fields with Applications in Coding Theory, Cryptography and Related Areas (with H. Stichtenoth and H. Tapia-Recillas), (2002). Springer.
Finite Fields (with A. Poli and H. Stichtenoth), Lect. Notes in Comp. Sci., Vol. 2948 (2004), Springer.
Finite Fields and Application (with D. Panario and I.E. Shparlinski), Contemporary Math., Amer. Math. Soc., Vol. 461 (2007).
Finite Fields and Application (with G. McGuire, D. Panario,
and I.E. Shparlinski),
Contemporary Math., Amer. Math. Soc., to appear.
INVITED BOOK REVIEW
Finite Fields by R. Lidl and H. Niederreiter, Bull. Amer. Math. Soc., 36 (1999), 99-101.