Papers: (Most papers are in preprint versions, so they may be slightly different
from the published versions.)
- Shuhong Gao, Frank Volny IV and Mingsheng Wang,
A New Algorithm for
Computing Grobner Bases,
Revised, July 2013.
PDF file (307k)
- Jang-Woo Park and Shuhong Gao,
Dynamics of x+x^{-1} via elliptic curves,
Revised, July 2011.
PDF file (221K)
- Mingfu Zhu, Guangran Jiang and Shuhong Gao,
Solving the 100 Swiss Francs Problem,
To appear in Mathematics in Computer Science.
PDF file (254K)
- Shuhong Gao, Yinhua Guan and Frank Volny IV,
A New Incremental Algorithm for
Computing Grobner Bases,
ISSAC 2010, pp. 13--19.
PDF file (168K)
- Shuhong Gao and Todd Mateer,
Additive Fast Fourier Transforms over Finite Fields,
IEEE Transactions on Information Theory 56 (2010), 6265--6272.
PDF file (164K)
- Shuhong Gao and Raymond Heindl,
Multivariate Public Key Cryptosystems from Diophantine Equations,
preprint 2009 .
PDF file (205K)
- Shuhong Gao, Daqing Wan and Mingsheng Wang,
Primary decomposition of zero-dimensional ideals over finite fields,
Mathematics of Computation, 78 (2009), 509--521.
PDF file (154K)
- Shuhong Gao and D. Frank Hsu,
Short containers in Cayley graphs,
Discrete Applied Mathematics, 157 (2009), 1354--1363.
- Jessie F. Burkhart, Neil J. Calkin, Justine C. Hyde-Volpe,
Shuhong Gao, Kevin James, Hiren Maharaj, Shelly Manber, Jared Ruiz and Ethan Smith,
Finite field towers and elements of high order arising from modular curves,
Designs, Codes and Cryptography, 51 (2009), 301--314.
- Guantao Chen, Shuhong Gao and Zhiquan Hu,
Linked Graphs with Restricted Lengths,
J. Combinatorial Theory, series B, 98 (2008), 735--751.
PDF file (282K)
- Shuhong Gao and Beth Novick,
Fault Tolerance of Cayley Graphs,
Annals of Combinatorics, 11 (2007), 161--171.
PDF file (186K)
- Jeff Farr and Shuhong Gao,
Gröbner bases and generalized Padé approximation,
Mathematics of Computation, 75 (2006), 461-473.
PSfile
(424K) or PDF
file (154K)
- Jeff Farr and Shuhong Gao, Computing
Grobner bases for vanishing ideals of finite sets of points,
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC16),
LNCS 3857, 2006, 118--127.
PSfile (390K)
or PDF file (144K)
- Jeffrey B. Farr and Shuhong Gao, Gröbner bases, Padé
approximation, and decoding of linear codes,
Coding Theory and Quantum Computing,
Contemporary Mathematics, vol.381, Amer. Math. Soc., Providence, RI, 3--18, 2005.
PSfile
(404K) or PDF
file (147K)
- Shuhong Gao, Erich Kaltofen, John P. May, Zhengfeng Yang and
Lihong Zhi, Approximate
factorization of multivariate polynomials via differential equations,
in Proceedings of the 2004
international symposium on Symbolic and algebraic computation
(ISSAC 2004, July 4-7, 2004, University of
Cantabria, Santander, Spain), pp. 167-174. PSfile
(232K) or PDF
file (171K) Maple software
available at Erich Kaltofen
or Lihong
Zhi
- Joel V. Brawley and Shuhong Gao, On
density of primitive elements for field extensions, preprint,
2004 (6 pages). PSfile
(280K) or PDF
file (94K)
- Shuhong Gao and Alan G.B. Lauder, Fast absolute irreducibility testing via
Newton polytopes, preprint 2004 (14 pages). PSfile
(273K) or PDF
file (95K)
- Fatima Abu Salem, Shuhong Gao and Alan G.B. Lauder, Factoring polynomials via polytopes,
in Proceedings of the 2004
international symposium on Symbolic and algebraic computation
(ISSAC 2004, July 4-7, 2004, University of Cantabria, Santander,
Spain), pp. 4-11. PSfile
(419K) or PDF
file (231K)
- Shuhong Gao, Virginia M. Rodrigues and Jeffrey Stroomer, Grobner basis structure of finite sets of
points, preprint, 2003 (16 pages). PSfile
(441K) or PDF
file (166K)
- Shuhong Gao, ``A new algorithm for decoding
Reed-Solomon codes'',
in Communications, Information and Network Security (V.
Bhargava,
H. V. Poor, V. Tarokh and S. Yoon, Eds.), Kluwer Academic Publishers,
2003,
pp. 55-68. (Correction: In Step 2 of Algorithm
1a, (d+1)/2 should be (d-1)/2 as pointed out by Professor Sergei
Fedorenko.) PSfile
(340K) or PDF
file (118K)
- Shuhong Gao and Virginia M. Rodrigues, ``Irreducibility of
polynomials
modulo p via Newton Polytopes'', Journal of Number Theory
101 (2003), 32-47. PS
file (411K) or PDF
file (142K)
- Richard P. Brent, Shuhong Gao and Alan G.B. Lauder, ``Random
Krylov
spaces over finite fields,'' SIAM J. on Discrete Mathematics 16
(2003), 276-287. PS
file (340K) or PDF
file (131K)
- Shuhong Gao, Erich Kaltofen and Alan G.B. Lauder, ``Deterministic
distinct
degree factorization for polynomials over finite fields'', to
appear in Journal of Symbolic
Computation (10 pages). PDF
file (210K)
- Shuhong Gao and D. Frank Hsu, ``Short containers in Cayley
graphs'',
DIMACS tech report 2001-18, May 2001. (15 pages) PS
file (336K) or PDF
file (135K)
- Shuhong Gao, ``Factoring multivariate polynomials via partial
differential
equations,'' Mathematics of Computation 72 (2003),
801-822. PS
file (502K) or PDF
file (201K)
- Shuhong Gao and Alan G.B. Lauder, ``Hensel lifting and
polynomial
factorisation,'' Mathematics
of Computation 71 (2002), 1663-1676. PS
file (396K) or PDF
file (150K)
- Huapeng Wu, M. Anwar Hasan, Ian F. Blake and Shuhong Gao, ``Finite
fields
multiplier using redundant representation'', IEEE Transactions
on
Computers, vol. 51, no. 11, 2002, 1306-1316. PDF
file (150K)
- Joel V. Brawley, Shuhong Gao and Donald Mills, ``Associative
Rational
functions in two variables'', in Finite Fields and
Their
Applications (D. Jungnickel and H. Niederreiter, Eds.),
Springer-Verlag,
2001, 43-56. PS
file (336K) or PDF
file (120K)
- Shuhong Gao, ``Abelian groups, Gauss periods, and normal bases,''
Finite
Fields and Their Applications 7 (2001), 149-164. PS
file (410K) or PDF
file (151K)
- Shuhong Gao, ``Absolute irreducibility of polynomials via
Newton
polytopes,'' J.
of Algebra 237 (2001), 501-520. gzipped
PS file (502K) or PDF
file (200K)
- Shuhong Gao, ``On the deterministic complexity of polynomial
factoring,'' J.
Symbolic Computation 31 (2001), 19-36. gzipped
PS file (680k) or PDF
file (238K)
- Shuhong Gao and Alan G.B. Lauder, ``Decomposition of polytopes
and
polynomials,'' Discrete
and Computational Geometry 26 (2001), 89-104. PS
file (365K) or PDF
file (133K)
- Shuhong Gao, Joachim von zur Gathen, Daniel Panario and Victor
Shoup, ``Algorithms
for exponentiation in finite fields,'' J. Symbolic Computation 29
(2000),
879-889. PDF
file (306K)
- Shuhong Gao and M. Amin Shokrollahi, ``Computing roots of
polynomials
over function fields of curves,'' in Coding Theory and
Cryptography:
From Enigma and Geheimschreiber to Quantum Theory (D. Joyner,
Ed.),
Springer,
2000, 214-228. gzipped
PS file (79K) or PDF
file (212K)
- Shuhong Gao, ``Elements of provable high orders in finite
fields,''
Proc. American Math. Soc.
127 (1999), 1615-1623. PDF
file (153K)
- Shuhong Gao and Jason Howell, ``A general polynomial sieve,''
Designs,
Codes and Cryptography 18 (1999), 149-157. gzipped
PS file (504K) or PDF
file (118K)
- Shuhong Gao, Jason Howell and Daniel Panario, ``Irreducible
polynomials
of given forms,'' Contemporary
Mathematics, vol. 225, Amer. Math.
Soc.,
Providence, RI, 1999, 43-54. PS
file (404K) or PDF
file (146K)
- Joel V. Brawley, Shuhong Gao and Donald Mills, ``Computing
composed
products of polynomials,'' Contemporary
Mathematics, vol. 225,
Amer.
Math. Soc., Providence, RI, 1999, 1-15.
- Shuhong Gao, Beth Novick and Ke Qiu, ``From Hall's matching
theorem
to optimal routing on hypercubes,'' J. Combinatorial Theory (B) 74
(1998), 291-301. PDF
file (138K)
- S. Gao and J. D. Key, ``Bases of minimum-weight vectors for
codes
from
designs,'' Finite Fields and
Their Applications 4 (1998),
1-15. PDF
file (187k)
- Shuhong Gao, Joachim von zur Gathen and Daniel Panario, ``Gauss
periods:
orders and cryptographical applications,'' Mathematics of
Computation,
67 (1998), 343-352. gzipped
PS file (46k)
- Shuhong Gao and Daniel Panario, ``Density of normal elements,''
Finite Fields and Their Applications 3 (1997), 141--150. PS
file (169k)
- Shuhong Gao and Daniel Panario, ``Tests and constructions of
irreducible
polynomials over finite fields,'' Foundations
of Computational
Mathematics
(F. Cucker and M. Shub, Eds.), Springer, 346-361, 1997. PDF
file (249k)
- Ian F. Blake, Shuhong Gao and Ronald C. Mullin, ``Specific
irreducible
polynomials with linearly independent roots over finite fields''
Linear
Algebra and Its Applications 253 (1997), 227--249. PDF
file (157k)
- Ian F. Blake, Shuhong Gao and Robert L. Lambert, ``Construction
and
distribution problems for irreducible trinomials over finite fields,''
in Applications of Finite
Fields (D. Gollmann, Ed.),
Oxford,
Clarendon Press, 1996, 19-32. PS
file (195k)
- Shuhong Gao and Scott A. Vanstone, ``On orders of optimal
normal
basis
generators,'' Mathematics of
Computation 64 (1995), 1227--1233. PDF
file (92k)
- Shuhong Gao, Joachim von zur Gathen and Daniel Panario, ``Gauss
periods
and fast exponentiation in finite fields,'' in Lecture Notes in
Computer
Science, vol. 911, Springer-Verlag, 1995, 311--322.
- Shuhong Gao and Gary L. Mullen, ``Dickson polynomials and
irreducible
polynomials over finite fields,'' J.
Number Theory 49 (1994),
118-132.
- Ian F. Blake, Shuhong Gao and Ronald C. Mullin, ``Normal and
self-dual
normal bases from factorization of cx q+1+dxq -ax
-b,'' SIAM J. on Discrete
Mathematics 7 (1994), 499-512. PDF
file (141k)
- Ian F. Blake, Shuhong Gao and Robert L. Lambert, ``Constructive
problems
for irreducible polynomials over finite fields,'' in Information
Theory
and Applications (A. Gulliver and N. Secord, eds), LNCS 793,
Springer-Verlag,
1994, 1-23.
- Shuhong Gao and Joachim von zur Gathen, ``Berlekamp's and
Niederreiter's
polynomial factorization algorithms,'' in Proc. 2nd
International
Conference on Finite Fields: Theory, Applications, and Algorithms,
Las
Vegas, 1993. Contemporary Mathematics, vol. 168, 1994, 101-116.
- Ian F. Blake, Shuhong Gao and Ronald C. Mullin, ``Explicit
factorization
of x2^k +1 over Fp with prime $p \equiv 3 \bmod 4$,''
Applicable Algebra in
Engineering, Communication and Computation 4
(1993),
89--94. PDF
file (81k)
- Shuhong Gao and Wan-Di Wei, ``On non-Abelian group difference
sets,''
Discrete Mathematics 123
(1993), 151-177. PDF
file (107k)
- Wan-Di Wei, Shuhong Gao and Benfu Yang,``Cyclic near
difference sets
of type 1,'' Discrete Mathematics
123 (1993), 151-177.
- Shuhong Gao and Hendrik W. Lenstra, Jr., ``Optimal normal
bases,''
Designs, Codes and Cryptography
2 (1992), 315-323. PDF
file (85k)
- Wan-Di Wei, Shuhong Gao and Qing Xiang, ``On extraneous
multipliers
of Abelian difference sets,'' Combinatorial
Designs and
Applications
(Huangshan, 1988, W. D. Wallis, H. Shen, W. Wei, L. Zhu, Eds.), Marcel
Dekker, Inc., 1990, 165--173.
- Shuhong Gao, ``On nonabelian group difference sets,''
Sichuan
Daxue
Xuebao 26 (1989), Special Issue, 136-139.
Books and Chapters: - Alfred J. Menezes (Editor), Ian F.
Blake, XuHong Gao, Ronald C.
Mullin,
Scott A. Vanstone and Tomic Yaghoobian, Applications
of Finite Fields, Kluwer Academic Publishers,
Boston/Dordrecht/Lancaster,
1993.
- ``Mathematical Models in Classical Cryptology,'' in Applied
Mathematical
Modeling: A Multidisciplinary Approach , (D. R. Shier and K. T.
Wallenius,
Eds.), Chapman & Hall/CRC, Boca Raton, 2000, p. 93--113. (With Joel
V. Brawley).
- ``Mathematical Models in Public-key Cryptology,'' Applied
Mathematical
Modeling: A Multidisciplinary Approach , (D. R. Shier and K. T.
Wallenius,
Eds.), Chapman & Hall/CRC, Boca Raton, 2000, p. 93--113. (With Joel
V. Brawley). HTML
, gzipped
PS file (522K) or PDF
file (178K)