Gregory V. Bard
Associate Professor of Mathematics
Preserving the lookandfeel of the World Wide Web as it was, in 1998.
Publications Page
Warning: This information has not been updated in a while. I
hope to remedy that soon, but in the meantime you can find links to the articles that
I have written by opening my CV. Each article is hotlinked. For a copy of my curriculum vitae, I offer a
short form (8 pages)
as well as a
long form (17 pages).
On this page, I am sharing preprints and reprints of my publications, patents, and
information about the books I have written.
Patents
 C. Gressel, N. Courtois, G. Bard, A. Hecht, R. Granot, T. J. Salmon, I. Mintz.
System and Methods for Encryption with Authentication Integrity.
Filed January 28, 2010. Published August 5, 2010. Awarded October 2, 2012.
US Patent No.: 8,280,056.
 C. Gressel, G. Bard, O. Dunkelman, A. Hecht, R. Granot.
System and Method to Preclude Message Modification in Data Authentications Systems
through Efficient Use of Feedback in Cryptographic Functions.
Filed Sept 6, 2007. Published March 13, 2008. Awarded October 2, 2012.
US Patent No.: 8,107,622.
Published and Submitted Articles
 Gregory Bard. "New
Practical Approximate Matrix Multiplication
Algorithms found via Solving a System of Cubic Equations."
The mathematical parts are done, but I am awaiting the results of a
potentially very long series of computations.
I am realeasing an early draft here, but still need to rephrase
my results in light of Igor Kaporin's work. When finished, this will
hopefully be submitted to The Mathematics of Computation.
 Gregory Bard, and David Hagman. "Are We Lying to Our Children? Conflating Real and
Nominal Rates of Return in Saving for Retirement." To be submitted soon
to the journal Teaching Mathematics and its Applications, Oxford
University Press.
 Nicolas Courtois, Gregory Bard, and Daniel Hulme.
"A new GeneralPurpose Method
to Multiply 3x3 Matrices Using Only 23 Multiplications." Under revision.
 Gregory Bard.
"Extending SATSolvers
to LowDegree Extension Fields of GF(2)."
Under revision.
 Gregory Bard.
"The Application
of Polynomials over the Field of Two Elements to
a problem in Intellectual Property."
Under revision. Hopefully to be submitted soon to the
International Journal of Digital Crime and Forensics.
 Gregory Bard, and Alexander Basyrov.
"Error
Bounds on Derivatives During Simulations." To be submitted soon to the journal Applied Numerical Mathematics.
 Martin Albrecht, Gregory Bard, and Clement Pernet.
"Effective
Dense Gaussian Elimination over the Finite Field
with Two Elements." Submitted to the journal ACM Transactions on
Mathematical Software.
 Nicolas Courtois, Gregory Bard, and Shaun van Ault.
"Statistics
of Random
Permutations and the Cryptanalysis of Periodic Block Ciphers."
The journal Cryptologia. Vol. 36 No. 3.
Pp 240262, 2012.
 Nicolas Courtois, and Gregory Bard.
"Random
Permutation Statistics and An Improved SlideDetermine Attack on KeeLoq."
Lecture Notes in Computer Science, Vol. 6805, Pp 3554, 2012.
 Michael Black, and Gregory Bard.
"SAT
Over BOINC: An ApplicationIndependent
Volutneer Grid Project."
Proceedings of the 12th IEEE/ACM International Conference
on Grid Computing (GRID'11), Lyon, France. IEEE Press, Pp 226227, 2011.
 Gregory Bard.
"Numerically
Estimating Derivatives During Simulations."
Proceedings of the 2011 International Conference on Modelling, Simulation, and
Visualization Methods (MSV'11). Las Vegas, Nevada. CSREA Press, Pp 341347, 2011.
 Gregory Bard, Nicolas Courtois, Jorge Nakahara Jr., Pouyan Sepehrdad,
and Bingsheng Zhang.
"Algebraic,
AIDA/Cube and SideChannel Analysis of
the KATAN family of Block Ciphers."
Progress in Cryptology (INDOCRYPT'10).
Hyderabad, India.
Lecture Notes in Computer Science,
Vol. 6498,
Pp 176196, 2010.
 Kenneth Wong, and Gregory Bard.
"Improved Algebraic
Cryptanalysis of QUAD, Bivium and Trivium
via Graph Partitioning on Equation Systems."
Proceedings of the
Australasian Conference on Information Security and Privacy (ACISP'10).
Sydney, Australia.
Lecture Notes in Computer Science,
Vol. 6168,
Pp. 1936, 2010.
 Gregory Bard.
"DEMOCRACY:
A Heuristic for Polynomial Systems of Equations over Finite
Fields." The journal ACM Communications in Computer Algebra.
Vol. 44 No. 1, Pg 25, 2010.
 Martin Albrecht, Gregory Bard, and Bill Hart.
"Algorithm
898: Efficient Multiplication of Dense Matrices over GF (2)."
The journal ACM Transactions on Mathematical Software.
Vol. 37, No. 1, 2009.
 Nicolas Courtois, Gregory Bard, and Andrey Bogdanov.
"Periodic
Ciphers with Small Blocks and Cryptanalysis of KeeLoq."
The journal Tatra Mountains Mathematical Publications.
(This is the mathematical Journal of the Slovak Academy of Sciences).
Vol. 41,
Pp. 167188,
2008.
 Nicolas Courtois, Gregory Bard, and David Wagner,
"Algebraic and Slide
Attacks on KeeLoq."
Fast Software Encryption (FSE'08).
Laussanne, Switzerland.
Lecture Notes in Computer Science,
Vol. 5086,
Pp. 97115,
2008.
 Nicolas Courtois, and Gregory Bard.
"Algebraic Cryptanalysis
of the Data Encryption Standard."
Proceedings of the IMA International Conference on Cryptography and Coding (IMACCC '07).
Cirencester, Wales.
Lecture Notes in Computer Science,
Vol. 4887,
Pp. 152169,
2008.
 Gregory Bard.
"Modes
of Encryption Secure Against BlockwiseAdaptive
ChosenPlaintext Attack."
Proceedings of the IMA International Conference on Cryptography and Coding (IMACCC '07).
Cirencester, Wales.
Lecture Notes in Computer Science,
Vol. 4887,
Pp. 129151,
2008.
 Gregory Bard.
"SpellingError
and Reordering Tolerant Passphrases
via the DamerauLevenshtein StringEdit Distance Metric."
Proceedings of the Australasian Information Security Workshop, (AISW '06).
ACM International Conference Proceeding Series,
Vol. 249,
Pp. 117124,
2007.
 Gregory Bard.
"A Challenging
but Feasible BlockwiseAdaptive ChosenPlaintext Attack on SSL."
Proceedings of the IEEEIACR joint International Conference on
Security and Cryptography, (SECRYPT '06).
Pp 99109,
2006.
 Note: The following is posted only for sentimental reasons, it was my first
research paper, and also the only bit of my work from my days at the National
Security Agency that I can talk about:
Gregory Bard.
"FLOWHUNT: An Attempt at
SpecificationBased Intrusion Detection using Neural Networks."
Proceedings of the 2nd Annual Computer Network Exploitation Conference,
(CNE '02). A conference limited to the US Department of Defense and
Intelligence Community, and the Ministries of Defense of certain allied
nations, but competitive and peerreviewed. While the proceedings were
classified, this paper was not.
Bachelors' and Master's Theses Supervised
 Joseph Bertino,
Honors Bachelor of Science in Mathematics and Economics,
from Fordham University.
Thesis: "Solving Systems of Polynomial Equations Using Gradient
Descent and Other Conjugate Gradient Methods, Enhanced by Darwinian and Evolutionary Methods."
Successfuly defended May of 2011.
 Kyle Kloster,
Honors Bachelor of Science in Mathematics,
from Fordham University.
Thesis: "
Factoring a semiprime n by estimating phi(n)."
Successfully defended on May 7th, 2010.
 Michael Levin,
Master of Science with Thesis in Computer Science,
from American University.
Thesis: "Darwinian
Gradient Descent."
Successfully defended on April 22, 2010.
 Seena Vali,
Bachelor of Science in Mathematics,
from Fordham.
(While Seena was not in the honors program, and so could not submit
a Bachelor's Thesis, he and I engaged in a great deal of research
that resulted in several conference presentations.
Expository and Technical Reports
 Gregory Bard. "DEMOCRACY:
A Heuristic for Polynomial systems of Equations over Finite Fields."
October 2010. (7 pp.)
 Gregory Bard. "Matrix Inversion,
LUPFactorization,
and System Solving, via the Method of Four Russians, in
Theta(n3 / log n) Time."
June 2009. (15 pp.)
 Kenneth Wong, Gregory Bard, and Robert Lewis.
"Partitioning Multivariate Polynomial Equations
via Vertex Separators for Algebraic Cryptanalysis and Mathematical
Applications." July 2009. (33 pp.)
 Gregory Bard, Carmi Gressel, Avi Hecht.
"Security
Analysis of the ZK Crypt Data Authenticator
and Stream Cipher against Algebraic Cryptanalysis,
Differential and Correlation Attacks." This is my
security analysis of the ZKCrypt hash function as part
of the NIST competition. January 2008. (20 pp.)
 Gregory Bard, Nicolas Courtois, and
Chris Jefferson. "Solution of
Sparse Polynomial Systems over GF(2)
via SATSolvers." January 2007. (14 pp.)
 Gregory Bard.
"Accelerating Cryptanalysis with
the Method of Four Russians." July 2006. (20 pp.)
 Gregory Bard. "Achieving
a log(n) Speed Up for Boolean Matrix Operations and Calculating the Complexity
of the Dense Linear Algebra step of Algebraic Stream Cipher Attacks and of
Integer Factorization Methods." January 2006. (20 pp.)
 Gregory Bard.
"Algorithms
for Fast Matrix Operations."
Scholarly paper for Master of Science (without Thesis) in Applied Mathematics
and Scientific Computation, December 2005. (13 pp.)
 Gregory Bard. "Vulnerability of SSL to
ChosenPlaintext Attack." March 2004. (10 pp.)
 Gregory Bard. "On
the Distributive Law."
This is only one page, but it is a favorite handout of mine in all elementary
classes. Any math teacher in high school or college who is tired of seeing
"(log a+b)" replaced with "(log a)+(log b)" should download this. I hereby
place the document "On the Distributive Law"
in the public domain, and waive all copyrights of this onepage document.
My Books, Lecture Notes, and My Dissertation
 For details on these, just click on "Books I Have Written" on the left of this page.
 Gregory Bard and Joseph Bertino.
Applied Finite and Financial Mathematics for University Freshmen.
In progress, scheduled to be finished in a few years.
 Gregory Bard. Sage for Undergraduates. To be published
by the American Mathematical Society in September 2014. Here is a link to the
electronic version,
which the AMS has permitted me to circulate for freeonly the printed version costs money.
 Gregory Bard. Algebraic Cryptanalysis. Published by Springer in 2009.
The book has
its own webpage.
 Gregory Bard.
"Group Theory."
Lecture Notes from when I taught Abstract Algebra.
These were intended to be a chapter of a manyauthor book project,
Mathematics, It's Not Just Calculus!, which is defunct. Intended for those
who have had a course in proofwriting or elementary notions (e.g. bijectivite,
injective, surjective, ...), but otherwise the text is as elementary as possible.
April 2008. (79 pp.)
 Gregory Bard.
"Algorithms
for Solving Linear and Polynomial Systems of
Equations over Finite Fields with Applications to Cryptanalysis."
(Submitted in fulfillment of the requirements of the degree
Doctor of Philosophy of Applied Mathematics and Scientific Computation,
under the supervision of Prof. Lawrence C. Washington.) 2007. (181 pp.)
Last updated October 23rd, 2013.
