ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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and Professor Angela Wu, who gave me much advice. The departmental secretary,Yana Shabaev was very kind to me on numerous occasions and I am grateful forall her help.I also received valuable teaching advice from my future father-inlaw,Emeritus Professor Albert Studdard, of the Philosophy Department at theUniversity of North Carolina at Pembroke.My admission to the Applied Mathematics and Scientific Computation (AMSC)Program at the University of Maryland was unusual. I had been working as a PhDstudent in Computer Engineering for several years when I decided to add a Mastersin AMSC as a side task. Later, I switched, and made it the focus of my PhD.Several faculty members were important in helping me make that transition, includingProfessor Virgil D. Gligor, Professor Richard Schwartz, and Professor JonathanRosenberg. But without the program chair, Professor David Levermore, and myadvisor, Professor Lawrence C. Washington, this would have been impossible. I amgrateful that many rules were bent.Several governmental agencies have contributed to my education. In reversechronological order,ˆ The National Science Foundation (NSF) VIGRE grant to the University ofMaryland provided the Mathematics Department with funds for “DissertationCompletion Fellowships.” These are exceedingly generous semester-long giftsand I had the pleasure to receive two of them. I am certain this dissertationwould look very different without that valuable support.ˆ I furthermore received travel funding and access to super-computing under thevii
NSF grant for the Sage project [sag], Grant Number 0555776.ˆ The European Union established the ECRYPT organization to foster researchamong cryptographers. The ECRYPT foundation was kind enough to supportme during my research visit in Paris. Much of the work of this dissertationwas done while working there.ˆ The Department of the Navy, Naval Sea Systems Command, Carderock Station,supported me with a fascinating internship during the summer of 2003.They furthermore allowed me to continue some cryptologic research towardmy graduate degrees while working there.ˆ The National Security Agency was my employer from May of 1998 untilSeptember of 2002. Not only did I have the chance to work alongside andlearn from some of the most dedicated and professional engineers, computerscientists and mathematicians who have ever been assembled into one organization,but also I was given five semesters of financial support, plus severalsummer sessions, which was very generous, especially considering that two ofthose semesters and one summer were during wartime. Our nation owes anenormous debt to the skill, perseverance, self-discipline, and creativity of theemployees of the National Security Agency. Several employees there were supportersof my educational hopes but security regulations and agency traditionswould forbid my mentioning them by name.Professor Virgil D. Gligor, Professor Jonathan Katz, and Professor WilliamR. Franklin (of RPI), have all been my academic advisor at one time or anotherviii
Page 1 and 2: ABSTRACTTitle of dissertation:ALGOR
Page 3 and 4: ALGORITHMS FOR SOLVING LINEAR AND P
Page 5 and 6: PrefacePigmaei gigantum humeris imp
Page 7 and 8: ForewordThe ignoraunte multitude do
Page 9: AcknowledgmentsI am deeply indebted
Page 13 and 14: for many years and taught me matric
Page 15 and 16: 3 Converting MQ to CNF-SAT 353.1 Su
Page 17 and 18: 6.4.4 The Conservative Algorithm .
Page 19 and 20: B.1 Algorithms and Performance, for
Page 21 and 22: List of Algorithms1 Method of Four
Page 23 and 24: Chapter 1SummaryGenerally speaking,
Page 25 and 26: adopted the author’s software lib
Page 27 and 28: Part IPolynomial Systems5
Page 29 and 30: 2.1 Special Acknowledgment of Joint
Page 31 and 32: Figure 2.1: The Keeloq Circuit Diag
Page 33 and 34: around regions of ones of size 32,
Page 35 and 36: the bit generated in the 528th and
Page 37 and 38: β = aeSince the non-linear functio
Page 39 and 40: were an obvious failure, and so we
Page 41 and 42: 2.4.3 The Consequences of Fixed Poi
Page 43 and 44: points. If it is the case that f k
Page 45 and 46: 2.4.6 Fraction of Plainspace Requir
Page 47 and 48: Then we can apply the probability d
Page 49 and 50: e the case for all block ciphers. T
Page 51 and 52: Thus h has p fixed points with prob
Page 53 and 54: expected points. Since we require f
Page 55 and 56: too long, we may elect to “guess
Page 57 and 58: Chapter 3Converting MQ to CNF-SAT3.
Page 59 and 60: Another NP-hard problem is finding
Page 61 and 62:
3.3 Notation and DefinitionsAn inst
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Step One: From a Polynomial System
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x 1 + x 2 + x 3 + y 1 = 0y 1 + x 6
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Table 3.1 CNF Expression Difficulty
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that a satisfying solution is found
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andomness is seeded from a hash of
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Figure 3.1: The Distribution of Run
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takes only a few seconds. Furthermo
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3.5.4 Comparison with MAGMA, Singul
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systems of equations over finite fi
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Table 3.4 CNF Expression Difficulty
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polynomial equation. The number of
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of variables, and the operators fro
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variables). However, the original v
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of variables that it contains.When
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If the contradiction stack becomes
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enough to produce a contradiction.I
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those two simple examples).Over the
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Chapter 5The Method of Four Russian
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matrix without calculating the othe
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exactly one bit in one position fro
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5.3.1 Role of the Gray CodeThe Gray
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Then substitute k = γ log n and ob
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and putting matrix A in unit upper
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(see Section 5.5 on page 92) that s
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= (1 − 2 −n )(1 − 2 −n+1 )
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Table 5.3 Probabilities of a Fair-C
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time was trivially perturbed if the
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Table 5.6 Percentage Error for Offs
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so there may be no work to perform
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calculation proceeds exactly as in
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ank is 1 − l −2c . Since there
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ond, 10, 000 × 10, 000, where M4RI
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algorithm (the 18 matrix additions,
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⎡⎤⎡⎤A =⎢⎣1 0 0 01 0 1 0
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and P can be kept invertible. The d
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Table 5.9 Trials between M4RI and G
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Chapter 6An Impractical Method of A
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6.2 The Algorithm over a Finite Rin
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c(n/b) ω multiplies in the ring M
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which is to be compared with∼ 4cn
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yielding a time requirement of[ (ω
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kind. (The all zero matrix has been
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(∼ (1 + log 2 n)n 2 1 + o(1) )log
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Table 6.2 Memory Required (in bits)
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While the cases of matrix rings, fi
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Appendix ASome Basic Facts about Li
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not orthogonal. Thus the algorithm
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Appendix BA Model for the Complexit
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B.1.3Success and FailureThe model d
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example, in a matrix multiplication
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listed in Algorithm 10 on page 142,
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allowing one to calculate, for a la
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Appendix COn the Exponent of Certai
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C.2.1Figure C.1: The Relationship o
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If the original matrix is unit lowe
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we can write[ ] [ ] [ ] [ ]Im/2 0 I
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Proof: If A = LUP then det(A) = det
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[BH74]J. Bunch and J. Hopcroft. Tri
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[FMM03]C. Fiorini, E. Martinelli, a
Page 181:
[Rys57][sag][San79][Sch81]H. J. Rys
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