ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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DedicationWith pleasure, I dedicate this dissertation to my parents. To my father, whotaught me much in mathematics, most especially calculus, years earlier than I wouldhave been permitted to see it. And to my mother, who has encouraged me in everyendeavor.v
AcknowledgmentsI am deeply indebted to my advisor, Professor Lawrence C. Washington, whohas read and re-read many versions of this document, and every other research paperI have written to date, provided me with countless hours of his time, and given mesound advice, on matters technical, professional and personal. He has had manystudents, and all I have spoken to are in agreement that he is a phenomenal advisor,and it is humbling to know that we can never possibly equal him. We can only tryto emulate him.I have been very fortunate to work with Dr. Nicolas T. Courtois, currentlySenior Lecturer at the University College of London. It suffices to say that the fieldof algebraic cryptanalysis has been revolutionized by his work.He permitted meto study with him for four months in Paris, and later even to stay at his home inEngland. Much of my work is joint work with him.My most steadfast supporter has been my best friend and paramour, PatrickStuddard. He has read every paper I have written to date, even this dissertation.He has listened to me drone on about topics like group theory or ring theory andhas tolerated my workaholism. My life and my work would be much more emptywithout him.My first autonomous teaching experience was at American University, a majorstepping stone in my life. The Department of Computer Science, Audio Technologyand Physics had faith in me and put their students in my hands. I am so very gratefulfor those opportunities. In particular I would like to thank Professor Michael Gray,vi
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: ForewordThe ignoraunte multitude do
Page 11 and 12: NSF grant for the Sage project [sag
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
Page 77 and 78:
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
Page 97 and 98:
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
Page 179 and 180:
[FMM03]C. Fiorini, E. Martinelli, a
Page 181:
[Rys57][sag][San79][Sch81]H. J. Rys
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