ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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and I thank them for their guidance. Each has taught me from their own view ofcomputer security and it is interesting how many points of view there really are.Our field is enriched by this diversity of scholarly background. Likewise Dr HaroldSnider gave me important advice during my time at Oxford. My Oxford days werean enormous influence on my decision to switch into Applied Mathematics fromComputer Engineering.My alma mater, the Rensselaer Polytechnic Institute, has furnished me withmany skills. I cannot imagine a better training ground. My engineering degreewas sufficiently mathematically rigorous that I succeeded in a graduate program inApplied Mathematics, without having a Bachelor’s degree in Mathematics. In fact,only two courses, Math 403, Abstract Algebra, and Math 404 Field Theory, weremy transition. Yet I was practical enough to hired into and promoted several timesat the NSA, from a Grade 6 to a Grade 11 (out of a maximum of 15), including severalcommendations and a Director’s Star. I am particularly indebted to the facultythere, and its founder, Stephen Van Rensselaer, former governor of New York and(twice) past Grand Master of Free & Accepted Masons of the State of New York. Itwas at Rensselaer that I learned of and was initiated into Freemasonry, and the Institute’smission of educating “the sons of farmers and mechanics” with “knowledgeand thoroughness” echoes the teachings of Freemasonry. My two greatest friends,chess partners, and Brother Freemasons, Paul Bello and Abram Claycomb, wereinitiated there as well, and I learned much in discussions with them.I learned of cryptography at a very young age, perhaps eight years old or so.Dr. Haig Kafafian, also a Freemason and a family friend, took me aside after Churchix
for many years and taught me matrices, classical cryptography, and eventually,allowed me to be an intern at his software development project to make devices toaid the handicapped. There I met Igor Kalatian, who taught me the C language. Iam grateful to them both.x
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 and 10: AcknowledgmentsI am deeply indebted
Page 11: NSF grant for the Sage project [sag
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
Page 63 and 64:
Step One: From a Polynomial System
Page 65 and 66:
x 1 + x 2 + x 3 + y 1 = 0y 1 + x 6
Page 67 and 68:
Table 3.1 CNF Expression Difficulty
Page 69 and 70:
that a satisfying solution is found
Page 71 and 72:
andomness is seeded from a hash of
Page 73 and 74:
Figure 3.1: The Distribution of Run
Page 75 and 76:
takes only a few seconds. Furthermo
Page 77 and 78:
3.5.4 Comparison with MAGMA, Singul
Page 79 and 80:
systems of equations over finite fi
Page 81 and 82:
Table 3.4 CNF Expression Difficulty
Page 83 and 84:
polynomial equation. The number of
Page 85 and 86:
of variables, and the operators fro
Page 87 and 88:
variables). However, the original v
Page 89 and 90:
of variables that it contains.When
Page 91 and 92:
If the contradiction stack becomes
Page 93 and 94:
enough to produce a contradiction.I
Page 95 and 96:
those two simple examples).Over the
Page 97 and 98:
Chapter 5The Method of Four Russian
Page 99 and 100:
matrix without calculating the othe
Page 101 and 102:
exactly one bit in one position fro
Page 103 and 104:
5.3.1 Role of the Gray CodeThe Gray
Page 105 and 106:
Then substitute k = γ log n and ob
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and putting matrix A in unit upper
Page 109 and 110:
(see Section 5.5 on page 92) that s
Page 111 and 112:
= (1 − 2 −n )(1 − 2 −n+1 )
Page 113 and 114:
Table 5.3 Probabilities of a Fair-C
Page 115 and 116:
time was trivially perturbed if the
Page 117 and 118:
Table 5.6 Percentage Error for Offs
Page 119 and 120:
so there may be no work to perform
Page 121 and 122:
calculation proceeds exactly as in
Page 123 and 124:
ank is 1 − l −2c . Since there
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ond, 10, 000 × 10, 000, where M4RI
Page 127 and 128:
algorithm (the 18 matrix additions,
Page 129 and 130:
⎡⎤⎡⎤A =⎢⎣1 0 0 01 0 1 0
Page 131 and 132:
and P can be kept invertible. The d
Page 133 and 134:
Table 5.9 Trials between M4RI and G
Page 135 and 136:
Chapter 6An Impractical Method of A
Page 137 and 138:
6.2 The Algorithm over a Finite Rin
Page 139 and 140:
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
Page 145 and 146:
kind. (The all zero matrix has been
Page 147 and 148:
(∼ (1 + log 2 n)n 2 1 + o(1) )log
Page 149 and 150:
Table 6.2 Memory Required (in bits)
Page 151 and 152:
While the cases of matrix rings, fi
Page 153 and 154:
Appendix ASome Basic Facts about Li
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not orthogonal. Thus the algorithm
Page 157 and 158:
Appendix BA Model for the Complexit
Page 159 and 160:
B.1.3Success and FailureThe model d
Page 161 and 162:
example, in a matrix multiplication
Page 163 and 164:
listed in Algorithm 10 on page 142,
Page 165 and 166:
allowing one to calculate, for a la
Page 167 and 168:
Appendix COn the Exponent of Certai
Page 169 and 170:
C.2.1Figure C.1: The Relationship o
Page 171 and 172:
If the original matrix is unit lowe
Page 173 and 174:
we can write[ ] [ ] [ ] [ ]Im/2 0 I
Page 175 and 176:
Proof: If A = LUP then det(A) = det
Page 177 and 178:
[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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