ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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first was a very carefully coded GF(2)-matrix operations and linear algebra library,that included the Method of Four Russians. This library was adopted by Sage,and is written in traditional ANSI C. The second relates to Sat-solvers, and isa Java library for converting polynomials into Cnf-Sat problems. I have decidedthat these two libraries are to be made publicly available, as soon as the formalitiesof submitting the dissertation are completed. (the first is already available underthe GPL—Gnu Public License).4
Part IPolynomial Systems5
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 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: adopted the author’s software lib
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
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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
Page 91 and 92:
If the contradiction stack becomes
Page 93 and 94:
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
Page 99 and 100:
matrix without calculating the othe
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exactly one bit in one position fro
Page 103 and 104:
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
Page 109 and 110:
(see Section 5.5 on page 92) that s
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= (1 − 2 −n )(1 − 2 −n+1 )
Page 113 and 114:
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
Page 127 and 128:
algorithm (the 18 matrix additions,
Page 129 and 130:
⎡⎤⎡⎤A =⎢⎣1 0 0 01 0 1 0
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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
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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
Page 157 and 158:
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
Page 171 and 172:
If the original matrix is unit lowe
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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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