ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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5.2 Rapid Subspace Enumeration . . . . . . . . . . . . . . . . . . . . . . 785.3 The Four Russians Matrix Multiplication Algorithm . . . . . . . . . . 805.3.1 Role of the Gray Code . . . . . . . . . . . . . . . . . . . . . . 815.3.2 Transposing the Matrix Product . . . . . . . . . . . . . . . . . 825.3.3 Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.4 A Quick Computation . . . . . . . . . . . . . . . . . . . . . . 825.3.5 M4RM Experiments Performed by SAGE Staff . . . . . . . . . 835.4 The Four Russians Matrix Inversion Algorithm . . . . . . . . . . . . . 845.4.1 Stage 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.2 Stage 2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.3 Stage 3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.4 A Curious Note on Stage 1 of M4RI . . . . . . . . . . . . . . . 875.4.5 Triangulation or Inversion? . . . . . . . . . . . . . . . . . . . . 905.5 Experimental and Numerical Results . . . . . . . . . . . . . . . . . . 915.6 Exact Analysis of Complexity . . . . . . . . . . . . . . . . . . . . . . 965.6.1 An Alternative Computation . . . . . . . . . . . . . . . . . . . 975.6.2 Full Elimination, not Triangular . . . . . . . . . . . . . . . . . 985.6.3 The Rank of 3k Rows, or Why k + ɛ is not Enough . . . . . . 995.6.4 Using Bulk Logical Operations . . . . . . . . . . . . . . . . . . 1015.6.5 M4RI Experiments Performed by SAGE Staff . . . . . . . . . 1025.6.5.1 Determination of k . . . . . . . . . . . . . . . . . . . 1025.6.5.2 Comparison to Magma . . . . . . . . . . . . . . . . . 1025.6.5.3 The Transpose Experiment . . . . . . . . . . . . . . 1035.7 Pairing With Strassen’s Algorithm for Matrix Multiplication . . . . . 1035.8 The Unsuitability of Strassen’s Algorithm for Inversion . . . . . . . . 1055.8.1 Bunch and Hopcroft’s Solution . . . . . . . . . . . . . . . . . 1075.8.2 Ibara, Moran, and Hui’s Solution . . . . . . . . . . . . . . . . 1086 An Impractical Method of Accelerating Matrix Operations in Rings of FiniteSize 1136.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.1.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2 The Algorithm over a Finite Ring . . . . . . . . . . . . . . . . . . . . 1156.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.3 Taking Advantage of z ≠ 1 . . . . . . . . . . . . . . . . . . . . 1186.2.4 The Transpose of Matrix Multiplication . . . . . . . . . . . . 1186.3 Choosing Values of b . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3.1 The “Conservative” Algorithm . . . . . . . . . . . . . . . . . . 1196.3.2 The “Liberal” Algorithm . . . . . . . . . . . . . . . . . . . . . 1206.3.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4 Over Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4.1 Complexity Penalty . . . . . . . . . . . . . . . . . . . . . . . . 1236.4.2 Memory Requirements . . . . . . . . . . . . . . . . . . . . . . 1236.4.3 Time Requirements . . . . . . . . . . . . . . . . . . . . . . . . 124xiii
6.4.4 The Conservative Algorithm . . . . . . . . . . . . . . . . . . . 1246.4.5 The Liberal Algorithm . . . . . . . . . . . . . . . . . . . . . . 1256.5 Very Small Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . 1256.6 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.7.1 Ring Additions . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.7.2 On the Ceiling Symbol . . . . . . . . . . . . . . . . . . . . . . 129A Some Basic Facts about Linear Algebra over GF(2) 131A.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.2 Boolean Matrices vs GF(2) Matrices . . . . . . . . . . . . . . . . . . 131A.3 Why is GF(2) Different? . . . . . . . . . . . . . . . . . . . . . . . . . 131A.3.1 There are Self-Orthogonal Vectors . . . . . . . . . . . . . . . . 132A.3.2 Something that Fails . . . . . . . . . . . . . . . . . . . . . . . 132A.3.3 The Probability a Random Square Matrix is Singular or Invertible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B A Model for the Complexity of GF(2)-Operations 135B.1 The Cost Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135B.1.1 Is the Model Trivial? . . . . . . . . . . . . . . . . . . . . . . . 136B.1.2 Counting Field Operations . . . . . . . . . . . . . . . . . . . . 136B.1.3 Success and Failure . . . . . . . . . . . . . . . . . . . . . . . . 137B.2 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.3 To Invert or to Solve? . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.4 Data Structure Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.4.1 Dense Form: An Array with Swaps . . . . . . . . . . . . . . . 139B.4.2 Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . 139B.5 Analysis of Classical Techniques with our Model . . . . . . . . . . . . 140B.5.1 Naïve Matrix Multiplication . . . . . . . . . . . . . . . . . . . 140B.5.2 Matrix Addition . . . . . . . . . . . . . . . . . . . . . . . . . 140B.5.3 Dense Gaussian Elimination . . . . . . . . . . . . . . . . . . . 140B.5.4 Strassen’s Algorithm for Matrix Multiplication . . . . . . . . . 142C On the Exponent of Certain Matrix Operations 145C.1 Very Low Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 145C.2 The Equicomplexity Theorems . . . . . . . . . . . . . . . . . . . . . . 146C.2.1 Starting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 147C.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147C.2.3 A Common Misunderstanding . . . . . . . . . . . . . . . . . . 153Bibliography 154xiv
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: 3 Converting MQ to CNF-SAT 353.1 Su
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
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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
Page 99 and 100:
matrix without calculating the othe
Page 101 and 102:
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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