ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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verts polynomial systems into larger linear systems. We introduce a new complexitymodel and data structures for GF(2)-matrix operations. This is discussed in AppendixB but applies to all of Part II. Chapter 5 contains an analysis of “the Methodof Four Russians” for multiplication and a variant for matrix inversion, which islog n faster than Gaussian Elimination, and can be combined with Strassen-like algorithms.Chapter 6 contains an algorithm for accelerating matrix multiplicationover small finite fields. It is feasible but the memory cost is so high that it is mostlyof theoretical interest. Appendix A contains some discussion of GF(2)-linear algebraand how it differs from linear algebra in R and C. Appendix C discusses algorithmsfaster than Strassen’s algorithm, and contains proofs that matrix multiplication,matrix squaring, triangular matrix inversion, LUP-factorization, general matrix inversionand the taking of determinants, are equicomplex. These proofs are alreadyknown, but are here gathered into one place in the same notation.
ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIALSYSTEMS OF EQUATIONS OVER FINITE FIELDS,WITH APPLICATIONS TO CRYPTANALYSISbyGregory V. BardDissertation submitted to the Faculty of the Graduate School of theUniversity of Maryland, College Park in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy2007Advisory Committee:Professor Lawrence C. Washington, Chair/AdvisorProfessor William Adams,Professor Steven Tretter,Professor William Gasarch,Assistant Professor Jonathan Katz.
Page 1: ABSTRACTTitle of dissertation:ALGOR
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 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
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too long, we may elect to “guess
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Chapter 3Converting MQ to CNF-SAT3.
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Another NP-hard problem is finding
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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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