ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

More documents

Recommendations

Info

systems but with polynomial systems. Also, it deals not only with dense systems(as Part II does), but with sparse systems also. Since it is known that solving aquadratic system of equations is NP-hard, and solving the Cnf-Sat problem is NPhard,and since all NP-Complete problems are polynomially reducible to each other,it makes sense to look for a method to use one in solving the other. This chaptershows how to convert quadratic systems of equations into Cnf-Sat problems, andthat using off-the-shelf Sat-solvers is an efficient method of solving this difficultproblem.Chapter 4 describes in general terms how Sat-solvers work. This tool is oftenviewed as a black box, which is unfortunate. There is no novel work in this chapter,except the author does not know of any other exposition on how these tools operate,either for experts or a general audience.The second part begins with Chapter 5, and contains the Method of FourRussians, which is an algorithm published in the 1970s, but mostly forgotten since,for calculating a step of the transitive closure of a digraph, and thus also squaringboolean matrices. Later it was adapted to matrix multiplication. This chapterprovides an analysis of that algorithm, but also shows a related algorithm for matrixinversion that was anecdotally known among some French cryptanalysts. However,the algorithm was not frequently used because it was unclear how to eliminate theprobability of abortion, how to handle memory management, and how to optimizethe algorithm. The changes have made negligible the probability of abortion, andhave implemented the algorithm so that it outperforms Magma [mag] in some cases.The software tool Sage [sag], which is an open source competitor to Magma, has2
adopted the author’s software library (built on the Method of Four Russians) for itsdense GF(2)-matrix operations, and this code is now deployed.Chapter 6 has an algorithm of theoretical interest, but which may find somepractical application as well. This algorithm is for finite-ring matrix multiplication,with special attention and advantage to the finite-field case. The algorithm takesany “baseline” matrix multiplication algorithm which works over general rings, ora class of rings that includes finite rings, and produces a faster version tailored toa specific finite ring. However, the memory required is enormous. Nonetheless, it isfeasible for certain examples. The algorithm is based on the work of Atkinson andSantoro [AS88], but introduces many more details, optimizations, techniques, anddetailed analysis.This chapter also modifies Atkinson and Santoro’s complexitycalculations.Three appendices are found, which round out Part II. Appendix A containssome discussion of GF(2)-linear algebra and how it differs from linear algebra in Rand C. These facts are well-known.We introduce a new complexity model and data structures for GF(2)-matrixoperations. This is discussed in Appendix B but applies to all of Part II.Appendix C discusses algorithms faster than Strassen’s algorithm, and containsproofs that matrix multiplication, matrix squaring, triangular matrix inversion,LUP-factorization, general matrix inversion and the taking of determinants,are equicomplex. These proofs are already known, but are here gathered into oneplace in the same notation.Finally, two software libraries were created during the dissertation work. The3
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: Chapter 1SummaryGenerally speaking,
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
Page 107 and 108:
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
Page 125 and 126:
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
Page 141 and 142:
which is to be compared with∼ 4cn
Page 143 and 144:
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
Page 155 and 156:
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
show all

ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

Create successful ePaper yourself

Delete template?

Save as template?