ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

More documents

Recommendations

Info

After translating from cubic into quadratic format, it becomes 384 equationsand 384 unknowns. This is much smaller than the 3168 equations and 3168 unknownswe had before. In each case, ElimLin, Magma, Singular, and the methodsof Chapter 3 solved the system for k 0 , . . . , k 63 in time too short to measure accurately(i.e. less than 1 minute).It should be noted that we require two fixed points, not merely one, to makethe attack work.One fixed point alone is not enough of a constraint to narrowthe keyspace sufficiently. However, two fixed points was sufficient each time it wastried. Therefore, we will assume f has two or more fixed points, and adjust ourprobabilities of success accordingly.One way to look at this is to say that onlythose keys which result in two or more fixed points are vulnerable to our attack.However, since the key changes rapidly in most applications (See Section 2.6 onpage 34), and since approximately 26.42% of random functions GF(2) 32 → GF(2) 32have this property (See Section 2.4.8 on page 29), we do not believe this to be amajor drawback.2.4.4 How to Find Fixed PointsObviously a fixed point of f k is a fixed point of f (8)kas well, but the reverseis not necessarily true. Stated differently, the set of fixed points of f (8)kwill containthe set of all fixed points of f k .We will first calculate the set of fixed points of f (8)k, which will be very small.We will try the attack given in the previous subsection, using every pair of fixed20
points. If it is the case that f k has two or more such fixed points, then one such pairwhich we try will indeed be a pair of fixed points of f k . This will produce the correctsecret key. The other pairs will produce spurious keys or inconsistent systems ofequations. But this is not a problem because spurious keys can be easily detectedand discarded.The running time required to solve the system of equations is too short toaccurately measure, with a valid or invalid pair. Recall, that this is 384 equationsand 384 unknowns as compared to 3168, as explained in Section 2.4.3 on page 20.There are probably very few fixed points of f (8) , which we will prove below.kAnd thus the running time of the entire attack depends only upon finding the setof fixed points of f (8)k. One approach would be to iterate through all 232 possibleplaintexts, using the f (8)koracle. This would clearly uncover all possible fixed pointsof f (8)kand if f k has any fixed points, they would be included. However, this is notefficient.Instead, one can simply try plaintexts in sequence using the f (8)koracle. Whenthe ith fixed point x i is found, one tries the attack with the i−1 pairs (x 1 , x i ), . . . , (x i−1 , x i ).If two fixed points of f k are to be found in x 1 , . . . , x i , the attack will succeed at thispoint, and we are done. Otherwise, continue until x i+1 is found and try the pairs(x 1 , x i+1 ,. . . , x i , x i+1 ), and so forth.21
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 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: 2.4.3 The Consequences of Fixed Poi
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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?