ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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integer, between 0 and 31.Then a non-linear function is applied, which will bedescribed shortly (denoted NLF).Meanwhile the key is placed initially in a 64-bit shift-register, which is alsoshifted one bit to the right at each iteration. The new bit introduced at the left isthe bit formerly at the right, and so the key is merely rotating.The least significant bit of the key, the output of the non-linear function, andtwo particular bits of the 32 bit shift-register are XORed together (added in GF(2)).The 32-bit shift-register is shifted right and the sum is now the new bit to be insertedinto the leftmost spot in the 32-bit shift-register.After 528 rounds, the contents of the 32 bit shift-register form the ciphertext.2.3.4 Modeling the Non-linear FunctionThe non-linear function NLF (a, b, c, d, e) is denoted NLF 3A5C742E . This meansthat if (a, b, c, d, e) is viewed as an integer i between 0 and 31, i.e. as a 5-bit number,then the value of NLF (a, b, c, d, e) is the ith bit of the 32-bit hexadecimal value3A5C742E.The following formula is a cubic polynomial and gives equivalent output tothe NLF for all input values, and was obtained by a Karnaugh map. In the case,the Karnaugh map is a grid with (for five dimensions) two variables in rows (i.e.4 rows), and three variables in columns (i.e. 8 columns). The rows and columnsare arranged via the Gray Code. This is a simple technique to rapidly arrive at thealgebraic normal form (i.e. polynomial), listed below, by first trying to draw boxes10
around regions of ones of size 32, 16, 8, 4, 2, and finally 1. See a text such as [Bar85,Ch. 3] for details.NLF (a, b, c, d, e) = d ⊕ e ⊕ ac ⊕ ae ⊕ bc ⊕ be ⊕ cd ⊕ de ⊕ ade ⊕ ace ⊕ abd ⊕ abc2.3.5 I/O Relations and the NLFAlso note that while the degree of this function is 3, there is an I/O relation ofdegree 2, below. An I/O relation is a polynomial in the input variables and outputvariables of a function, such that no matter what values are given for input to thefunction, the I/O relation always evaluates to zero. Note y signifies the output ofthe non-linear function.(e ⊕ b ⊕ a ⊕ y)(c ⊕ d ⊕ y) = 0This can be thought of as a constraint that the function must always satisfy.If there are enough of these, then the function is uniquely defined. What makesthem cryptanalyticly interesting is that the degree of the I/O relations can be muchlower than the degree of the function itself.Since degree impacts the difficultyof polynomial system solving dramatically, this is very useful. The I/O degree ofa function is the lowest degree of any of its I/O relations, other than the zeropolynomial.Generally, low I/O-degree can be used for generating attacks but that is notthe case here, because we have only one relation, and this above relation is true11
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: Figure 2.1: The Keeloq Circuit Diag
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
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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
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
Page 179 and 180:
[FMM03]C. Fiorini, E. Martinelli, a
Page 181:
[Rys57][sag][San79][Sch81]H. J. Rys
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