ALGORITHMS FOR SOLVING LINEAR AND POLYNOMIAL ...

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Chapter 2An Extended Example: The Block-Cipher KeeloqThe purpose of this chapter is to supply a new, feasible, and economicallyrelevant example of algebraic cryptanalysis. The block cipher “Keeloq” 1 is used inthe keyless-entry system of most luxury automobiles. It has a secret key consistingof 64 bits, takes a plaintext of 32 bits, and outputs a ciphertext of 32 bits. Thecipher consists of 528 rounds. Our attack is faster than brute force by a factor ofaround 2 14.77 as shown in Section 2.4.7 on page 26. A summary will be given inSection 2.5 on page 32.This attack requires around 0.6×2 32 plaintexts, or 60% of the entire dictionary,as calculated in Section 2.4.5 on page 22. This chapter is written in the “chosenplaintext attack” model, in that we assume that we can request the encryption ofany plaintext and receive the corresponding ciphertext as encrypted under the secretkey that we are to trying guess. This will be mathematically represented by oracleaccess to E k ( ⃗ P ) = ⃗ C.However, it is easy to see that random plaintexts wouldpermit the attack to proceed identically.1 This is to be pronounced “key lock.”6
2.1 Special Acknowledgment of Joint WorkThe work described in this chapter was performed during a two-week visitwith the Information Security Department of the University College of London’sIpswich campus. During that time the author worked withNicolas T. Courtois.The content of this chapter is joint work. Nonetheless, the author has rewrittenthis text in his own words and notation, distinct from the joint paper [CB07]. Someproofs are found here which are not found in the paper.2.2 Notational Conventions and TerminologyEvaluating the function f eight times will be denoted f (8) .For any l-bit sequence, the least significant bit is numbered 0 and the mostsignificant bit is numbered l − 1.If h(x) = x for some function h, then x is a fixed point of h. If h(h(x)) = xbut h(x) ≠ x then x is a “point of order 2” of h. In like manner, if h (i) (x) = x buth (j) (x) ≠ x for all j < i, then x is a “point of order i” of h. Obviously if x is a pointof order i of h, thenh (j) (x) = x if and only if i|j7
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: Part IPolynomial Systems5
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
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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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