Verfügbarkeit: Algebraic cryptanalysis

Algebraic cryptanalysis:

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1. Verfasser:	Bard, Gregory V. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Dordrecht [u.a.] Springer 2009
Schlagworte:	Cryptography Data encryption (Computer science) Kryptoanalyse
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXXIII, 356 S. graph. Darst.
ISBN:	9780387887579 9780387887562

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Datensatz im Suchindex

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adam_text	Contents List of Ta ¡ .....................................................xxvii List of Fi; ;s ....................................................xxix List of AI ¡thms..................................................xxxi List of Al viations .............................................xxxi і і 1 Intre :tion: How to Use this Book .............................. 1 Part I Ci analysis 2 The . :k Cipher Keeloq and Algebraic Attacks ................... 9 2.1 iát is Algebraic Cryptanalysis? ............................. 10 .1 The CSP Model ..................................... 10 2.2 ; Keeloq Specification .................................... 10 2.3 deling the Non-linear Function ............................ 11 .1 I/O Relations and the NLF ............................ 12 2.4 scribing the Shift-Registers ................................ 12 . 1 Disposing of the Secret Key Shift-Register ............... 13 2.4.2 Disposing of the Plaintext Shift-Register ................. 13 2.5 The Polynomial System of Equations .......................... 13 2.6 Variable and Equation Count ................................. 14 2.7 Dropping the Degree to Quadratic ............................. 14 2.8 Fixing or Guessing Bits in Advance ........................... 15 2.9 The Failure of a Frontal Assault .............................. 16 3 The Fixed-Point Attack ......................................... 17 3.1 Overview ................................................. 17 3.1.1 Notational Conventions ............................... 17 3.1.2 The Two-Function Representation ...................... 17 3.1.3 Acquiring an fk -oracle .............................. 18 Contents 3.2 The Consequences of Fixed Points ............................ 18 3.3 How to Find Fixed Points .................................... 19 3.4 How far must we search? .................................... 20 3.4.1 With Analytic Combinatorics .......................... 21 3.4.2 Without Analytic Combinatorics ....................... 23 3.5 Comparison to Brute Force .................................. 23 3.6 Summary ................................................. 24 3.7 Other Notes ............................................... 25 3.7.1 A Note about Keeloq s Utilization ...................... 25 3.7.2 RPA vs KPA vs СРА ................................. 26 3.8 Wagner s Attack ........................................... 26 3.8.1 Later Work on Keeloq ................................ 27 Iterated Permutations .......................................... 29 4.1 Applications to Cryptography ................................ 29 4.2 Background ............................................... 30 4.2.1 Combinatorial Classes ................................ 30 4.2.2 Ordinary and Exponential Generating Functions .......... 30 4.2.3 Operations on OGFs .................................. 31 4.2.4 Examples ........................................... 34 4.2.5 Operations on EGFs .................................. 36 4.2.6 Notation and Definitions .............................. 39 4.3 Strong and Weak Cycle Structure Theorems .................... 40 4.3.1 Expected Values ..................................... 41 4.4 Corollaries ................................................ 43 4.4.1 On Cycles in Iterated Permutations ..................... 45 4.4.2 Limited Cycle Counts ................................ 46 4.4.3 Monomial Counting .................................. 47 4.5 Of Pure Mathematical Interest ................................ 47 4.5.1 The Sigma Divisor Function ........................... 48 4.5.2 The Zeta Function and Apery s Constant ................. 48 4.5.3 Greatest Common Divisors and Cycle Length ............ 49 4.6 Highly Iterated Ciphers ...................................... 49 4.6.1 Distinguishing Iterated Ciphers ........................ 50 4.6.2 A Key Recovery Attack ............................... 52 Stream Ciphers ................................................ 55 5.1 The Stream Ciphers Bivium and Trivium ....................... 55 5.1.1 Background ......................................... 55 5.1.2 Bivium as Equations ................................. 61 5.1.3 An Excellent Trick ................................... 64 5.1.4 Bivium-A ........................................... 65 5.1.5 A Notational Issue ................................... 65 5.1.6 For Further Reading .................................. 65 5.2 The Stream Cipher QUAD ................................... 66 Contents xjx 5.2.1 How QUAD Works .................................. 66 5.2.2 Proof of Security .................................... 67 5.2.3 The Yang-Chen-Bernstein-Chen Attack against QUAD ___ 72 5.2.4 Extending to GF(16) ................................. 75 5.2.5 For Further Reading .................................. 77 5.3 Conclusions for QUAD ..................................... 78 Part II Linear Systems Mod 2 6 Some Basic Facts about Linear Algebra over GF(2) ................ 81 6.1 Sources ................................................... 81 6.2 Boolean Matrices vs GF(2) Matrices .......................... 81 6.2.1 Implementing with the Integers ........................ 82 6.3 Why is GF(2) Different? .................................... 82 6.3.1 There are Self-Orthogonal Vectors ...................... 82 6.3.2 Something that Fails .................................. 83 6.3.3 The Probability a Random Square Matrix Singular or Invertible ........................................... 84 6.4 Null Space from the RREF ................................... 85 6.5 The Number of Solutions to a Linear System ................... 86 7 The Complexity of GF(2)-Matrix Operations ...................... 89 7.1 The Cost Model ............................................ 89 7.1.1 A Word on Architecture and Cross-Over ................. 90 7.1.2 Is the Model Trivial? ................................. 91 7.1.3 Counting Field Operations ............................ 91 7.1.4 Success and Failure .................................. 92 7.2 Notational Conventions ..................................... 92 7.3 To Invert or to Solve? ....................................... 93 7.4 Data Structure Choices ...................................... 94 7.4.1 Dense Form: An Array with Swaps ..................... 94 7.4.2 Permutation Matrices ................................. 94 7.5 Analysis of Classical Techniques with our Model ................ 96 7.5.1 Naïve Matrix Multiplication ........................... 96 7.5.2 Matrix Addition ..................................... 96 7.5.3 Dense Gaussian Elimination ........................... 96 7.5.4 Back-Solving a Triangulated Linear System .............. 98 7.6 Strassen s Algorithms ....................................... 99 7.6.1 Strassen s Algorithm for Matrix Multiplication ........... 100 7.6.2 Misunderstanding Strassen s Matrix Inversion Formula ---- 101 7.7 The Unsuitability of Strassen s Algorithm for Inversion ........... 101 7.7.1 Strassen s Approach to Matrix Inversion ................. 102 7.7.2 Bunch and Hopcroft s Solution ......................... 103 7.7.3 Ibara, Moran, and Hui s Solution ....................... 103 Contents On the Exponent of Certain Matrix Operations 107 8.1 Very Low Exponents ........................................107 8.2 The Equicompl exity Theorems ...............................108 8.2.1 Starting Point .......................................109 8.2.2 Proofs ..............................................109 8.3 Determinants and Matrix Inverses .............................118 8.3.1 Background .........................................118 8.3.2 The Baur-Strassen-Morgenstern Theorem ................120 8.3.3 Consequences for the Determinant and Inverse ...........132 The Method of Four Russians ...................................133 9.0.4 The Fair Coin Assumption ............................134 9.1 Origins and Previous Work ..................................134 9.1.1 Strassen s Algorithm .................................135 9.2 Rapid Subspace Enumeration ................................135 9.3 The Four Russians Matrix Multiplication Algorithm .............137 9.3.1 Role of the Gray Code ................................137 9.3.2 Transposing the Matrix Product ........................138 9.3.3 Improvements .......................................138 9.3.4 A Quick Computation ................................139 9.3.5 M4RM Experiments Performed by S AGE Staff ...........139 9.3.6 Multiple Gray-Code Tables and Cache Management .......141 9.4 The Four Russians Matrix Inversion Algorithm ..................141 9.4.1 Stage 1:............................................141 9.4.2 Stage 2:............................................142 9.4.3 Stage 3:............................................142 9.4.4 A Curious Note on Stage 1 of M4RI ....................143 9.4.5 Triangulation or Inversion? ............................145 9.5 Exact Analysis of Complexity ................................145 9.5.1 An Alternative Computation ...........................146 9.5.2 Full Elimination, not Triangular ........................147 9.5.3 The Rank of 3* Rows, or Why к + ε is not Enough ........148 9.5.4 Using Bulk Logical Operations .........................149 9.6 Experimental and Numerical Results ..........................149 9.7 M4RI Experiments Performed by S AGE Staff ...................151 9.7.1 Determination of к ...................................151 9.7.2 The Transpose Experiment ............................151 9.8 Pairing With Strassen s Algorithm for Matrix Multiplication ......151 9.8.1 Pairing M4RI with Strassen ...........................152 9.9 Higher Values of q ..........................................152 9.9.1 Building the Gray Code over GF(^) ....................152 9.9.2 Other Modifications ..................................153 9.9.3 Running Time .......................................153 9.9.4 Implementation ......................................154 Contents XXI 10 The Quadratic Sieve ............................................] 59 10.1 Motivation ................................................159 10.1.1 A View of RSA from 60,000 feet .......................160 10.1.2 Two Facts from Number Theory ........................161 10.1.3 Reconstructing the Private Key from the Public Key .......161 10.2 Trial Division ..............................................163 10.2.1 Other Ideas .........................................165 10.2.2 Sieve of Eratosthenes .................................167 10.3 Theoretical Foundations .....................................169 10.4 The Naïve Sieve ............................................170 10.4.1 An Extended Example ................................171 10.5 The Godei Vectors ..........................................171 10.5. 1 Benefits of the Notation ...............................172 10.5.2 Unlimited-Dimension Vectors ..........................173 10.5.3 The Master Stratagem ................................173 10.5.4 Historical Interlude ................................... 1 73 10.5.5 Review of Null Spaces ................................174 10.5.6 Constructing a Vector in the Even-Space .................175 10.6 The Linear Sieve Algorithm ..................................176 10.6.1 Matrix Dimensions in the Linear & Quadratic Sieve .......176 10.6.2 The Running Time ...................................178 10.7 The Example, Revisited .....................................178 10.8 Rapidly Generating Smooth Squares ...........................180 10.8.1 New Strategy ........................................181 10.9 Further Reading ............................................183 lO.lOHistorical Notes ............................................183 Part III Polynomial Systems and Satisfiability 11 Strategies for Polynomial Systems ................................187 11.1 Why Solve Polynomial Systems of Equations over Finite Fields? ... 187 11.2 Universal Maps ............................................189 11.3 Polynomials over GF(2) .....................................191 11.3.1 Exponents: x2=x ....................................191 11.3.2 Equivalent versus Identical Polynomials .................191 11.3.3 Coefficients .........................................192 11.3.4 Linear Combinations .................................192 11.4 Degree Reduction Techniques ................................192 11.4.1 An Easy but Hard-to-State Condition ....................193 11.4.2 An Algorithm that meets this Condition .................194 11.4.3 Interpretation ........................................195 11.4.4 Summary ........................................... l96 11.4.5 Detour: Asymptotics of the Choose Function ...........196 11.4.6 Complexity Calculation ...............................197 11.4.7 Efficiency Note ......................................198 xxii Contents 11.4.8 The Greedy Degree-Dropper Algorithm .................198 11.4.9 Counter-Example for Linear Systems ...................199 11.5 NP-Completeness of MP ....................................199 11.6 Measures of Difficulty in MQ ................................203 11.6.1 The Role of Over-Definition ...........................203 11.6.2 Ultra-Sparse Quadratic Systems ........................203 11.6.3 Other Views of Sparsity ...............................205 11.6.4 Structure ...........................................205 11.7 The Role of Guessing a Few Variables .........................206 11.7.1 Measuring Infeasible Running Times ....................206 11.7.2 Fix-XL .............................................207 12 Algorithms for Solving Polynomial Systems .......................209 12.1 A Philosophical Point on Complexity Theory ...................209 12.2 Gröbner Bases Algorithms ...................................210 12.2.1 Double-Exponential Running Time .....................210 12.2.2 Remarks about Gröbner Bases .........................210 1 2.3 Linearization ..............................................211 12.4 The XL Algorithm ..........................................213 12.4.1 Complexity Analysis .................................215 12.4.2 Sufficiently Many Equations ...........................216 12.4.3 Jumping Two Degrees ................................216 12.4.4 Fix-XL .............................................217 12.5 ElimLin ..................................................219 12.5.1 Why is this useful? ...................................220 12.5.2 How to use ElimLin ..................................221 12.5.3 On the Sub-Space of Linear Equations in the Span of a Quadratic System of Equations .........................223 12.5.4 The Weight of the Basis ...............................224 12.5.5 One Last Trick for GF(2)-only .........................225 12.5.6 Notes on the Sufficient Rank Condition ..................226 12.6 Comparisons between XL and F4 .............................227 12.7 SAT-Solvers ...............................................228 12.8 System Fragmentation ......................................228 12.8.1 Separability .........................................229 12.8.2 Gaussian Elimination is Not Enough ....................230 12.8.3 Depth First Search ...................................230 12.8.4 Nearly Separable Systems .............................231 12.8.5 Removing Multiple vertices ...........................232 12.8.6 Relation to Menger s Theorem .........................232 12.8.7 Balance in Vertex Cuts ................................233 12.8.8 Applicability ........................................233 12.9 Resultants .................................................234 12.9.1 The Univariate Case ..................................234 12.9.2 The Bivariate Case ...................................235 Contents xxiii 12.9.3 Multivariate Case.................................... 236 12.9.4 Further Reading .....................................238 12.10ТІ1Є Raddum-Semaev Method ................................238 12.10.1 Building the Graph ...................................238 12.10.2 Agreeing ...........................................239 12.10.3Propigation .........................................240 12.10.4Termination .........................................240 12.10.5Gluing............................................. 240 12. lO.óSplitting ............................................242 ^.lOJSummary ...........................................242 12.1 lThe Zhuang-Zi Algorithm ...................................243 12.1 2Homotopy Approach ........................................243 13 Converting MQ to CNF-SAT ....................................245 13.1 Summary .................................................245 13.2 Introduction ...............................................246 13.2.1 Application to Cryptanalysis ...........................247 13.3 Notation and Definitions .....................................247 13.4 Converting MQ to SAT ......................................248 1 3.4.1 The Conversion ......................................248 13.4.2 Measures of Difficulty ................................250 13.4.3 Preprocessing .......................................252 13.4.4 Fixing Variables in Advance ...........................253 13.4.5 SAT-Solver Used ....................................254 13.5 Experimental Results .......................................255 13.5.1 The Source of the Equations ...........................255 13.5.2 Note About the Variance ..............................255 13.5.3 The Log-Normal Distribution of Running Times ..........256 13.5.4 The Optimal Cutting Number ..........................257 13.6 Cubic Systems .............................................258 13.6.1 Do All Possible Monomials Appear? ....................258 13.6.2 Measures of Efficiency ...............................260 13.7 Further Reading ............................................260 13.7.1 Previous Work .......................................260 13.7.2 Further Work ........................................261 13.8 Conclusions ...............................................262 14 How do SAT-Solvers Operate? ...................................263 14.1 The Problem Itself ..........................................263 14.1.1 Conjunctive Normal Form .............................264 14.2 Solvers like Walk-SAT ......................................264 14.2.1 The Search Space ....................................265 14.2.2 Papadimitriou s Algorithm ............................265 14.2.3 Greedy SAT or G-SAT ................................266 14.2.4 Walk-SAT ..........................................267 xxiv Contents 14.2.5 Walk-SAT versus Papadimitriou ........................268 14.2.6 Where Heuristic Methods Fail .........................268 14.2.7 Closing Thoughts on Heuristic Methods .................269 14.3 Back-Tracking .............................................269 14.4 Chaff and its Descendants ...................................272 14.4.1 Variable Management ................................272 14.4.2 Unit Propagation .....................................273 14.4.3 The Method of Watched Literals .......................273 14.4.4 Absent Literals ......................................274 14.4.5 Summary ...........................................274 14.5 Enhancements to Chaff ......................................275 14.5.1 Learned Clauses .....................................275 14.5.2 The Alarm Clock ....................................275 14.5.3 The Third Finger ....................................276 14.6 Economic Motivations ......................................276 14.7 Further Reading ............................................277 15 Applying SAT-Solvers to Extension Fields of Low Degree ...........279 15.1 Introduction ...............................................279 15.2 Solving GF(2) Systems via SAT-Solvers .......................280 15.2.1 Sparsity ............................................280 15.3 Overview .................................................281 15.4 Polynomial Systems over Extension Fields of GF(2) .............281 15.4.1 Extensions of the Coefficient Field ......................282 15.4.2 Difficulty in Bits .....................................282 15.5 Finding Efficient Arithmetic Representations via Matrices ........282 15.6 Using the Algebraic Normal Forms ............................286 15.6.1 Remarks on the Special Forms .........................287 15.6.2 Remarks on Degree ..................................287 15.6.3 Remarks on Coefficients ..............................288 15.6.4 Solving with Gröbner Bases ...........................288 15.7 Experimental Results .......................................289 15.7.1 Computers Used .....................................291 15.7.2 Polynomial Systems Used .............................291 15.8 Inverses and Determinants ...................................292 15.8.1 Determinants ........................................292 15.8.2 Inverses ............................................292 15.8.3 Rijndael and the Para-Inverse Operation .................293 15.9 Conclusions ...............................................294 l5.10Review of Extension Fields ..................................295 15.10.1Constructing the Field ................................295 lS.lO^Regular Representation ...............................297 15.11 Reversing the Isomorphism: The Existence of Dead Give-Aways ... 298 Contents xxv A On the Philosophy of Block Ciphers With Small Blocks .............301 A. 1 Definitions ................................................301 A.2 Brute-Force Generic Attacks on Ciphers with Small Blocks .......302 A.3 Key Recovery vs. Applications of Ciphers with Small Blocks ......303 A.4 The Keeloq Code-book—Practical Considerations ...............306 A. 5 Conclusions ...............................................307 В Formulas for the Field Multiplication law for Low-Degree Extensions of GF(2) ............................................309 B.I For GF(4) .................................................309 B.2 For GF(8) .................................................309 B.3 For GF(16) ................................................310 B.4 For GF(32) ................................................311 B.5 For GF(64) ................................................312 С Polynomials and Graph Coloring, with Other Applications .........315 C.I A Very Useful Lemma ......................................315 C.2 Graph Coloring ............................................316 C.2.1 The с φ pn Case .....................................316 C.2.2 Application to GF(2) Polynomials ......................316 C.3 Related Applications ........................................317 C.3.1 Radio Channel Assignments ...........................317 C.3.2 Register Allocation ...................................318 C.4 Interval Graphs ............................................318 C.4.1 Scheduling an Interval Graph Scheduling Problem ........319 C.4.2 Comparison to Other Problems .........................320 C.4.3 Moral of the Story ...................................321 D Options for Very Sparse Matrices ................................323 D. 1 Preliminary Points ..........................................323 D. 1.1 Accidental Cancellations ..............................323 D.I.2 Solving Equations by Finding a Null Space ..............324 D.1.3 Data Structures and Storage ...........................324 D.2 Naive Sparse Gaussian Elimination ...........................325 D.2. 1 Sparse Matrices can have Dense Inverses ................326 D.3 Markowitz s Algorithm ......................................326 D.4 The Block Wiedemann Algorithm .............................326 D.5 The Block Lanczos Algorithm ................................327 D.6 The Pomerance-Smith Algorithm .............................327 D.6.1 Overview ...........................................328 D.6.2 Inactive and Active Columns ..........................329 D.6.3 The Operations ......................................329 D.6.4 The Actual Algorithm ................................331 D.6.5 Fill-in and Memory Management .......................332 D.6.6 Technicalities .......................................333 xxvi Contents D.6.7 Cremona s Implementation............................334 D.6.8 Further Reading .....................................335 E Inspirational Thoughts, Poetry and Philosophy ....................337 References .........................................................339 Index .............................................................351
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discipline	Informatik Mathematik
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id	DE-604.BV035726051
illustrated	Illustrated
indexdate	2024-07-09T21:53:04Z
institution	BVB
isbn	9780387887579 9780387887562
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-018002747
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owner	DE-739 DE-703 DE-12 DE-898 DE-BY-UBR
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physical	XXXIII, 356 S. graph. Darst.
publishDate	2009
publishDateSearch	2009
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publisher	Springer
record_format	marc
spelling	Bard, Gregory V. Verfasser aut Algebraic cryptanalysis Gregory V. Bard Dordrecht [u.a.] Springer 2009 XXXIII, 356 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cryptography Data encryption (Computer science) Kryptoanalyse (DE-588)4830502-9 gnd rswk-swf Kryptoanalyse (DE-588)4830502-9 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018002747&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Bard, Gregory V. Algebraic cryptanalysis Cryptography Data encryption (Computer science) Kryptoanalyse (DE-588)4830502-9 gnd
subject_GND	(DE-588)4830502-9
title	Algebraic cryptanalysis
title_auth	Algebraic cryptanalysis
title_exact_search	Algebraic cryptanalysis
title_full	Algebraic cryptanalysis Gregory V. Bard
title_fullStr	Algebraic cryptanalysis Gregory V. Bard
title_full_unstemmed	Algebraic cryptanalysis Gregory V. Bard
title_short	Algebraic cryptanalysis
title_sort	algebraic cryptanalysis
topic	Cryptography Data encryption (Computer science) Kryptoanalyse (DE-588)4830502-9 gnd
topic_facet	Cryptography Data encryption (Computer science) Kryptoanalyse
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018002747&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT bardgregoryv algebraiccryptanalysis

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