Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

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2.3.7 Small Decryption Exponent Attack . . . . . . . . . . . . . . 25 2.4 Lattice and LLL Algorithm . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Solving Modular Polynomials . . . . . . . . . . . . . . . . . . . . . 31 2.5.1 Coppersmith’s Method . . . . . . . . . . . . . . . . . . . . . 32 2.5.2 General Method by Jochemsz and May . . . . . . . . . . . . 35 2.6 Solving Integer Polynomials . . . . . . . . . . . . . . . . . . . . . . 38 2.6.1 Coron’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6.2 General Method by Jochemsz and May . . . . . . . . . . . . 40 2.7 Analysis of the Root Finding Techniques . . . . . . . . . . . . . . . 43 2.7.1 Time Complexity Analysis . . . . . . . . . . . . . . . . . . . 43 2.8 Experimental Framework . . . . . . . . . . . . . . . . . . . . . . . . 44 2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 A class of Weak Encryption Exponents in RSA 47 3.1 Our Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Improvements over Existing Work . . . . . . . . . . . . . . . . . . . 52 3.2.1 The Improvement in the Bounds of X,Y . . . . . . . . . . . 52 3.2.2 Further Improvement in the Bounds . . . . . . . . . . . . . 53 3.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 53 3.3 A New Class of Weak Keys . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Estimation of Weak Keys . . . . . . . . . . . . . . . . . . . 59 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Cryptanalysis of RSA with more than one Decryption Exponent 63 4.1 Theoretical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 viii
5 Reconstruction of Primes given few of its Bits 75 5.1 LSB Case: Combinatorial Analysis of Existing Work . . . . . . . . 76 5.1.1 The Reconstruction Algorithm . . . . . . . . . . . . . . . . . 76 5.1.2 Growth of the Search Tree . . . . . . . . . . . . . . . . . . . 78 5.1.3 Known Prime Bits: Complementary Sets for p,q . . . . . . . 81 5.1.4 Known Prime Bits: Distributed at Random . . . . . . . . . 82 5.1.5 Known Prime Bits: Distributed in a Pattern . . . . . . . . . 83 5.2 LSB Case: Lattice Based Technique . . . . . . . . . . . . . . . . . . 85 5.3 MSB Case: Our Method and its Analysis . . . . . . . . . . . . . . . 87 5.3.1 The Reconstruction Idea . . . . . . . . . . . . . . . . . . . . 87 5.3.2 The Reconstruction Algorithm . . . . . . . . . . . . . . . . . 88 5.3.3 Analysis of the Reconstruction Algorithm . . . . . . . . . . 89 5.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6 Implicit Factorization 95 6.1 Implicit Factoring of Two Large Integers . . . . . . . . . . . . . . . 96 6.1.1 The General Result . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.2 The MSB Case . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1.3 The LSB Case . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1.4 Implicit Factorization problem when k = 3 . . . . . . . . . . 104 6.2 Two Primes with Shared Contiguous Portion of Bits at the Middle . 107 6.3 Exposing a Few MSBs of One Prime . . . . . . . . . . . . . . . . . 111 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7 Approximate Integer Common Divisor Problem 115 7.1 Finding q −1 mod p ≡ Factorization of N . . . . . . . . . . . . . . . 116 7.2 Finding Smooth Integers in a Short Interval . . . . . . . . . . . . . 118 7.3 Extension of Approximate Common Divisor Problem . . . . . . . . 120 ix
Page 1: some results on Cryptanalysis of RS
Page 5 and 6: Abstract In this thesis, we propose
Page 7 and 8: Acknowledgements There are many peo
Page 9: To Thaakuma (Grandmother), the firs
Page 14 and 15: 7.4 The General Solution for EPACDP
Page 17 and 18: List of Figures 1.1 Communication o
Page 19 and 20: Chapter 1: Introduction 2 The motiv
Page 21 and 22: Chapter 1: Introduction 4 there is
Page 23 and 24: Chapter 1: Introduction 6 Discrete
Page 25 and 26: Chapter 1: Introduction 8 Chapter 5
Page 28 and 29: Chapter 2 Mathematical Preliminarie
Page 30 and 31: 13 2.2 RSA Cryptosystem are impleme
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Page 34 and 35: 17 2.2 RSA Cryptosystem istic prima
Page 36 and 37: 19 2.2 RSA Cryptosystem Afterfindin
Page 38 and 39: 21 2.2 RSA Cryptosystem integer x <
Page 40 and 41: 23 2.3 Cryptanalysis of RSA to be h
Page 42 and 43: 25 2.3 Cryptanalysis of RSA The att
Page 44 and 45: 27 2.4 Lattice and LLL Algorithm No
Page 46 and 47: 29 2.4 Lattice and LLL Algorithm Wh
Page 48 and 49: 31 2.5 Solving Modular Polynomials
Page 56 and 57: 39 2.6 Solving Integer Polynomials
Page 58 and 59: 41 2.6 Solving Integer Polynomials
Page 60 and 61: 43 2.7 Analysis of the Root Finding
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45 2.9 Conclusion 2.9 Conclusion No
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Chapter 3: A class of Weak Encrypti
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Chapter 4: Cryptanalysis of RSA wit
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Chapter 5 Reconstruction of Primes
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77 5.1 LSB Case: Combinatorial Anal
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85 5.2 LSB Case: Lattice Based Tech
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Chapter 6 Implicit Factorization In
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99 6.1 Implicit Factoring of Two La
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107 6.2 Two Primes with Shared Cont
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109 6.2 Two Primes with Shared Cont
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111 6.3 Exposing a Few MSBs of One
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113 6.3 Exposing a Few MSBs of One
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Chapter 7 Approximate Integer Commo
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117 7.1 Finding q −1 mod p ≡ Fa
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119 7.2 Finding Smooth Integers in
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121 7.3 Extension of Approximate Co
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123 7.4 The General Solution for EP
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133 7.5 Sublattice and Generalized
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139 7.6 Improved Results for Larger
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141 7.6 Improved Results for Larger
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143 7.7 EGACDP The approach in this
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145 7.7 EGACDP Sinceinthiscasethema
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147 7.8 Conclusion poly{loga,k}. Th
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Chapter 8: Conclusion 150 Chapter 3
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Chapter 8: Conclusion 152 p 1 ,p 2
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Chapter 8: Conclusion 154 represent
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Chapter 8: Conclusion 156 Final Wor
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BIBLIOGRAPHY 158 [9] J. Blömer and
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BIBLIOGRAPHY 160 [31] B. de Weger.
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BIBLIOGRAPHY 162 [54] M.J.Hinek. Cr
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BIBLIOGRAPHY 164 [76] A. K. Lenstra
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BIBLIOGRAPHY 166 [97] A. Nitaj. Cry
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BIBLIOGRAPHY 168 [121] C. L. Siegel
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