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

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Chapter 5 Reconstruction of Primes given few of its Bits An extensive amount of research has been done in RSA factorization and we refer the reader to the survey papers by Boneh [11] and May [84] for a complete account. One major class of RSA attacks exploit partial knowledge of the RSA secret keys or the primes. Rivest and Shamir [109] pioneered these attacks using Integer Programming and factored RSA modulus given two-third of the LSBs of a factor. Later, a seminal paper [24] by Coppersmith proved that factorization of the RSA modulus can be achieved given half of the MSBs of a factor. His method used LLL [77] lattice reduction technique to solve for small solutions to modular equations. This method triggered a host of research in the field of lattice based factorization, e.g., the works by Howgrave-Graham [59], Jochemsz and May [65]. These results require knowledge of contiguous blocks of bits of the RSA secret keysortheprimes. However,inanactualpracticalscenarioofside-channelattacks, it is more likely that an adversary will gain the knowledge of random bits of the RSA parameters instead of contiguous blocks. In fact, the cold-boot attack proposed by Halderman et al [46] in 2009 was mounted to recover random bits of RSA secret parameters exploiting data remanence in the computer memory. Thus the motivation comes from side channel attack on RSA where some bits of p and q are revealed but not the entire key. In this model, the application of the earlier factorization methods prove insufficient, and one requires a way to extract more information out of the random bits obtained via the side channel attacks. In [51], it has been shown how N can be factored with the knowledge of a random subset 75
Chapter 5: Reconstruction of Primes given few of its Bits 76 of the bits (distributed over small contiguous blocks) in one of the primes. Later, a similar result has been studied by Heninger and Shacham [50] to reconstruct the RSA private keys given a certain fraction of the bits, distributed at random. This is the work [50] where the random bits of both the primes are considered unlike the earlier works (e.g., [16,24,51]) where knowledge of the bits of a single prime have been exploited. This chapter studies how the least (respectively most) significant halves of the RSA primes can be completely recovered from some amount of randomly chosen bits from the least (respectively most) significant halves of the same. Thereafter one can exploit the existing lattice based results towards factoring the RSA modulus N = pq when p,q are of the same bitsize. It is possible to factor N in any one of the following cases in poly(logN) time: (i) when the most significant half of any one of the primes is known [24, Theorem 4], (ii) when the least significant half of any one of the primes is known [16, Corollary 2.2]. 5.1 LSB Case: Combinatorial Analysis of Existing Work In this section, we analyze the reconstruction algorithm by Heninger and Shacham [50, Section 3] from combinatorial point of view. Though the algorithm extends to all relations among the RSA secret keys, we shall concentrate our attention to the primary relation N = pq for the sake of factorization. The algorithm is a smart brute-force method on the total search space of unknown bits of p and q, which prunes the solutions those are infeasible given the knowledge of N and some random bits of the primes. 5.1.1 The Reconstruction Algorithm Definition 5.1. Let us define X[i] to be the i-th bit of X with X[0] being the LSB. Also define X i to be the partial approximation of X through the bits 0 to i. Then Algorithm 7 creates all possible pairs (p i ,q i ) by appending (p[i],q[i]) to the partial solutions (p i−1 ,q i−1 ) and prunes the incorrect ones by checking the validity of the available relation. A formal outline of Algorithm 7, which retrieves
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some results on Cryptanalysis of RS
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Abstract In this thesis, we propose
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Acknowledgements There are many peo
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To Thaakuma (Grandmother), the firs
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2.3.7 Small Decryption Exponent Att
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7.4 The General Solution for EPACDP
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List of Figures 1.1 Communication o
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Chapter 1: Introduction 2 The motiv
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Chapter 1: Introduction 4 there is
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Chapter 1: Introduction 6 Discrete
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Chapter 1: Introduction 8 Chapter 5
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Chapter 2 Mathematical Preliminarie
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13 2.2 RSA Cryptosystem are impleme
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15 2.2 RSA Cryptosystem • private
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17 2.2 RSA Cryptosystem istic prima
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19 2.2 RSA Cryptosystem Afterfindin
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21 2.2 RSA Cryptosystem integer x <
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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
Page 62: 45 2.9 Conclusion 2.9 Conclusion No
Page 65 and 66: Chapter 3: A class of Weak Encrypti
Page 81 and 82: Chapter 4: Cryptanalysis of RSA wit
Page 94 and 95: 77 5.1 LSB Case: Combinatorial Anal
Page 102 and 103: 85 5.2 LSB Case: Lattice Based Tech
Page 104 and 105: 87 5.3 MSB Case: Our Method and its
Page 112 and 113: Chapter 6 Implicit Factorization In
Page 114 and 115: 97 6.1 Implicit Factoring of Two La
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Page 126 and 127: 109 6.2 Two Primes with Shared Cont
Page 128 and 129: 111 6.3 Exposing a Few MSBs of One
Page 130 and 131: 113 6.3 Exposing a Few MSBs of One
Page 132 and 133: Chapter 7 Approximate Integer Commo
Page 134 and 135: 117 7.1 Finding q −1 mod p ≡ Fa
Page 136 and 137: 119 7.2 Finding Smooth Integers in
Page 138 and 139: 121 7.3 Extension of Approximate Co
Page 140 and 141: 123 7.4 The General Solution for EP
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125 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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