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

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87 5.3 MSB Case: Our Method and its Analysis of bits to be known is 135 (shown in Table 5.2), instead of 120, as we use limited lattice dimensions in the experiments. In all the cases mentioned above, we miss the first τl N LSBs of the primes. If we miss the information of the bits of the prime in a contiguous block of size τl N somewhere in the middle, after the i-th level say, then this method offers similar solution if we have η > 2τ +2i/l N . 5.3 MSB Case: Our Method and its Analysis In this section, we put forward a novel idea of reconstructing the most significant half of the primes p,q given the knowledge of some blocks of bits. To the best of our knowledge, this has not been studied in a disciplined manner in the existing literature. As before, N = pq, and the primes p,q are of the same bitsize. For this section of MSB reconstruction, let us propose the following definition to make notations simpler. Definition 5.4. Let us define X[i] to be the i-th most significant bit of X with X[0] being the MSB. Also define X i to be the partial approximation of X where X i shares the most significant bits 0 through i with X. 5.3.1 The Reconstruction Idea Theideaforreconstructingthemostsignificanthalvesoftheprimesisquitesimple. We shall use the basic relation N = pq. If one of the primes, p say, is known, the other one is easy to find by q = N/p. Now, if a few MSBs of one of the primes, p say, is known, then we may obtain an approximation p ′ of p. This allows us to construct an approximation q ′ = ⌈N/p ′ ⌉ of the other prime q as well. Our idea is to use the known blocks of bits of the primes in a systematic order to repeat this approximation process until we recover half of one of the primes. A few obvious questions may be as follows. • How accurate are the approximations? • How probable is the success of the reconstruction process? • How many bits of the primes do we need to know?
Chapter 5: Reconstruction of Primes given few of its Bits 88 We answer these questions by describing the reconstruction algorithm in Section 5.3.2 and analyzing the same in Section 5.3.3. But first, let us present an outline of our idea. Suppose that we have the knowledge of MSBs {0,...,a} of prime p. This allows us to construct an approximation p a of p, and hence an approximation q ′ = ⌈N/p a ⌉ of q. Lemma 5.5, discussed later in Section 5.3.3, tells us that q ′ matches q through MSBs {0,...,a−t−1}, i.e, q ′ = q a−t−1 , with some probability depending on t. Now, if one knows the MSBs {a−t,...,2a} of q, then a better approximation q 2a may be constructed using q a−t−1 and these known bits. Again, q 2a facilitates the construction of a new approximation p ′ = ⌈N/q 2a ⌉, which by Lemma 5.5, satisfies p ′ = p 2a−t−1 with some probability depending on t. With the knowledge of MSBs {2a − t,...,3a} of p, it is once again possible to construct a better approximation p 3a of p. This process of constructing approximations is recursed until one reconstructs the most significant half of one of the primes. A graphical representation of the reconstruction process is illustrated in Figure 5.4. p 0 p a p 2a−t p 3a q a−t ≈ N/p a p 2a−t ≈ N/q 2a q 3a−t ≈ N/p 3a q 0 q a−t q 2a q 3a−t Figure 5.4: The feedback mechanism in MSB reconstruction. 5.3.2 The Reconstruction Algorithm Let S = {0,...,T} denote the set of bit indices from the most significant halves of the primes. Let us assume that k = ⌊T/a⌋ is odd in this case. Consider U,V ⊆ S such that U = {0,...,a}∪{2a−t,...,3a}∪···∪{(k −1)a−t,...,ka}, V = {a−t,...,2a}∪{3a−t,...,4a}∪···∪{ka−t,...,T}. Also consider that p[i]’s are known for i ∈ U and q[j]’s are known for j ∈ V. Then, Algorithm 8 reconstructs T many contiguous most significant bits of the prime q. The subroutine CORRECT used in Algorithm 8 (and Algorithm 10 later) takes as input a partial approximation Y of X and a set of contiguous known bits, X[i] for i ∈ Σ, say. It outputs a better approximation Z of X by correcting the bits of the partial approximation Y using the knowledge of the known bits. Formally,
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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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23 2.3 Cryptanalysis of RSA to be h
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25 2.3 Cryptanalysis of RSA The att
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27 2.4 Lattice and LLL Algorithm No
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29 2.4 Lattice and LLL Algorithm Wh
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31 2.5 Solving Modular Polynomials
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137 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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