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

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79 5.1 LSB Case: Combinatorial Analysis of Existing Work p[i] = 0 p[i] = 0 (0, 0) (1, 1) (1, 0) (0, 1) p[i]+q[i] ≡ 0 (mod 2) p[i]+q[i] ≡ 1 (mod 2) Figure 5.2: Branching when exactly one bit of p[i],q[i] is known. and otherwise we do not. In the case where neither of the possibilities for p i ,q i generated from p i−1 ,q i−1 satisfy the relation, we discard the whole subtree rooted atp i−1 ,q i−1 . Thus, thepruningprocedurenotonlydiscardsthewrongonesatlevel i, but also discards subtrees from level i−1, thereby narrowing down the search tree. An example case (p[i] = 0 and q[i] = 1 are known, say) may be presented as in Figure 5.3. p[i] = 0 q[i] = 1 p[i] = 0 q[i] = 1 (0, 0) (1, 1) (1, 0) (0, 1) p[i]+q[i] ≡ 1 (mod 2) p[i]+q[i] ≡ 1 (mod 2) Figure 5.3: Branching when both the bits p[i],q[i] are known. Based on our discussion so far, let us try to model the growth of the search tree following Algorithm 7. As both p,q are odd, we have p[0] = 1 and q[0] = 1. Thus the tree starts from W 0 = 1 and the expansion or contraction of the tree at each level can be modeled as follows. • p[i] = UNKNOWN, q[i] = UNKNOWN: W i = 2W i−1 . • p[i] = KNOWN, q[i] = UNKNOWN: W i = W i−1 . • p[i] = UNKNOWN, q[i] = KNOWN: W i = W i−1 . • p[i] = KNOWN, q[i] = KNOWN: W i = γ i W i−1 .
Chapter 5: Reconstruction of Primes given few of its Bits 80 Here, we assume that the tree narrows down to a γ i fraction (0 < γ i ≤ 1) from the earlier level if both the bits of the primes are known. One may note that Heninger and Shacham [50, Conjecture 4.3] conjectures the average value of γ i (call it γ) to be 1 . We shall discuss this in more details later. 2 Suppose that randomly chosen α fraction of bits of p and β fraction of bits of q are known (by some side channel attack, e.g., cold boot). Then the joint probability distribution table for the bits of the primes will be as follows. ↓ q[i], p[i] → UNKNOWN KNOWN UNKNOWN (1−α)(1−β) α(1−β) KNOWN (1−α)β αβ As shown before, the growth of the search tree depends upon the knowledge of the bits in the primes. Hence, we can model the growth of the tree as a recursion on the level index i: W i = (1−α)(1−β)2W i−1 +α(1−β)W i−1 +(1−α)βW i−1 +αβγ i W i−1 = (2−α−β +αβγ i )W i−1 . IfwewanttorestrictW i (thatisthegrowthofthetree)asapolynomialofi(thatis thenumberoflevel),wewouldlike(roughlyspeaking)thevalueof(2−α−β+αβγ i ) close to 1 on an average. Considering the average value γ (instead of γ i at each level), we get, 2 − α − β + αβγ ≈ 1 which implies 1 − α − β + αβγ ≈ 0. If we assume that the same fraction of bits are known for p and q, then α = β and we get 1 − 2α + α 2 γ ≈ 0 ⇒ α ≈ 1−√ 1−γ . If we assume [50, Conjecture γ 4.3], then γ ≈ 0.5 and hence α ≈ 2 − √ 2 ≈ 0.5858, as obtained in [50, Section 4.4]. One may note that our idea is simpler compared to the explanation in [50]. This simplification is achieved here by using average value for γ i in the recurrence relation of W i . The most natural strategy is to first apply Algorithm 7 to retrieve the least significant half of any one of the primes and then apply the result of Boneh et al [16, Corollary 2.2] to factorize N. One may note that [50] utilizes their prime reconstruction algorithm to reconstruct the whole primes p,q whereas our idea is to use lattice based results after reconstructing just one half of any prime. This is more practical as it requires the knowledge of lesser number of random bits of the primes, namely, just about 0.5858×0.5 ≈ 0.3 fraction of bits (from the LSB half) insteadof0.5858fractionoftheprimesasexplainedin[50]. Moreover,factorization
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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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Page 60 and 61: 43 2.7 Analysis of the Root Finding
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129 7.4 The General Solution for EP
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131 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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