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

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155 8.3 Open Problems Problem 8.3. Is there any polynomial time algorithm to factor N with encryption exponents of order N if d pi > N 0.073 and d qi > N 0.073 for 1 ≤ i ≤ k? 8.3.3 Reconstruction of Primes The terms {p,q,d,d p ,d q ,q −1 (mod p)} are stored as a part of the secret key in PKCS #1 [99] to expedite the decryption in CRT-RSA. Now, one may note from Chapter 5 that Heninger and Shacham [50] used random known bits of {p,q,d,d p ,d q }, and our work uses bits of {p,q} to factorize N. None of the methods could utilize the knowledge of q −1 mod p to factor N. In Chapter 7, we have proved that knowing q −1 mod p completely is equivalent to factoring N. But, we do not have any results if one knows some random bits of q −1 mod p. In the presentation of the paper [50] at Crypto 2009, this problem was also asked. In this line, let us present the following two open questions. Problem 8.4. Can one use some known random bits of q −1 mod p to factor N? Problem 8.5. Does the knowledge of random bits of q −1 mod p reduce the required number of bits to be known for other private keys in case of factoring N? In Chapter 5, we studied the case when random bits are known from the lower half of p and q. However consider the situation when random bits are known from the upper half of p and q. Hence, we have the following question. Problem 8.6. Can one factor N when random bits are known from the upper half of p and q? 8.3.4 Implicit Factorization In PKC 2009, May and Ritzenhofen [86] introduced the problem of implicit factorization. They presents some results when few LSBs of the secret primes are same. They also extend their results for balanced RSA moduli. That is for the case N 1 = p 1 q 1 ,N 2 = p 2 q 2 ,...,N k = p k q k , with p i and q i are of same bitsize and few LSBs of p 1 ,...,p k are same. In Chapter 6 and Chapter 7 we analyze the situation when p 1 ,...,p k share their MSBs. However, we can not extend our ideas for the balanced case. Hence, we leave the following open question. Problem 8.7. Can one factor k balanced RSA moduli N 1 ,...,N k in polynomial time when p 1 ,...,p k share their MSBs?
Chapter 8: Conclusion 156 Final Words This thesis discusses a variety of cryptanalytic results on RSA and its variants, mainly by the use of lattice based root finding techniques. Analysis of certain versions of factorization problem has also been studied. Efficient algorithms to find small roots of modular and bivariate integer polynomials have been exploited alotduringthelastdecadeinconnectionwithRSAcryptanalysisaswellasinteger factorization, and this proves to be a potent line of research for years to come.
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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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39 2.6 Solving Integer Polynomials
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41 2.6 Solving Integer Polynomials
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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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87 5.3 MSB Case: Our Method and its
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Chapter 6 Implicit Factorization In
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97 6.1 Implicit Factoring of Two La
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Page 175 and 176: BIBLIOGRAPHY 158 [9] J. Blömer and
Page 177 and 178: BIBLIOGRAPHY 160 [31] B. de Weger.
Page 179 and 180: BIBLIOGRAPHY 162 [54] M.J.Hinek. Cr
Page 181 and 182: BIBLIOGRAPHY 164 [76] A. K. Lenstra
Page 183 and 184: BIBLIOGRAPHY 166 [97] A. Nitaj. Cry
Page 185: BIBLIOGRAPHY 168 [121] C. L. Siegel
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