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

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7 1.5 Organization of the Thesis 1.5 Organization of the Thesis The thesis presents several cryptanalytic results against the RSA cryptosystem when parts of the secret key are known or known in disguise. Most of our results use the LLL algorithm to obtain the secret key, and the main idea for each application is how to phrase the problem as a lattice problem and how to compute the dimension and determinant of the lattice. Lattice based root-finding ideas for polynomials is the theme for most of the results presented here. Moreover, all the results and discussions are targeted towards finding weaknesses in the RSA cryptosystem and its implementation. There are several attempts at RSA cryptanalysis in the same vein, and this thesis also illustrates how these results improve on the previous state of the art. It is recommended that one reads the chapters in the order they are presented. However, the reader may choose to browse quickly to a chapter of his/her choice and refer back to Chapter 2 for the mathematical background. A short summary for each chapter is presented as follows. Chapter 1: In the current chapter, we have discussed some introductory materials regarding cryptography, and its major classifications. We also present the goal and structure of this thesis. Chapter 2: In the next chapter, we start with some basic mathematical definitions and an overview of the RSA cryptosystem. In section 2.5, we introduce lattice based root finding techniques for modular polynomials. In section 2.6, we discuss the approach to find roots of a polynomial over integers. We will use these root finding techniques frequently in the thesis. Chapter 3: In this chapter, we discuss our work to identify encryption exponents for which RSA becomes weak. The materials of this chapter are based on our publication [81]. Chapter 4: Here we study the vulnerabilities of RSA in terms of its decryption exponent. We go through the model of RSA in the presence of many decryption exponents, anddiscussourworkregardingcryptanalysisofRSAwithinthismodel. The materials of this chapter are based on our publications [114,115].
Chapter 1: Introduction 8 Chapter 5: In addition to the weaknesses of RSA due to weak keys, it may also be vulnerable due to leaked information about the RSA primes. In this chapter, we discuss the factorization of the RSA modulus N by reconstructing the primes from randomly known bits. In Sections 5.1 and 5.2, we analyze the fact that N = pq can be factored in reasonable time complexity when a few bits of the primes are known from the least significant halves. In Section 5.3, we analyze the same when a few bits are known from the most significant halves of the primes. The materials of this chapter are based on our publication [82]. Chapter 6: At times, one may not obtain explicit bitwise information about the primes, but have some implicit knowledge regarding those. An attempt at RSA factorization based on this implicit knowledge is of interest as well. This chapter deals with the analysis of a situation when one can factor RSA moduli N 1 = p 1 q 1 ,N 2 = p 2 q 2 ,...,N k = p k q k in polynomial time if p 1 ,p 2 ,...,p k share a few bits. This problem is called the implicit factorization problem, introduced by May and Ritzenhofen in [86]. In Section 6.1, we present the implicit factorization strategy for two or three large integers when they share MSBs and/or LSBs. Next, in Section 6.2, we analyze the same problem when the primes p 1 ,p 2 share a (contiguous) portion of bits at the middle. The materials of this chapter are based on our publications [113,116]. Chapter 7: In this chapter, we present two generalizations, Extended Partially Approximate Common Divisor Problem (EPACDP) and Extended General Approximate Common Divisor Problem (EGACDP), of the ‘approximate common divisor problem’ introduced by Howgrave-Graham [61]. We also propose two applications of ‘approximate common divisor problem’. In Sections 7.4 and 7.6, we propose two methods to solve EPACDP, and in Section 7.7, we discuss the solution of EGACDP. Most importantly, continuing from Chapter 6, we discuss the applications of EPACDP for implicit factorization when p 1 ,p 2 ,...,p k share some MSBs andorLSBs. Thematerialsofthischapterarebasedonourpublications[112,116]. Chapter 8: This chapter concludes the thesis. Here we present a comprehensive summary of our work that has been discussed throughout the thesis. We analyze andcompareourworkwiththecontemporaryadvancesinthefieldofcryptography and also discuss open problems which might be interesting for further investigation along this line of research.
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
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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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Chapter 6 Implicit Factorization In
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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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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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