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

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5 1.3 Asymmetric Key Cryptography this communication. Now, in case of such a dispute, Bob will have to prove to an unbiased arbitrator (may be the judge or jury at the court) that the message was in fact sent by Alice. Now, the arbitrator can not verify who actually encrypted/signed that particular plaintext, as both Alice and Bob are capable of encrypting/signing it using the same (shared) key. This problem of authentication is known as repudiation. This problem can not be solved using symmetric key systems, as the concept of non-repudiation using signatures can not be accomplished. To overcome these problems, Diffie and Hellman [33] introduced the notion of ‘asymmetric key’ cryptosystems in 1976. Invention of asymmetric key cryptosystems is arguably the most celebrated breakthrough in modern day cryptography. However, one should note that public key cryptosystems are much slower than symmetric key cryptosystems in general. 1.3 Asymmetric Key Cryptography In asymmetric key cryptography, the encryption and the decryption keys are different, but they are related by some mathematical relation. These type of cryptosystems are also called ‘public key’ cryptosystems, as the encryption key is made public while the decryption key is kept secret. Hence, it is required for the security of the system that, finding the decryption key from the encryption key is computationally infeasible. The public key cryptosystems are most often based on some computationally hard mathematical problem. A list of well known hard problems is as follows. Integer Factorization Problem (IFP). The problem is to find a proper factor of a given positive integer N > 1. The first usable public key cryptosystem RSA [110] is based on the hardness of integer factorization problem. Rabin cryptosystem [104] is also based on integer factorization problem. Quadratic Residue Problem (QRP). The problem is to decide whether an element x ∈ Z N has a modular square root where N is a composite number. It can be proved that if factorization of N is known then QRP in Z N is no longer hard. Goldwasser-Micali [45] cryptosystem is based on QRP.
Chapter 1: Introduction 6 Discrete Logarithm Problem (DLP). Let G be a large cyclic group with generator g. Then given any element y = g a in G, the problem of finding the exponent a is called DLP over G. El Gamal system [35,36] is a public key cryptosystem based on DLP. In an elliptic curve group over a finite field, a similar problem of ECDLP can be formulated using the additive structure of the group. Based on the hardness of ECDLP and ECDHP (elliptic curve Diffie-Hellman problem), the notion of Elliptic Curve Cryptography was proposed independently by Koblitz [71] and Miller [91]. Knapsack Problem. Givenasetofnpositiveintegers{k i }andapositiveinteger N, the problem is to find whether it is possible to represent N = ∑ n i=1 a ik i where each a i is either 0 or 1. This is also called the subset sum problem. Many cryptosystems like Merkle-Hellman cryptosystem [88] were proposed based on this subset sum problem, and most of them have been broken. One may refer to [117,118] for an account on such attacks. Shortest Vector Problem (SVP). The problem is to find the shortest nonzero vector in a high dimensional lattice. This is hard in general and a few cryptosystems like NTRU [58], Ajtai-Dwork system [4,5] are based on this problem. It is worth noting that neither IPF nor DLP is hard under the quantum computation model, but SVP continues to remain hard in the quantum era. 1.4 Goal of this Thesis The main goal of this thesis is Cryptanalysis of RSA modulus N = pq and related Factorization problems. It is still unknown whether there is an efficient (polynomial time) algorithm to solve the ‘Integer Factorization Problem (IFP)’ in the classical model. The best known algorithm to solve this problem is the Number Field Sieve (NFS) [76], which has runtime greater than exp(log 1/3 N). However, if one obtains certain information about the RSA parameters, there are algorithms which can factor N quite efficiently. Our intention is to identify such weaknesses of the RSA cryptosystem and also to look into certain versions of factorization problem (in this thesis, the implicit factorization problem) that can be solved efficiently.
Page 1: some results on Cryptanalysis of RS
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Chapter 3: A class of Weak Encrypti
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Chapter 5 Reconstruction of Primes
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Chapter 6 Implicit Factorization In
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101 6.1 Implicit Factoring of Two L
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109 6.2 Two Primes with Shared Cont
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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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121 7.3 Extension of Approximate Co
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123 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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Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

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