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

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3 1.2 Symmetric Key Cryptography encrypts the plaintext M to get the ciphertext C = E(M) and sends C to Bob. When Bob receives C, using his decryption function D, he decrypts the ciphertext C to get back the plaintext M = D(C). Sender Alice M → E(M) = C Ciphertext C Receiver Bob C → D(C) = M Charles Adversary Figure 1.1: Communication over an Insecure Channel. Intuitively, one must have the decryption function D to be the inverse of the encryption function E. Kerckhoffs’ law [87] in cryptography mandates that the security of a cryptosystem should not depend on the secrecy of either algorithm E or D. The security should only depend on some secret parameter(s) used in the process of encryption and/or decryption. This secret parameter is called the Key of the system in cryptographic terms. Thus, in a secure cryptographic protocol, Alice will encrypt plaintext M using themodifiedencryptionfunctionC = E(e k ,M), whichisdependentontheencryptionkeye k . Likewise,BobwillusethemodifieddecryptionfunctionM = D(d k ,C), dependent on the decryption key d k . For correct encryption and decryption using these key-dependent functions, some mathematical relation between e k and d k must exist in such a system. Depending on this relation between the encryption key and the decryption key, the study of modern cryptography can be classified under two major heads, namely Symmetric Key Cryptography and Asymmetric Key Cryptography. 1.2 Symmetric Key Cryptography In symmetric key cryptography, the encryption key is trivially related to the decryption key. So either the encryption key e k is same as the decryption key d k , or
Chapter 1: Introduction 4 there is a simple transformation between them. These type of cryptosystems are also called ‘secret key’ cryptosystems, as the keys are kept hidden from unauthorized users. There are two major types of symmetric key cryptosystems, namely Block Ciphers and Stream Ciphers. We can broadly characterize them as follows. Block Ciphers. The goal of block ciphers is to scramble the plaintext block-byblock using the basic tools of confusion and diffusion applied to the plaintext over multiple rounds. A generic setup for a block cipher would use complex substitution and permutation rounds on the plaintext block, using a secret key, to create the effect of confusion and diffusion. Stream Ciphers. On the other hand, the inherent goal of a stream cipher is to generate a pseudo-random stream of data using a randomly chosen key of fixed (preferably short) length. After the generation of such a pseudorandom stream, the plaintext is simply XOR-ed with the stream to obtain the ciphertext. One may say that the security of a block cipher depends upon the amount of confusion and diffusion created over rounds, whereas that of a stream cipher depends upon the indistinguishability of its output stream from an actual random stream of bytes. The reader may refer to [126] for further details. One of the most well known Block Ciphers at present is the Advanced Encryption Standard (AES) [1]. Before this, the Data Encryption Standard (DES) [32] was the most popular one. RC4 [108], SNOW [34], TURING [111] etc. are some of the popular stream ciphers. One may refer to the eStream project [39] for recent developments in the area of stream cipher design. Although symmetric key cryptosystems are very fast in practice, there are few drawbacks, as follows. Key distribution problem: A secure and authenticated secret channel should be needed to distribute the secret keys beforehand. Key management problem: In a network of n users, every pair of users must share a secret key, for a total of ( ) n 2 = n(n−1) keys. If n is large, then the number 2 of keys becomes unmanageable. Signature problem: Consider the situation where Alice sends a message (encrypted/signed using some symmetric key cipher) to Bob, and later refuses
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
Page 9: To Thaakuma (Grandmother), the firs
Page 12 and 13: 2.3.7 Small Decryption Exponent Att
Page 14 and 15: 7.4 The General Solution for EPACDP
Page 17 and 18: List of Figures 1.1 Communication o
Page 19: Chapter 1: Introduction 2 The motiv
Page 23 and 24: Chapter 1: Introduction 6 Discrete
Page 25 and 26: Chapter 1: Introduction 8 Chapter 5
Page 28 and 29: Chapter 2 Mathematical Preliminarie
Page 30 and 31: 13 2.2 RSA Cryptosystem are impleme
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Page 36 and 37: 19 2.2 RSA Cryptosystem Afterfindin
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Page 40 and 41: 23 2.3 Cryptanalysis of RSA to be h
Page 42 and 43: 25 2.3 Cryptanalysis of RSA The att
Page 44 and 45: 27 2.4 Lattice and LLL Algorithm No
Page 46 and 47: 29 2.4 Lattice and LLL Algorithm Wh
Page 48 and 49: 31 2.5 Solving Modular Polynomials
Page 56 and 57: 39 2.6 Solving Integer Polynomials
Page 58 and 59: 41 2.6 Solving Integer Polynomials
Page 60 and 61: 43 2.7 Analysis of the Root Finding
Page 62: 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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99 6.1 Implicit Factoring of Two La
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101 6.1 Implicit Factoring of Two L
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107 6.2 Two Primes with Shared Cont
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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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