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

More documents

Recommendations

Info

Chapter 8 Conclusion In this chapter, we conclude the thesis. We have studied weaknesses of RSA cryptosystem, and results related to integer factorization, throughout Chapters 3 to 7. We exploited the vulnerabilities of RSA arising due to weak encryption/decryption keys, information about the bits of RSA primes, and implicit knowledge about the same. Most of our results are based on lattice basis reduction techniques applied to finding solutions to integer and modular polynomials. We revisit the chapters one-by-one to summarize the thesis. We mention the existing results and prior work (if any) in the direction. Most importantly, we present the crux of the chapters, that is our contributions, improvements and extensions to existing methods. Finally, we also discuss the future scope for research and potential open problems in respective field of study. 8.1 Summary of Technical Results Chapter 1 provided the introduction to the thesis, while Chapter 2 covered some mathematical topics the reader should know before reading the thesis comfortably. The main technical results of the thesis are discussed in Chapters 3 to 7, and the highlights of these chapters are as follows. 149
Chapter 8: Conclusion 150 Chapter 3: Class of Weak Encryption Exponents Blömer and May [9] have shown that N can be factored in polynomial time if the public exponent e satisfies ex+y ≡ 0 (mod φ(N)), with x ≤ 1 3 N 1 4 and |y| = O(N −3 4ex). Someextensionsconsideringthedifferencep−q havealsobeenstudied. The number of such weak keys has been estimated as N 3 4 −ǫ . In a similar direction, more weak keys are presented by Nitaj [96]. Nitaj proved that [ if e satisfies ] eX − (p−u)(q −v)Y = 1 with 1 ≤ Y < X < 2 −1 4N 4, 1 |u| < N 1 4, v = − qu , and if all p−u the prime factors of p−u or q−v are less than 10 50 , then N can be factored from the knowledge of N,e. The number of such weak exponents is estimated as N 1 2 −ǫ . We concentrate on the cases when e(= N α ) satisfies eX − ZY = 1, given |N −Z| = N τ . Using the idea of Boneh and Durfee [14,15], we show that the LLL algorithm can be efficiently applied to get Z when |Y| = N γ and γ < 4ατ ⎛ √ ( ⎝ 1 4τ + 1 1 12α − 4τ + 1 ) 2 + 1 12α 2ατ ( 1 12 + τ 24α − α ) ⎞ ⎠. 8τ This idea substantially extends the class of weak keys presented in [96] when Z = ψ(p,q,u,v) = (p−u)(q −v). Further, we consider Z = ψ(p,q,u,v) = N −pu−v to provide a new class of weak keys in RSA. This idea does not require any kind of factorization as in [96]. A very conservative estimate for the number of such weak exponents is N 0.75−ǫ , where ǫ > 0 is arbitrarily small for suitably large N. Chapter 4: More than one Decryption Exponent From the results of Howgrave-Graham et al [62] and Hinek et al [55] we know that in the presence of n many decryption exponents, one can factor N when the decryption exponents d i < N δ , for 1 ≤ i ≤ n, where δ < ⎧ ⎪⎨ ⎪⎩ { } (2n+1)·2 n −(2n+1)( n min 2 ) (2n−2)·2 n +(4n+2)( n 2 ) ,0.5 min { (2n+1)·2 n −4n·( n−1 n−1 ) 2 (2n−2)·2 n +8n·( n−1 n−1 2 ),0.5 } if n is even if n is odd We improved this bound by showing that if n many decryption exponents (d 1 ,...,d n ) are used with the same N, then RSA is insecure when d i < N 3n−1 4n+4 , for
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 and 20:
Chapter 1: Introduction 2 The motiv
Page 21 and 22:
Chapter 1: Introduction 4 there is
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
Page 32 and 33:
15 2.2 RSA Cryptosystem • private
Page 34 and 35:
17 2.2 RSA Cryptosystem istic prima
Page 36 and 37:
19 2.2 RSA Cryptosystem Afterfindin
Page 38 and 39:
21 2.2 RSA Cryptosystem integer x <
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 50 and 51:
Page 52 and 53:
Page 54 and 55:
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
Page 65 and 66:
Chapter 3: A class of Weak Encrypti
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Chapter 4: Cryptanalysis of RSA wit
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 92 and 93:
Chapter 5 Reconstruction of Primes
Page 94 and 95:
77 5.1 LSB Case: Combinatorial Anal
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
85 5.2 LSB Case: Lattice Based Tech
Page 104 and 105:
87 5.3 MSB Case: Our Method and its
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Chapter 6 Implicit Factorization In
Page 114 and 115: 97 6.1 Implicit Factoring of Two La
Page 116 and 117: 99 6.1 Implicit Factoring of Two La
Page 118 and 119: 101 6.1 Implicit Factoring of Two L
Page 124 and 125: 107 6.2 Two Primes with Shared Cont
Page 126 and 127: 109 6.2 Two Primes with Shared Cont
Page 128 and 129: 111 6.3 Exposing a Few MSBs of One
Page 130 and 131: 113 6.3 Exposing a Few MSBs of One
Page 132 and 133: Chapter 7 Approximate Integer Commo
Page 134 and 135: 117 7.1 Finding q −1 mod p ≡ Fa
Page 136 and 137: 119 7.2 Finding Smooth Integers in
Page 138 and 139: 121 7.3 Extension of Approximate Co
Page 140 and 141: 123 7.4 The General Solution for EP
Page 150 and 151: 133 7.5 Sublattice and Generalized
Page 156 and 157: 139 7.6 Improved Results for Larger
Page 158 and 159: 141 7.6 Improved Results for Larger
Page 160 and 161: 143 7.7 EGACDP The approach in this
Page 162 and 163: 145 7.7 EGACDP Sinceinthiscasethema
Page 164: 147 7.8 Conclusion poly{loga,k}. Th
Page 169 and 170: Chapter 8: Conclusion 152 p 1 ,p 2
Page 171 and 172: Chapter 8: Conclusion 154 represent
Page 173 and 174: Chapter 8: Conclusion 156 Final Wor
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?