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

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33 2.5 Solving Modular Polynomials n dimensional lattice L satisfies ||r 1 (xX)|| < 2 n−1 4 (det(L)) 1 n . Thus, if 2 n−1 4 (det(L)) 1 n < Nm √ n , then the LLL reduction gives us a polynomial h(x) = r 1 (x) such that ||h(xX)|| < Nm √ n and hence h(x 0 ) = 0. This h(x) is our desired polynomial to find x 0 using common root finding algorithms. We can extend the idea for multivariate case as well. But in that case, we need some assumptions, and hence the method becomes heuristic. Consider a multivariate polynomial f N (x 1 ,...,x t ) with root (x (0) 1 ,...,x (0) t ). Similar to the univariate case, we can construct a lattice L using shift polynomials g i1 ,...,i t,k(x 1 ,...,x t ) = x i 1 1 ···x it t (f N (x 1 ,...,x t )) k N m−k , for some fixed m, k = 0,...,m and i 1 ,...,i t non-negative integers. Clearly, g i1 ,...,i t,k(x (0) 1 ,...,x (0) t ) ≡ 0 (mod N m ). Now one can extend Theorem 2.22 for the multivariate case as follows. Theorem 2.23. Let h(x 1 ,...,x t ) ∈ Z[x 1 ,...,x t ] be the sum of at most ω monomials. Suppose that h(x (0) 1 ,...,x (0) t ) ≡ 0 (mod N m ) where |x (0) 1 | ≤ X 1 ,...,|x (0) t | ≤ X t and ||h(x 1 X 1 ,...,x t X t )|| < Nm √ ω . Then h(x (0) 1 ,...,x (0) t ) = 0 over integers. Therefore, combining Lemma 2.20 and Theorem 2.23, we can say that if 2 ω(ω−1) 4(ω+1−t) det(L) 1 ω+1−t < N m √ ω , (2.1) then the LLL basis reduction algorithm will produce r 1 (x (0) 1 ,...,x (0) t ) = ··· = r t (x (0) 1 ,...,x (0) t ) = 0. One can now collect the root (x (0) 1 ,...,x (0) t ) from the polynomialsr 1 (x 1 ,...,x t ),...,r t (x 1 ,...,x t )iftheyarealgebraicallyindependent. Wesay polynomials r 1 ,...,r t are algebraically independent if and only if P(r 1 ,...,r t ) = 0 ⇔ P ≡ 0 for any polynomial P defined over Q[x 1 ,...,x t ]. However, all we know that the basis vectors r 1 ,...,r t generated by LLL are linearly independent, and in general one can not prove their algebraic independence. If r 1 ,...,r t are algebraically independent, one can find the common root using the method of resultants [30, Section 3]. For our purpose we need the following. Proposition 2.24. Suppose that r 1 (x 1 ,...,x t ) and r 2 (x 1 ,...,x t ) share the common root (x (0) 1 ,...,x (0) t ). Let R(r 1 ,r 2 ) be the resultant of r 1 ,r 2 with respect to x t . Then, we have • R(r 1 ,r 2 ) is a polynomial in t−1 variables x 1 ,...,x t−1 ,
Chapter 2: Mathematical Preliminaries 34 • R(r 1 ,r 2 ) ≠ 0 if r 1 ,r 2 are algebraically independent, i.e., gcd(r 1 ,r 2 ) = 1, and • R(r 1 ,r 2 )(x (0) 1 ,...,x (0) t−1) = 0. Using the method of resultants recursively, we can find a polynomial in x 1 (with root x (0) 1 ) in t −1 iterations. We can then find x (0) 1 easily using some root finding algorithm. Furthermore, using back-substitution t−1 times, we can find x (0) 2 ,...,x (0) t as well, one in each step of back-substitution. Similarly one can use the technique of Gröbner Basis [30] to find the roots. The Gröbner Basis G = {g 1 ,g 2 ,...,g l } is a set of polynomials such that g 1 (x (0) 1 ,...,x (0) t ) = g 2 (x (0) 1 ,...,x (0) t ) = ··· = g l (x (0) 1 ,...,x (0) t ) = 0. Each polynomial g i can be computed with respect to some ordering that eliminates the variables. So it is easy to extract the desired root. However, the elimination of variables fails if the variety V(I) of the ideal I generated by r 1 ,r 2 ,...,r t is not zero-dimensional. An interested reader may refer to [30] for more details regarding Gröbner Basis. Occasionally, one can collect the roots by examining special structure in the polynomials as well. For example, if we get a polynomial f(x,y) = ax − by + b where a,b are constants, then (b,a+1) is quite clearly a root of f(x,y). From our discussion so far, we can conclude that in case of solving multivariate modular polynomials, one needs the following assumption before attempting a lattice based technique. Assumption 1: The common root (x (0) 1 ,...,x (0) t ) can be efficiently collected from the polynomials r 1 ,...,r t using the method of resultants, Gröbner Basis technique or exploiting the structure of the polynomials. Once we assume the above mentioned statement, (2.1) reduces the condition for solving a modular polynomial using LLL to the following: 2 ω(ω−1) 4(ω+1−t) det(L) 1 ω+1−t < N m √ ω . If we treat the number of variables t, and the dimension of lattice L as constants, then we can simply state the condition as det(L) < N mω . This technique of finding
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 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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Page 92 and 93: Chapter 5 Reconstruction of Primes
Page 94 and 95: 77 5.1 LSB Case: Combinatorial Anal
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83 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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