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

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43 2.7 Analysis of the Root Finding Techniques 2.7 Analysis of the Root Finding Techniques In the previous sections we have studied a few root finding techniques for modular and integer polynomials using lattice based methods. We have so far analyzed the correctness and method of operation of these techniques. We shall now analyze the time complexity of the generic approach by Coppersmith [22], taking it as a representative of all similar techniques. 2.7.1 Time Complexity Analysis The running time Coppersmith’s root finding approach can be split into its two major components: lattice reduction, and root extraction using resultant method or Gröbner Basis calculation. The individual analysis of the two is as follows. LLL Lattice Reduction The runtime of Coppersmith’s method is dominated by the lattice basis reduction which runs in time polynomial in the dimension ω of the lattice and bitsize of the entries of the lattice. The time complexity of lattice reduction algorithm by Nguyen and Stehlé [93] is O(ω 5 (ω +A)A), where A is the maximum bitsize of an entry in the lattice. In Coppersmith’s construction, dim(L) depends only upon ǫ. Also, the bitsize of the entries in L are bounded by poly(log 2 N). Hence we can find the LLL reduced basis vectors in time poly(log 2 N). Root Extraction It is also worth noting that the degrees of r 1 ,...,r ω are fixed as they depend only upon ǫ, and the coefficients of r 1 ,...,r ω are bounded by a polynomial in log 2 N. If r 1 (x 1 ,x 2 ,...,x t ) and r 2 (x 1 ,x 2 ,...,x t ) are two polynomials with degrees of x 1 equal to d 1 ,d 2 respectively, then to calculate the resultant R(r 1 ,r 2 ) with respect to x 1 , one needs to find the determinant of a matrix of size (d 1 +d 2 )×(d 1 +d 2 ). One can calculate the determinant of a matrix of size D in O(D 3 ) time. For detailed calculation of the resultant, the reader may consult [30, Chapter 3]. Similarly, the Gröbner Basis calculation is in general double-exponential in the degree of the polynomial, which in case of Coppersmith’s method, is dependent only on ǫ.
Chapter 2: Mathematical Preliminaries 44 Cumulative Runtime From the earlier discussion, one may deduce that for a fixed positive number ǫ, and for a fixed number of variables, the time complexity of Coppersmith’s lattice based root finding techniques is poly(logN). Although lattice reduction can be done in polynomial time, it is almost impossible to reduce a lattice of large dimension (larger than 400, say) that has large entries. The same problem occurs for the calculation of resultant or Gröbner Basis when the number of variables is too large. 2.8 Experimental Framework Throughout this thesis, we have furnished numerous experimental results supportingourclaims. Theseexperiments, inmostofthecases, areimplementationsofthe lattice based root finding techniques and the algorithms that we have proposed. We have performed almost all (except for a few in Chapter 5) experiments using the following computing framework. • Symbolic Computation Package: Sage [124] (versions 2.10 through 4.2) • Operating System: Linux Ubuntu (8.10 through 9.10) • System Configuration: Dual Core Intel R○ Pentium R○ D CPU 1.83 GHz with 2 GB RAM and 2 MB Cache For the examples of prime reconstruction from random known bits of the primes from the LSB side, as presented in Chapter 5, we have implemented the reconstruction algorithm in the following framework. • Coding Platform: C/C++ with gcc/g++ compiler using GMP library [43]. • Operating System: Linux Ubuntu 9.04 • System Configuration: Intel R○ Pentium R○ 4 CPU 1.7 GHz with 1 GB RAM and 2 MB Cache
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some results on Cryptanalysis of RS
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Abstract In this thesis, we propose
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Acknowledgements There are many peo
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93 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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