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

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153 8.2 Overview of Root Finding Techniques have thoroughly discussed the lattice based root finding methods that are present in the current literature. Our discussion covered the earlier techniques of finding small roots of modular polynomials by Coppersmith [23], as well as the general result by Jochemsz and May [65]. We have presented Coron’s [28] method to find roots of bivariate integer polynomials, and have also discussed the general version of the same proposed by Jochemsz and May [65]. Case I: Modular Polynomial f N Monomials of f N Bound Ref. 1, x,y,xy X ·Y < N 2 3 Sec. 5.2 Case II: Integer Polynomial f Monomials of f Bound Ref. 1,x 1 ,...,x n+1 , X n+1( 1 1 n+2 +τ n) 1 ·(X 2···X n+1 ) ( n( 1 1 n+2 + n+1)) τ Sec. 4.1 x 2 x n+2 ,...,x n+1 x n+2 ( 1 2(n+2) + τ n+1 +τ2 2n Xn+2 < W ( 1 ) n+1( 1 n+2 +τ n)) 1,x,y,yz,xyz X (1 2 +3 2 τ) ·Y (5 6 +3 2 τ) ·Z (1 2 +3 2 τ+τ2 ) < W ( 1 3 +τ) Sec. 6.1.1 1,y,z,xyv,xv,xzt X 5 24 ·Y 1 8 ·Z 1 6 ·V 1 8 ·T 1 12 < W 1 12 Sec. 6.1.4 1,x,y,yz,yv,xyz,xyv X (1 4 +τ 1+τ 2 + 3 2 τ 1τ 2) ·Y ( 11 24 +4 3 (τ 1+τ 2 )+ 3 2 τ 1τ 2) Sec. 6.2 ·Z (1 6 +2 3 (τ 1+τ 2 )+ 3 4 τ2 1 +3 2 τ 1τ 2 +τ1 2τ 2) ·V (1 6 +2 3 (τ 1+τ 2 )+ 3 4 τ2 2 +3 2 τ 1τ 2 +τ 1 τ2) 2 < W (1 8 +1 2 (τ 1+τ 2 )+τ 1 τ 2) Table 8.1: Bounds for finding roots of a polynomial. Now, thegeneralrootfindingmethodsbyJochemszandMay[65]havebeenthe most effective ones in case of the polynomials we have encountered. In Table 8.1, we give an overview of the results obtained by using the general root finding techniques of [65] to the specific polynomials in our case. The idea of this tabular
Chapter 8: Conclusion 154 representation is motivated by similar tables presented in [64, Page 44], and the results we present here extend the array of results provided in [64]. The reader may refer back to the respective sections to obtain the relevant details. 8.3 Open Problems In this section, we propose a few open problems related to our work. These may lead to new interesting research topics in the related field. 8.3.1 Weak Encryption Keys One may note that Blömer and May [9] and our work in Chapter 3 present a different classes of weak encryption exponents of cardinality N 0.75−ǫ . For fixed N, there are O(N) many encryption exponents e which are less than N. Hence most of the encryption exponents are not weak based on the current knowledge in this field. It is unknown whether there is a class of weak encryption exponents of cardinality substantially greater than N 0.75−ǫ . So, we have the following open problem. Problem 8.1. Is there any new class of weak encryption exponents of cardinality substantially greater than N 0.75−ǫ ? 8.3.2 More than one Decryption Key Since the time complexity of the work of Howgrave-Graham and Seifert [62] as well as our approach in Chapter 4 are exponential in n, we can not get better experimental results for large n. Hence we have the following question. Problem 8.2. Is there any algorithm with runtime polynomial in (log 2 N,n) which can improve the bound achieved by our method? Furthermore,considerthesamesituationforCRT-RSA.Supposethate 1 ,...,e k are k encryption exponents and (d p1 ,d q1 ),...,(d pk ,d qk ) are the corresponding decryption exponents for the same CRT-RSA modulus N. We know when e is of order N, then one can factor N in polynomial time [66] if d p ,d q < N 0.073 . So, we can pose the following problem in the same direction.
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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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To Thaakuma (Grandmother), the firs
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2.3.7 Small Decryption Exponent Att
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7.4 The General Solution for EPACDP
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List of Figures 1.1 Communication o
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Chapter 1: Introduction 2 The motiv
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Chapter 1: Introduction 4 there is
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Chapter 1: Introduction 6 Discrete
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Chapter 1: Introduction 8 Chapter 5
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Chapter 2 Mathematical Preliminarie
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13 2.2 RSA Cryptosystem are impleme
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15 2.2 RSA Cryptosystem • private
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23 2.3 Cryptanalysis of RSA to be h
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25 2.3 Cryptanalysis of RSA The att
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27 2.4 Lattice and LLL Algorithm No
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29 2.4 Lattice and LLL Algorithm Wh
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31 2.5 Solving Modular Polynomials
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39 2.6 Solving Integer Polynomials
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41 2.6 Solving Integer Polynomials
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43 2.7 Analysis of the Root Finding
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45 2.9 Conclusion 2.9 Conclusion No
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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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97 6.1 Implicit Factoring of Two La
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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
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