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

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51 3.1 Our Basic Technique • For a fixed α, the value of γ decreases when τ increases, and • given a fixed τ, the value of γ increases as α increases. In Table 3.1, we present the numerical values of γ corresponding to α following Theorem 3.2 for three different values of τ, namely 1 4 , 1 2 and 3 4 . α 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.875 τ = 1 4 0.482 0.555 0.629 0.704 0.780 0.856 0.934 1.012 1.091 1.150 τ = 1 2 0.284 0.347 0.412 0.477 0.544 0.612 0.681 0.751 0.821 0.875 τ = 3 4 0.131 0.188 0.245 0.305 0.365 0.427 0.489 0.553 0.618 0.667 Table 3.1: The numerical upper bounds of γ (in each cell) following Theorem 3.2, given different values of α and τ. In the work of [96], the value of τ has been taken as 1 . Thus we discuss some 2 cases when τ = 1 to highlight the improvements we achieve over [96]. Note that 2 for randomly chosen e with e < φ(N), the value of e will be O(N) in most of the cases. In such a case, putting α = 1 and τ = 1 , we get that γ < 0.284. 2 When α < 1 and τ = 1 , the bound on γ will decrease and it will become 0 2 at α = 1 . However, for randomly chosen e with e < φ(N), this will happen in 2 negligibly small proportion of cases. Most interestingly, the bound of γ will increase further beyond 0.284 when α > 1. Wiener’s attack [130] becomes ineffective when e > N 1.5 and the attack proposedbyBonehandDurfee[15]becomesineffectivewhene > N 1.875 . Similarto theresultof[15], equationsoftheformeX−ZY = 1cannotbeusedfore > N 1.875 , since in such case no X will exist given the bound on Y. For τ = 1 , we have 2 presented the theoretical results for e reaching N 1.875 in Table 3.1. Experimental results will not reach this bound as we work with small lattice dimensions in practice, but even then the experimental results for e reach close to the value N 1.875 as we demonstrate results for N 1.774 in Section 3.2.3 for 1000-bit N. Note that here we present a theoretical estimate on the bound of γ. These bounds may not be achievable in practice due to the large lattice dimensions. However, the experimental results, presented in Sections 3.2.3, 3.3.1, are close to the theoretical estimates.
Chapter 3: A class of Weak Encryption Exponents in RSA 52 3.2 Improvements over Existing Work In this section we present various improvements over the work of [96]. For this, first we present an outline of the strategy in [96]. Consider that [ e satisfies ] eX − (p−u)(q −v)Y = 1 with 1 ≤ Y < X < 2 −1 4N 1 4, |u| < N 4, 1 v = − qu . If all the p−u 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 are estimated as N 1 2 −ǫ . The flow of the algorithm in [96] is as follows. 1. Continued Fraction algorithm is used to find the unknowns X,Y among the convergents of e N . 2. Then, the Elliptic Curve Factorization Method (ECM [79]) is used to partially factor eX−1 Y , i.e., into the factors (p−u)(q −v). 3. Next, an integer relation detection algorithm (LLL [77]) is used to find the divisors of B ecm -smooth part of eX−1 Y in a small interval. 4. Finally, if p−u or q −v is found, the method due to [24] is applied. After knowing (p−u)(q −v), if one gets the factorization of p−u or q −v, then it is possible to identify p − u or q − v efficiently and the overall complexity is dominated by the time required for factorization. According to [96], if ECM [79] is used for factorization, and if all prime factors of p−u or q−v are less than 10 50 , then getting p−u or q −v is possible in moderate time. Once p−u or q −v is found, as u,v are of the order of N 1 4, using the technique of [24], it is possible to find p or q efficiently. 3.2.1 The Improvement in the Bounds of X,Y In [96] the bounds of X and Y are given as 1 ≤ Y < X < 2 −1 4N 1 4. Since, u,v are of O(N 1 4), we get that (p − u)(q − v) is O(N). When e is O(N 1+µ ), µ > 0 and X is O(N ν ), 0 < ν ≤ 1 4 , the value of eX is O(N1+µ+ν ). In such a case, Y will be O(N µ+ν ), which is not possible as Y < X. Thus the values of e are bounded by O(N) in the work of [96]. Next we generalize the bounds on X,Y. The method of [96] requires 1 ≤ Y < X < 2 −1 4N 1 4. For τ = 1 , our result 2 in Lemma 3.1 implies that it is enough to have 2XY < N 2, 1 which gives better bounds than [96] due to the following reasons.
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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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Chapter 7 Approximate Integer Commo
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145 7.7 EGACDP Sinceinthiscasethema
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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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