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[13] Whitfield Diffie and Martin E. Hellman, New Directions in Cryptography, IEEE Transactions on Information Theory (1976). [14] ANDREJ DUJELLA, Continued Fractions and RSA with Small Secret Ex- ponent, Tatra Mt. Math. Publ (2004). [15] D Boneh G Durfee and Y Frankel, Exposing an RSA Private Key Given a Small Fraction of its Bits, Journal of Cryptology (2001). [16] Glenn Durfee, Cryptanalysis of RSA Using Algebraic and Lattice Methods, Ph.D. thesis, Stanford University, 2002. [17] M. K. Franklin and M. K. Reiter, A Linear Protocol Failure for RSA with Exponent Three, CRYPTO ’95 rump session (1995). [18] J. Hastad, On using RSA with Low Exponent in a Public Key Network, Lecture Notes in Computer Science (1985). [19] M Jason Hinek, Low Public Exponent Partial Key and Low Private Expo- nent Attacks on Multi-prime RSA, Master’s thesis, University of Waterloo, 2002. [20] M. Jason Hinek, Another Look at Small RSA Exponents, 2006. [21] M Jason Hinek, On the Security of Some Variants of RSA, Ph.D. thesis, University of Waterloo, 2007. [22] M. Jason Hinek, Cryptanalysis of RSA and its Variants, Chapman Hall, 2010. [23] M. Jason Hinek and Charles C. Y. Lam, Common Modulus Attacks on Small Private Exponent RSA and Some Fast Variants (in Practice), IACR Cryptology ePrint Archive 2009 (2009), 37. [24] Abdullah Darwish Imad Khaled Salah and Saleh Oqeili, Mathematical At- tacks on RSA Cryptosystem, Journal of Computer Science (2006). [25] S. Katzenbeisser, Recent Advances in RSA Cryptography, Klewer Academic Publishers (2001). [26] A.K. Lenstra, H.W.jun. Lenstra, and Lászlo Lovász, Factoring Polynomials With Rational Coefficients, Math. Ann. 261 (1982), 515–534. 53
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Page 1 and 2:
A Survey of Cryptanalytic Attacks o
Page 3 and 4:
Palavras Chave: RSA, criptanálise,
Page 5 and 6:
Contents 1 Introduction 1 1.1 Crypt
Page 7 and 8:
List of Tables 1 Notation . . . . .
Page 9 and 10: messages, Alice and Bob needed a se
Page 11 and 12: key, have lead to restrictions over
Page 13 and 14: 1.5.2 Time Complexity In cryptograp
Page 15 and 16: Definition 8. Given an integer N >
Page 17 and 18: If we want to solve a system of lin
Page 19 and 20: 1.5.5 Continued Fractions The conti
Page 21 and 22: 1.6 RSA Definition We are now in co
Page 23 and 24: calculates these roots (though, as
Page 25 and 26: not the case regarding RSA as we ha
Page 27 and 28: 1.8.3 Common Prime RSA For the Comm
Page 29 and 30: So if we use contrapositive of this
Page 31 and 32: The consequence of this theorem is
Page 33 and 34: 2.1.4 AKS Test In 2004, a major bre
Page 35 and 36: We will begin with an old method cr
Page 37 and 38: Theorem 18. Let N > 1 be an integer
Page 39 and 40: e reduced. Regarding the GNFS, it s
Page 41 and 42: factorization of N. That is, each u
Page 43 and 44: Table 3: DeLaurentis Attack’s Exp
Page 45 and 46: Notice that gi(m) ∼ = 0 (mod Ni)
Page 47 and 48: ecovery exponent l to reveal the pl
Page 49 and 50: y fN(x) = 2 −k1e ((m22 k2 + x2 k1
Page 51 and 52: Theorem 25. Let two plain texts m1,
Page 53 and 54: Theorem 28. Given an RSA modulus N
Page 55 and 56: increment m by one and start again.
Page 57 and 58: avoided. It should be noted that, o
Page 59: 5 Bibliography [1] Kamilah Abdullah
Page 63 and 64: A Implementations of the attacks fr
Page 65 and 66: q = RandomPrime[{2∧ (nbits − 1)
Page 67 and 68: While[i ≤ 23, n0 = Min[n0, nlist[
Page 69 and 70: B Implementations of the attacks fr
Page 71 and 72: C Implementation of Wiener’s atta
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