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[40] Burt Rosenberg, The Solovay-Strassen Primality Test, October 1993. [41] R G Salembier and Paul Southerington, An Implementation of the AKS Primality Test, Computer Engineering (2005). [42] G. J. Simmons, A ”Weak” Privacy Protocol using the RSA Crypto Algo- rithm, Cryptologia (1983). [43] G. J. Simmons and M. J. Norris, Preliminary Comments on the MIT Public-key Cryptosystem, Cryptologia (1977). [44] William Stein, Elementary number theory. Primes, congruences, and se- crets. A computational approach., New York, NY: Springer, 2009 (English). [45] M Trott, The Mathematica Guidebook for Symbolics, Springer-Verlag, 2006. [46] M J Wiener, Cryptanalysis of Short RSA Secret Exponents, IEEE Trans- actions on Information Theory (1990). [47] Oded Yacobi and Yacov Yacobi, A New Related Message Attack on RSA, Essays in Memory of Shimon Even, 2006, pp. 187–195. [48] Song Y. Yan, Cryptanalytic Attacks on RSA, Springer, 2008. 55
A Implementations of the attacks from section 3.2 A.1 Common Modulus Attack rcm = {}; i = 3; While[i < 13, nbits = 2∧ nbits = 2 i; ∧ nbits = 2 i; ∧i; p = RandomPrime[{2∧ (nbits/2 − 1), 2∧ p = RandomPrime[{2 (nbits/2) − 1}]; ∧ (nbits/2 − 1), 2∧ p = RandomPrime[{2 (nbits/2) − 1}]; ∧ (nbits/2 − 1), 2∧ (nbits/2) − 1}]; q = RandomPrime[{2∧ (nbits/2 − 1), 2∧ q = RandomPrime[{2 (nbits/2) − 1}]; ∧ (nbits/2 − 1), 2∧ q = RandomPrime[{2 (nbits/2) − 1}]; ∧ (nbits/2 − 1), 2∧ (nbits/2) − 1}]; While[q == p, q = RandomPrime[{2∧ (nbits/2 − 1), 2∧ While[q == p, q = RandomPrime[{2 (nbits/2) − 1}]]; ∧ (nbits/2 − 1), 2∧ While[q == p, q = RandomPrime[{2 (nbits/2) − 1}]]; ∧ (nbits/2 − 1), 2∧ (nbits/2) − 1}]]; n = p ∗ q; e1 = n; While[e1 == n, e1 = RandomPrime[n]; ] e2 = n; While[e2 == ne2 ne2 ne2 == e1, e2 = RandomPrime[n]; ] m = RandomPrime[n]; c1 = PowerMod[m, e1, n]; c2 = PowerMod[m, e2, n]; r = Append[rcm, {nbits, Timing[ l = ExtendedGCD[e1, e2]; a1 = l[[2, 1]]; a2 = l[[2, 2]]; dec = Mod[PowerMod[c1, a1, n] ∗ PowerMod[c2, a2, n], n] ][[1]]}]; Print[rcm]; i++; ] 56
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 and 60: 5 Bibliography [1] Kamilah Abdullah
Page 61: [27] Arjen K. Lenstra, Primality Te
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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