j = p + f; m0 = 0; While[j
C Implementation of Wiener’s attack rwi = {}; i = 5; While[i < 14, 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; lambdan = LCM[(p − 1), (q − 1)]; d = RandomPrime[Floor[n∧ d = RandomPrime[Floor[n (1/4)]]; ∧ d = RandomPrime[Floor[n (1/4)]]; ∧ (1/4)]]; While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n∧ While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n (1/4)]]; ]; ∧ While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n (1/4)]]; ]; ∧ (1/4)]]; ]; g0 = n; k0 = 1; l = 1; While[d > p ∗ q/(2(p + q − 1)g0 ∗ k0), 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; lambdan = LCM[(p − 1), (q − 1)]; d = RandomPrime[Floor[n∧ d = RandomPrime[Floor[n (1/4)/l]]; ∧ d = RandomPrime[Floor[n (1/4)/l]]; ∧ (1/4)/l]]; While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n∧ While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n (1/4)/l]]; ]; ∧ While[GCD[d, lambdan] = 1, d = RandomPrime[Floor[n (1/4)/l]]; ]; ∧ (1/4)/l]]; ]; e = PowerMod[d, −1, lambdan]; k = (e ∗ d − 1)/lambdan; g = GCD[p − 1, q − 1]; g0 = g/GCD[g, k]; k0 = k/GCD[k, g]; l++; ]; rwi = Append[rwi, {nbits, Timing[ conv = Convergents[e/n]; candidates = Table[Floor[e/conv[[i]]], {i, 2, Length[conv]}]; m = 0; 64
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A Survey of Cryptanalytic Attacks o
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Palavras Chave: RSA, criptanálise,
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Contents 1 Introduction 1 1.1 Crypt
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List of Tables 1 Notation . . . . .
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messages, Alice and Bob needed a se
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key, have lead to restrictions over
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1.5.2 Time Complexity In cryptograp
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Definition 8. Given an integer N >
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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 and 62: [27] Arjen K. Lenstra, Primality Te
- 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: B Implementations of the attacks fr
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