dissertacao.pdf

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3 Cryptanalysis of RSA 33 3.1 Kind of Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Some Misuses of RSA . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Common Modulus Attack . . . . . . . . . . . . . . . . . . 33 3.2.2 Hastad’s Broadcast Attack . . . . . . . . . . . . . . . . . 36 3.2.3 Cycling Attack . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Recovering Plain texts Encrypted With Small Public Exponent . 40 3.3.1 Stereotyped Message Attack . . . . . . . . . . . . . . . . . 41 3.3.2 Related Message Attack . . . . . . . . . . . . . . . . . . . 43 3.3.3 Random Padding Attack . . . . . . . . . . . . . . . . . . 44 3.3.4 Leaking Information . . . . . . . . . . . . . . . . . . . . . 45 3.4 Factoring the modulus of RSA with Small Private Exponent d . 46 3.4.1 Wiener’s Continuous Fractions Attack . . . . . . . . . . . 47 3.4.2 Improving Wiener’s Attack . . . . . . . . . . . . . . . . . 48 4 Conclusions 51 5 Bibliography 52 A Implementations of the attacks from section 3.2 56 A.1 Common Modulus Attack . . . . . . . . . . . . . . . . . . . . . . 56 A.2 DeLaurentis Attack . . . . . . . . . . . . . . . . . . . . . . . . . . 57 A.3 Hastad’s Common Plaintext Attack . . . . . . . . . . . . . . . . 57 A.4 Related Plaintext Attack . . . . . . . . . . . . . . . . . . . . . . . 59 B Implementations of the attacks from section 3.3 62 B.1 Stereotyped Message Attack . . . . . . . . . . . . . . . . . . . . . 62 B.2 Related Message Attack . . . . . . . . . . . . . . . . . . . . . . . 63 C Implementation of Wiener’s attack 64 v
List of Tables 1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Common Modulus Attack’s Experimental Results . . . . . . . . . 34 3 DeLaurentis Attack’s Experimental Results . . . . . . . . . . . . 36 4 Common Plain text Attack’s Experimental Results . . . . . . . . 37 5 Related Plain text Attack’s Experimental Results . . . . . . . . . 38 6 Cycling Attack’s Experimental Results . . . . . . . . . . . . . . . 39 7 Stereotyped Message Attack’s Experimental Results . . . . . . . 42 8 Wiener’s Attack’s Experimental Results . . . . . . . . . . . . . . 48 vi
Page 1 and 2: A Survey of Cryptanalytic Attacks o
Page 3 and 4: Palavras Chave: RSA, criptanálise,
Page 5: Contents 1 Introduction 1 1.1 Crypt
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 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 and 70:
B Implementations of the attacks fr
Page 71 and 72:
C Implementation of Wiener’s atta
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