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Acknowledgements I am deeply thankful to my supervisor Professor Vilius Stakenas. Not only for making me so interested in the RSA cryptosystem in particular and investi- gation in general, but also for his extreme kindness, sensitivity and helpfulness. I would like to sincerely thank to my supervisor Professor Carlos Caleiro for his helpful remarks, suggestions and reflections about this work. My family has been a great support since the beginning of my life, so I would like to thank them most of all, since it is because of them that I stand here today. Finally, I would like to thank all my friends who helped me to endure this task both in Vilnius and in Lisbon. I am extremely grateful to them and I hope I have the opportunity to show it after a reasonable amount of time. iii
Contents 1 Introduction 1 1.1 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Public Key Cryptography . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The RSA cryptosystem . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Mathematical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.2 Time Complexity . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.3 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . 7 1.5.4 Useful Algorithms and Results . . . . . . . . . . . . . . . 10 1.5.5 Continued Fractions . . . . . . . . . . . . . . . . . . . . . 12 1.5.6 Results from Coppersmith . . . . . . . . . . . . . . . . . . 12 1.6 RSA Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 RSA Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Variants of RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8.1 CRT-RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8.2 Multi-Prime RSA . . . . . . . . . . . . . . . . . . . . . . . 19 1.8.3 Common Prime RSA . . . . . . . . . . . . . . . . . . . . . 20 2 Primality Tests and Factorization Algorithms 21 2.1 Primality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Fermat’s Primality Test . . . . . . . . . . . . . . . . . . . 21 2.1.2 Solovay-Strassen Test . . . . . . . . . . . . . . . . . . . . 23 2.1.3 Miller-Rabin Primality Test . . . . . . . . . . . . . . . . . 24 2.1.4 AKS Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Factoring Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Fermat’s Factorization . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Pollard’s ρ Algorithm . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Elliptic Curve Method . . . . . . . . . . . . . . . . . . . . 29 2.2.4 General Number Field Sieve . . . . . . . . . . . . . . . . . 31 2.3 Overview of the Different Methods . . . . . . . . . . . . . . . . . 31 iv
Page 1 and 2: A Survey of Cryptanalytic Attacks o
Page 3: Palavras Chave: RSA, criptanálise,
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 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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