dit-upm RSA Cybersecurity Cryptography

Transcription

1 -upm Cybersecurity Cryptography José A. Mañas < Information Technology Department Universidad Politécnica de Madrid 4 october 2018

2 public key (asymmetric) public key secret key data encrypt decrypt data home 2

3 wrapped encryption public key session key secret key encrypt decrypt data encrypt decrypt data home 3

4 signed hash public key private key data data hash hash validation sign signature valid? 4

5 e=13 n= 55 d= 37 n= 55 toy data m e mod n x d mod n data

6 cryptoanalysis e = 13 n = 55 brute force try every option! is there a better path? red = 3 black = pow(red, e, n) for d in range(0, n): x = pow(black, d, n) if x == red: print(d) 6

7 encrypt: (1978) Ron Rivest, Adi Shamir & Leonard Adleman (MIT) Set up choose prime numbers p, q public key: <n, e> n= p q φ n = (p 1)(q 1) e gcd(e, φ n ) = 1 private key: <n, d> e d 1 mod φ n Encryption c = m e mod n Decryption m = c d mod n e may be simple: 0x11 0x10001 attack: guess p y q out of n integer number factorization 7

8 preparation encrypt: : example m= 688 p= 47 n= p*q= 47*71= 3337 q= 71 Z= (p-1)*(q-1)= 3220 e= 79; prime w.r.t. Z e*d= 1 mod Z d= 1019 public <e= 79, n= 3337> encrypt c= m e mod n = mod 3337 private <d=1019, n= 3337> decrypt m= c d mod n= mod 3337 black <c= 1570> m= 688 8

9 Machine 9

10 maths Fermat (1636): A B 1 1 mod B if A > 0, B is prime, A prime to B (1978): M (p 1)(q 1) 1 (mod pq) fermat (A = M p 1, B = q) M (p 1)(q 1) 1 (mod q) M (p 1)(q 1) 1 = kq, that is, q X fermat (A = M q 1, B = p) M (p 1)(q 1) 1 (mod p) M (p 1)(q 1) 1 = kp, that is, p X q X and p X and p prime and q prime pq X M (p 1)(q 1) 1 = kpq M (p 1)(q 1) 1 (mod pq) 10

11 maths encrypt then decrypt M ed = M k p 1 q 1 +1 = M M (p 1)(q 1) k = M 1 k = M 11

12 example # return (g, x, y) a*x + b*y = gcd(x, y) def egcd(a, b): if a == 0: return (b, 0, 1) else: g, x, y = egcd(b % a, a) return (g, y - (b // a) * x, x) # x = mulinv(b) mod n, (x * b) % n == 1 def mulinv(b, n): g, x, _ = egcd(b, n) if g == 1: return x % n p = 5 q = 11 n = p * q z = (p-1)*(q-1) e = 13 d = mulinv(e, z) m = pow(m, e, n) pow(pow(m, e, n), d, n) extended euclidean algorithm 12

13 more general private key: <n, d> e d 1 mod lcm p 1, q 1 ; 0 < d < n def lcm(a, b): return a * b / math.gcd(a, b) m = lcm(p-1, q-1) d = mulinv(e, m) d + m d + 2*m d + 3 * m 32 bits p = bd19 q = f2a7 e = d = 1b63fb39 d = d1 d = 57226a69 d = 7501a201 d = 92e0d999 d = b0c

14 Ron Rivest, Adi Shamir & Leonard Adleman (MIT) set up let prime numbers p, q public key: <n, e> n= p q e prime w.r.t. φ n = (p 1)(q 1) private key: <n, d> e d 1 mod φ n sign: (1978) sign send verify s = H(m) d mod n <m, s> compare equality H(m) s e mod n 14

15 sign: : example preparation p= 5 n= p*q= 5*11= 55 q= 11 Z= (p-1)*(q-1)= 4*10= 40 e= 13; prime w.r.t. Z e*d= 1 mod Z d= 37 private <d=37, n= 55> signed s= m d mod n= 3 37 mod 55= 53 public <e= 13, n= 55> verification s e mod n = mod 55 = 3 h(m) = 3 signed message <m, s= 53> h(m)= 3 15

16 how many primes are there? Estimates with 512 bits with bits dictionary attacks are unfeasible The first fifty million primes 16

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19 number of bits 19

20 references A Method for Obtaining Digital Signatures and Public-Key Cryptosystems Tools R.L. Rivest, A. Shamir, and L. Adleman Prime factorization tool Prime factors calculator Number Factorization 20

21 exercises 1. check validity of pub = (133, 40501), sec = (38797, 40501) 2. check validity of pub = (133, 40501), sec = (397, 40501) 3. check validity of pub = (133, 40501), sec = (16413, 40501) 4. design a key pair where p = 53, q = 113 choose minimal e 5. design a key pair where p = 193, q = 163 choose minimal e try e= 3, 5, 7, 9, 11, 13,, 17,, 25, idem p= 193, q=

22 exam Home automation. We need reduced keys. Specifically, we have a thermostat with this public reception key: {e = 7, n = 33}. The owner of the house sends it an order to set the temperature of the house to "31", where 31 is the desired temperature, encrypted with the previous key. It is requested: 1. Is the key correct? What is its strength in bits? 2. What temperature range can we mark? 3. Break the key; that is, find out the private part to decipher. 4. Find out at what temperature you want to set the thermostat. Decipher and verify. 22

23 exercise Alice uses the Crypto System to receive messages from Bob Alice publishes n = 299 and e = 35 Check that e = 35 is a valid exponent for the algorithm Compute d, the private exponent of Alice Bob wants to send to Alice the (encrypted) plaintext P=15 What does he send to Alice? Verify she can decrypt this message 23

24 exercise Alice publishes the following data n = pq = 221 and e = 13 Bob receives the message M = 65 and the corresponding digital signature S = 182. Verify the signature 24