ECE 646 Lecture 9. RSA: Genesis, operation & security

Transcription

1 ECE 646 Lecture 9 RSA: Genesis, operation & security

2 Required Reading (1) W. Stallings, "Cryptography and Network-Security," Chapter 8.1 Prime Numbers Chapter 8.2 Fermat's and Euler's Theorems Chapter 9.2 The RSA Algorithm Description of the Algorithm The Security of RSA The Factoring Problem Appendix 9A Proof of the RSA Algorithm Appendix 9B The Complexity of Algorithms

3 Required Reading (2) A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography Chapter 2.4 Number Theory The integers Chapter 8.2 RSA public-key encryption Description Security of RSA Chapter 2.3 Complexity theory Basic definitions Asymptotic notation Complexity classes (until 2.61 Example)

4 Recommended Reading Adi Shamir and Eran Tromer, "On the Cost of Factoring RSA-1024," CryptoBytes, Summer Factorization Announcements Steven Levy, "Crypto: How the Code Rebels Beat the Government - Saving Privacy in the Digital Age," Viking 2001, Chapter: "Prime Time".

5 Public Key (Asymmetric) Cryptosystems Public key of Bob - K B Private key of Bob - k B Network Alice Encryption Decryption Bob

6 Trap-door one-way function Whitfield Diffie and Martin Hellman New directions in cryptography, 1976 PUBLIC KEY X f(x) Y f -1 (Y) PRIVATE KEY

7 Professional (NSA) vs. amateur (academic) approach to designing ciphers 1. Know how to break Russian ciphers 2. Use only well-established proven methods 3. Hire 50,000 mathematicians 4. Cooperate with an industry giant 5. Keep as much as possible secret 1. Know nothing about cryptology 2. Think of revolutionary ideas 3. Go for skiing 4. Publish in Scientific American 5. Offer a $100 award for breaking the cipher

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11 Challenge published in Scientific American Ciphertext: Public key: N = e = 9007 Award $100 (129 decimal digits) 1977

12 Rivest estimation The best known algorithm for factoring a 129-digit number requires: trilion years = years assuming the use of a supercomputer being able to perform 1 multiplication of 129 decimal digit numbers in 1 ns Rivest s assumption translates to the delay of a single logic gate 10 ps Estimated age of the universe: 100 bln years = years

13 Lehmer Sieve Bicycle chain sieve [D. H. Lehmer, 1928] Computer Museum, Mountain View, CA

14 Machine à Congruences [E. O. Carissan, 1919]

15 Supercomputer Cray Computer Museum, Mountain View, CA

16 Early records in factoring large numbers Years 1974 Number of decimal digits 45 Number of bits 149 Required computational power (in MIPS-years)

17 Number of bits vs. number of decimal digits 10 #digits = 2 #bits #digits = (log 10 2) #bits 0.30 #bits 256 bits = 77 D 384 bits = 116 D 512 bits = 154 D 768 bits = 231 D 1024 bits = 308 D 2048 bits = 616 D

18 How to factor for free? A. Lenstra & M. Manasse, 1989 Using the spare time of computers, (otherwise unused) Program and results sent by (later using WWW)

19 Practical implementations of attacks Factorization, RSA Year Number of bits of N Number of decimal digits of N Method Estimated amount of computations QS 5000 MIPS-years GNFS 750 MIPS-years GNFS 2000 MIPS-years GNFS 8000 MIPS-years

20 Breaking RSA-129 When: Who: How: August April 1, 1994, 8 months D. Atkins, M. Graff, A. K. Lenstra, P. Leyland volunteers from the entire world 1600 computers from Cray C90, through 16 MHz PC, to fax machines Only 0.03% computational power of the Internet Results of cryptanalysis: The magic words are squeamish ossifrage An award of $100 donated to Free Software Foundation

21 Elements affecting the progress in factoring large numbers computational power increase of about 1500 times computer networks Internet better algorithms

22 RSA as a trap-door one-way function PUBLIC KEY M C = f(m) = M e mod N C M = f -1 (C) = C d mod N PRIVATE KEY N = P Q P, Q - large prime numbers e d 1 mod ((P-1)(Q-1))

23 RSA keys PUBLIC KEY PRIVATE KEY { e, N } { d, P, Q } N = P Q P, Q - large prime numbers e d 1 mod ((P-1)(Q-1))

24 Why does RSA work? (1)? M = C d mod N = (M e mod N) d mod N = M decrypted message original message e d 1 mod ((P-1)(Q-1)) e d 1 mod ϕ(n) Euler s totient function

25 Euler s totient (phi) function (1) ϕ(n) - number of integers in the range from 1 to N-1 that are relatively prime with N Special cases: 1. P is prime ϕ(p) = P-1 Relatively prime with P: 1, 2, 3,, P-1 2. N = P Q P, Q are prime ϕ(n) = (P-1) (Q-1) Relatively prime with N: {1, 2, 3,, P Q-1} {P, 2P, 3P,, (Q-1)P} {Q, 2Q, 3Q,, (P-1)Q}

26 Euler s totient (phi) function (2) Special cases: 3. N = P 2 P is prime ϕ(n) = P (P-1) Relatively prime with N: {1, 2, 3,, P 2-1} {P, 2P, 3P,, (P-1)P} In general If N = P e1 1 P e2 2 P e3 3 P et t t ϕ(n) = P i ei-1 (P i -1) i=1

27 The first 1000 values of φ(n)

28 Euler s Theorem Leonard Euler, a ϕ(n) 1 (mod N) a: gcd(a, N) = 1

29 Euler s Theorem - Justification (1) For N=10 For arbitrary N R = {1, 3, 7, 9} R = {x 1, x 2,, x ϕ(n) } Let a=3 S = { 3 1 mod 10, 3 3mod 10, 3 7 mod 10, 3 9 mod 10 } = {3, 9, 1, 7} Let us choose arbitrary a, such that gcd(a, N) = 1 S = {a x 1 mod N, a x 2 mod N,, a x ϕ(n) mod N} = rearranged set R

30 Euler s Theorem - Justification (2) For N=10 For arbitrary N R = S R = S x 1 x 2 x 3 x 4 (a x 1 ) (a x 2 ) (a x 3 ) (a x 4 ) mod N ϕ(n) i=1 ϕ(n) x i i=1 a x i (mod N) x 1 x 2 x 3 x 4 a 4 x 1 x 2 x 3 x 4 mod N ϕ(n) i=1 ϕ(n) x i a ϕ(n) i=1 x i (mod N) a 4 1 (mod N) a ϕ(n) 1 (mod N)

31 Why does RSA work? (2) M = C d mod N = (M e mod N) d mod N = = M e d mod N = e d 1 mod ϕ(n) e d = 1 + k ϕ(n) = = M 1+k ϕ(n) mod N = M (M ϕ(n) ) k mod N = = M (M ϕ(n) mod N) k mod N = = M 1 k mod N = M