Download presentation

Presentation is loading. Please wait.

Published byRosalyn Mason Modified about 1 year ago

1
Cryptography Lecture 8: Primality Testing and Factoring Piotr Faliszewski

2
Previous Class Attacks on RSA Known digits attacks Low exponent attacks Short plaintext attacks Timing attacks Practical RSA Exponentiation modulo n Primality testing

3
Factoring and Primality Testing Factoring Input: n N Output: nontrivial factor of n Primality testing Input n N Output: the number is composite the number is probably prime Is there a difference? Yes! – primality testing much easier! You do not need to factor the number to see it is composite

4
Miller-Rabin Test Generalization of the Fermat’s test Principle if p is a prime then x 2 = 1 (mod p) has only two solutions: x = 1 and x = -1 Why does the principle hold? Gist of the MR test find a number b such that b 2 = 1 (mod p) If b {-1,1} then composite

5
Miller-Rabin Test MR( int n ): let n-1 = 2 k m a random in {2, 3,..., n-2 } b 0 = a m (mod n) if b 0 = ±1 (mod n) then declare prime for j = 1 to k-1 do b j = b j-1 2 (mod n) if b j = 1 (mod n) then declare composite if b j = -1 (mod n) then declare prime declare composite What are we doing? b 0 = a m (mod n) b 1 = a 2m (mod n) b 2 = a 4m (mod n) ... b j = a 2 j m (mod n) ... b k-1 = a (n-1)/2 (mod n)

6
Miller-Rabin Test: Examples n = 5*7*11 = 385 n -1 = 384 = 2 7 *3 k = 7, m = 3 a = 9 b 0 = 9 3 = 344 (mod 385) b 1 = 9 3*2 = 141 (mod 385) b 2 = 9 3*2 2 = 246 (mod 385) b 3 = 9 3*2 3 = 71 (mod 385) b 4 = 9 3*2 4 = 36 (mod 385) b 5 = 9 3*2 5 = 141 (mod 385) n = 3*11*17 = 561 n -1 = 560 = 2 4 *35 k = 4, m = 35 a = 2 b 0 = 2 35 = 263 (mod 561) b 1 = 2 35*2 = 166 (mod 561) b 2 = 2 35*2 2 = 67 (mod 561) b 3 = 2 35*2 3 = 1 (mod 561) Proof of compositeness!

7
Miller-Rabin Test: Examples n = 401 n -1 = 400 = 2 4 *25 k = 4, m = 25 a = 3 b 0 = 3 25 = 268 (mod 401) b 1 = 3 25*2 = 45 (mod 401) b 2 = 3 25*2 2 = 20 (mod 401) b 3 = 3 25*2 3 = 400 (mod 401) = -1 (mod 401) n = 401 n -1 = 400 = 2 4 *25 k = 4, m = 25 a = 2 b 0 = 2 25 = 356 (mod 401) b 1 = 2 25*2 = 20 (mod 401) b 2 = 2 25*2 2 = 400 (mod 401) Evidence of primality!

8
Miller-Rabin Test if b 0 = ±1 (mod n) all b i ’s (i > 0) will be 1 can’t find nontrivial roots of 1 i {1,..., k-1} if b i = 1 (mod n) then b i-1 is neither 1 nor -1 b i-1 2 = 1 (mod n) we found a nontrivial root if b i = -1 (mod n) then b i+1 through b k are all 1 (mod n) can’t find nontrivial roots of 1 Why this works? n-1 = 2 k m b 0 = a m (mod n) b 1 = a 2m (mod n) b 2 = a 4m (mod n) ... b j = a 2 j m (mod n) ... b k-1 = a (n-1)/2 (mod n)

9
Miller-Rabin Test: Quality MR test is probabilistic Answer composite – the number is certainly composite prime – the number is prime with high probability Errors MR(n) says prime but n is composite Pr[error] ≤ ¼ Repeat the test to downgrade the prob. of error

10
Other Primality Tests Solovay-Strassen Test similar in nature to MR uses so called Jacobi symbol fast in practice probabilistic Deterministic test Agrawal, Kayal, and Saxena 2002 extremely slow Tests that prove primality MR tests compositeness! fairly slow needed in very few cases

11
Factoring Huge amount of work on factoring! we look at some simple algorithms Some best algorithms quadratic sieve elliptic curve number field sieve Assumption Factor an odd integer produce one factor how to get all of them? O(e (1+o(1))sqrt(lnn lnln n) ) O(e (1+o(1))sqrt(lnp lnln p) ) O(e (1.92+o(1))(lnn) 1/3 (lnlnn) 2/3 )

12
Factoring Input: n N Output: nontrivial factor of n There are about (n) = n / ln n primes ≤ n Trivial methods divide by all numbers in {2,..., n-1} or by all primes p p ≤ sqrt(n) These are exponential!

13
Fermat’s Method The principle express n as a difference of squares n = x 2 - y 2 n = (x-y)(x+y) The algorithm Compute: n + i 2 for i {1,2,... } Stop when n + i 2 is a square (i.e., x 2 = n+i 2 ) Then we have n = x 2 – i 2 Examples 15 = 4 2 – 1 2 = (4-1)(4+1) = 3*5 21 = 5 2 – 2 2 = = (5-2)(5+2) = 3*7

14
Fermat’s Method The principle express n as a difference of squares n = x 2 - y 2 n = (x-y)(x+y) The algorithm Compute: n + i 2 for i 2 {1,2,... } Stop when n + i 2 is a square (i.e., x 2 = n+i 2 ) Then we have n = x 2 – i 2 Performance depends on distance between x and y could be very slow! Conclusion for RSA p and q should differ by a large value

15
Pollard’s p-1 Method The method input: n choose a > 1 (e.g., a = 2) choose B let b = a B! (mod n) d = gcd( b - 1, n ) d is a factor of n Goal of the method factor n = pq... provided p-1 has only small prime factors Example n = 7 * 11 = 77 a = 2 B = 4, B! = 2*3*4 = 24 b = 2 24 = 71 (mod 77) gcd(b-1, n) = gcd(70,77) = 7

16
Pollard’s p-1 Method The method input: n choose a > 1 (e.g., a = 2) choose B let b = a B! (mod n) d = gcd( b - 1, n ) d is a factor of n Goal of the method factor n = pq... provided p-1 has only small prime factors Example n = 7 * 11 = 77 a = 2 B = 2, B! = 2 b = 2 2 = 4 (mod 77) gcd(b-1, n) = gcd(3,77) = 1

17
Pollard’s p-1 Method The method input: n choose a > 1 (e.g., a = 2) choose B let b = a B! (mod n) d = gcd( b - 1, n ) d is a factor of n Goal of the method factor n = pq... provided p-1 has only small prime factors Example n = 7 * 11 = 77 a = 2 B = 6, B! = 2*3*4*5*6 = 720 b = = 1 (mod 77) gcd(b-1, n) = gcd(0,77) = 77

18
Pollard’s p-1 Method In symbols: b 1 = a (mod n) b 2 = b 1 2 (mod n) ... b i = b i-1 i ... How to compute a B! B! – can be very big 5! = 120 6! = 720 10! = 20! = n! – about n log 2 n bits

19
Pollard’s p-1 Method In symbols: b 1 = a (mod n) b 2 = b 1 2 (mod n) ... b i = b i-1 i ... How to compute a B! a = 2,B = 4,n = 77 b 1 = 2 (mod 77) b 2 = 2 2 = 4 (mod 77) b 3 = 4 3 = 64 (mod 77) b 4 = 64 4 = = 71 (mod 77)

20
Pollard’s p-1 Method The method input: n choose a > 1 (e.g., a = 2) choose B let b = a B! (mod n) d = gcd( b - 1, n ) d is a factor of n Why does it work? p – prime factor of n suppose: p-1 has only small prime factors Then likely p-1 | B! Then B! = k(n-1) b = (a p-1 ) k (mod p) b = 1 (mod p) p | b - 1

21
Pollard’s p-1 Method Potential problems n = pq both p and q have small factors b = 1 (mod p) b = 1 (mod q) Method fails Choice of B too small method won’t work too big works slowly or fails Example n = 7 * 11 = 77 a = 2 B = 2 to small B = 4 worked B = 6 to big = 6 = 2*3 2! – does not contain 3 4! – contains 2 and 3 6! – contains 2,3 and 5 covers both factors!

22
Pollard’s p-1 Method Conclusions for RSA n = pq p-1 or q-1 has small prime factors? then RSA can be broken How to defend? p 0 chose a large prime e.g., p 0 > try numbers of the form: kp k – needs to be even! k > test kp 0 +1 for primality

23
Factoring Relation to squares n – an integer x,y – to integers s.t. x 2 = y 2 (mod n) x y (mod n) if such x, y exist then n is composite gcd( x-y, n ) is a nontrivial factor Examples 11 2 = 121 = 1 (mod 12) 5 2 = 25 = 1 (mod 12) 11 5 (mod 12) gcd(11-5, 12 ) = 6

24
Factoring Relation to squares n – an integer x,y – to integers s.t. x 2 ´ y 2 (mod n) x y (mod n) if such x, y exist then n is composite gcd( x-y, n ) is a nontrivial factor Examples 5 2 = 25 = 7 (mod 9) 14 2 = 196 = 7 (mod 9) 14 = 9+5 = 5 (mod 9) gcd(14-5, 9 ) = 9

25
Quadratic Sievie Idea try to apply the principle from the previous slide find x,y such that x 2 = y 2 (mod n) x y (mod n) x -y (mod n) finding such x, y not obvious Take “random” squares Reduce modulo n Factor (hope for small factors!) Try to build squares from what you get

26
Quadratic Sieve: Example n = = 5 5 19 (mod n) = 2 2 5 11 13 19 (mod n) = 3 2 13 3 (mod n) =2 6 3 2 11 (mod n) = 2 19 ( mod n) = 2 5 5 13 2 (mod n) =5 2 7 2 13 (mod n) (9398 1964 17078) 2 = 2 8 3 2 5 6 11 2 13 4 19 2 = (2 4 3 5 3 11 13 2 19) 2 (mod n) = (mod n) gcd( – , ) = 1093

27
Quadratic Sieve: Example n = = 5 5 19 (mod n) = 2 2 5 11 13 19 (mod n) = 3 2 13 3 (mod n) =2 6 3 2 11 (mod n) = 2 19 ( mod n) = 2 5 5 13 2 (mod n) =5 2 7 2 13 (mod n) (9398 8077 3397) 2 = 2 6 5 6 13 2 19 2 = (2 3 5 3 13 19) 2 (mod n) = (mod n) BUT: n – = = (mod n)

28
How to Find the Squares? What squares to use? we want small prime factors? so x 2 should be slightly above n Idea: Try integers close to: sqrt(i n) + j small j, various i (sqrt(i n) + j) 2 ≈ in + 2j sqrt(in) +j 2 approx: 2j sqrt(in) + j 2 (mod n)

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google