Randomness and Computation: Some Prime Examples

1. Randomness and Computation: Some Prime Examples

2. Did you get that file ok? Was the transmission accurate? How would I know? Earth has transferred a huge file X to Moon. Moon received Y. Earth: X Moon: Y

3. Let p(n) be the number of primes between 1 and n. I wonder how fast p(n) grows? Conjecture [1790s]: Legendre Gauss

4. Their estimates

5. Two independent proofs of the Prime Density Theorem [1896]: De la Vallée Poussin J-S Hadamard

6. The Prime Density Theorem • This theorem remains one of the celebrated achievements of number theory. In fact, an even sharper conjecture remains one of the great open problems of mathematics!

7. The Riemann Hypothesis [1859] Riemann

8. Slightly easier to show (n)/n ≥ 1/(2 log n) (We’ll use this, but won’t prove it here.)

9. Random (log n)-bit number is a random number from 1..n (just add one) (n) / n ≥ 1/(2log n) means that a random (log n)-bit number has at least a 1/2log n chance of being prime.

10. Random k-bit number is a random number from 1..2k (2k) / 2k≥ 1/2k means that a random k-bit number has at least a 1/2k chance of being prime.

11. Really useful fact • A random k-bit number has at least a 1/2k chance of being prime. So if we pick 2k random k-bit numbers the expected number of primes on the list is at least 1

12. Picking A Random Prime • Many modern cryptosystems (e.g., RSA) include the instructions: • “Pick a random n-bit prime.” • How can this be done efficiently?

13. Picking A Random Prime • “Pick a random n-bit prime.” • Strategy: • Generate random n-bit numbers • Test each one for primality • [more on this later in the lecture]

14. Tremendously Useful Inequality • x,1 + x ≤ ex (note: so for small x, 1 + x ≈ ex) Corollaries x,1 – x ≤ e-x x≠0,1 + 1/x ≤ e1/x x>0,(1 + 1/x)x ≤ e x≠0,1 - 1/x ≤ e-1/x x>0,(1 - 1/x)x ≤ 1/e

15. Picking A Random Prime • “Pick a random n-bit prime.” • Generate kn random n-bit numbers • Each trial has a ≥ 1/2n chance of being prime. • Pr[ all kn trials yield composites ] ≤(1-1/2n)kn = (1-1/2n)2n * k/2≤ 1/ek/2

16. Picking A Random Prime • “Pick a random n-bit prime.” • Strategy: • Generate random n-bit numbers • Test each one for primality For 1000-bit primes, if we try out 10000 random 1000-bit numbers, chance of failing ≤ e-5 ≤ .0068

17. Moral of the story • Picking a random prime is “almost as easy as”picking a random number. • (Provided we can check for primality.)

18. Checking for Primality • Fermat’s Little Theorem: • An integer n > 1 is prime if and only if • an-1 1 (mod n) • for all a such that 1 ≤ a ≤ n-1 “Fake Square Root of 1” Theorem: If a and n are positive integers such that a2 1 (mod n) but a 1 (mod n) then n is composite

19. Miller-Rabin Randomized Primality Test • If n > 2 and n is even, return “composite” • Pick a uniformly at random from {1,2,…,n-1} • If an-1  1 (mod n), return “composite” • Let n-1 = t2s for some s > 0 and odd t • For i = 1, 2, …, s • if a2it 1 (mod n) but a2i-1t 1 (mod n) • return “composite” • Return “passed test”

20. Monte Carlo Algorithm • The Miller-Rabin randomized primality test might return “passed test” even when the number is actually composite!

21. What does the test tell us? • If n is prime, the test says “passed test” • If n is composite, the test says • “composite” with probability at least ¾ • “passed test” with probability at most ¼ • I.e., the answer is incorrect with probability at most ¼ • If n is composite and the test is run k times, the probability that it says “passed test” each time is at most (1/4)k. If the test ever says “composite” the number is composite.

22. Did you get that file ok? Was the transmission accurate? How would I know? Earth has transferred a huge file X to Moon. Moon received Y Earth: X Moon: Y

23. p = random (2 log n)-bit prime Send (p, X mod p) Answer to “X  Y (mod p) ?” Are X and Y the same n-bit numbers? (assume no transmission errors either way) Earth: X Moon: Y

24. Why is this any good? • Easy case: • If X = Y, then X  Y (mod p) and answer to • “X  Y (mod p) ?” is Yes!

25. Why is this any good? • Harder case: • What if X  Y? We want answer to “X  Y (mod p) ?” to be No! • But answer is Yes! if X  Y (mod p), i.e., p | (X-Y) • How likely is this? • Define Z = (X-Y). To mess up, p must divide Z. • Z is an n-bit number  Z is at most 2n. • But each prime is ≥ 2. Hence Z has at most n prime divisors.

26. Almost there… • Z has at most n prime divisors. • How many (2log n)-bit primes are there? • Recall (2k) ≥ 2k /2k • at least 22logn/2*2log n = n2/(4 log n) >> 2n primes. At most half of them divide Z. Hence the probability that a random (2 log n)-bit prime divides Z is at most ½. Make mistake (answer Yes!) with probability at most ½.

27. Theorem: Let X and Y be distinctn-bit numbers. Let p be a random (2 log n)-bit prime. Then Prob [X = Y mod p] < 1/2 Earth-Moon protocol makes mistake with probability at most 1/2!

28. Are X and Y the same n-bit numbers? Pick k random (2 log n)-bit primes: P1, P2, .., Pk Send (X mod Pi) for 1 ≤ i ≤ k k answers to “X = Y mod Pi ?” (assume no transmission errors either way) EARTH: X MOON: Y

29. Exponentially smaller error probability • If X=Y, always accept. • If X  Y, • Prob [X = Y mod Pi for all i] ≤ (1/2)k