Randomness and Computation: Some Prime ExamplesChecking Our WorkGreat Idea: Evaluating on Random InputsEquality checking by random evaluationSlide 5Slide 6Slide 7Slide 8Slide 9What does “evaluate” mean?Slide 11Slide 12Slide 13“Random Fingerprinting”Earth has huge file X that she transferred to Moon. Moon gets Y.Slide 16Their estimatesSlide 18The Prime Density TheoremSlide 20Slide 21Slide 22Slide 23Really useful factPicking A Random PrimeSlide 26Slide 27Slide 28Moral of the storySlide 30Are X and Y the same n-bit numbers?Why is this any good?Slide 33Almost there…Slide 35Slide 36Exponentially smaller error probabilitySlide 38Primality Testing: Trial Division On Input nSlide 40Do the primes have a fast decision algorithm?Euclid gave us a fast GCD algorithm. Surely, he tried to give a faster primality test than trial division. But Euclid, Euler, and Gauss all failed!Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48ProofSlide 50Randomized Primality TestIs ECn non-empty for all primes n?The saving graceSlide 54Randomized AlgorithmsSlide 56Another Randomized Primality TestSlide 58Many Randomized TestsSlide 60Slide 61Slide 62Slide 63Slide 64Great Theoretical Ideas In Computer ScienceAnupam GuptaCS 15-251 Fall 2006Lecture 17 Oct 24, 2006 Carnegie Mellon UniversityRandomness and Computation: Some Prime ExamplesChecking Our WorkSuppose we want to check p(x) q(x) = r(x), where p, q and r are three polynomials. (x-1)(x3+x2+x+1) = x4-1If the polynomials are long, this requires n2 mults by elementary school algorithms -- or can do faster with fancy techniques like the Fast Fourier transform.Can we check if p(x) q(x) = r(x) more efficiently?Great Idea: Evaluating on Random InputsLet f(x) = p(x) q(x) – r(x). Is f zero?Idea: Evaluate f on a random input z.If we get f(z) = 0, this is evidence that f is zero everywhere.If f(x) is a degree 2n polynomial, it can only have 2n roots. We’re unlikely to guess one of these by chance!Equality checking by random evaluation1. Fix a sample space S={z1, z2,…, zm} with arbitrary points zi, for m=2n/ .2. Select random z from S with probability 1/m.3. Evaluate f(z) = p(z) q(z) – r(z)4. If f(z) = 0, output “equal” otherwise output “not equal”Equality checking by random evaluationWhat is the probability the algorithm outputs “not equal” when in fact f 0?Zero!If p(x)q(x) = r(x) , always correct!Equality checking by random evaluationWhat is the probability the algorithm outputs “equal” when in fact f 0?Let A = {z | z is a root of f}. Recall that |A| degree of f ≤ 2n.Therefore: P(A) 2n/m = . We can choose to be small.Equality checking by random evaluationBy repeating this procedure k times, we are “fooled” by the event f(z1) = f(z2) = … = f(zk) = 0 when actually f(x) 0with probability no bigger than P(A) (2n/m)k = kWow! That idea could be used for testing equality of lots of different types of “functions”!Yes! E.g., a matrix is just a Yes! E.g., a matrix is just a special kind of function. special kind of function. Suppose we do a matrix Suppose we do a matrix multiplication of two multiplication of two nxn matrices:matrices:AB = CThe idea of random The idea of random evaluation can be used to evaluation can be used to efficiently check the efficiently check the calculation.calculation.What does “evaluate” mean? Just evaluate the “function” C on a Just evaluate the “function” C on a random bit random bit vectorvector rr by taking the matrix-vector product by taking the matrix-vector product C C ×× r r1 0 3 -4 87 0 0 2 913 5 -6 0 -71 6 21 9 010110=09731So to test if AB = C we compute x = Br, y = Ax (= Abr), and z = CrIf y = z, we take this as evidence that the calculation was correct. The amount of work is only O(n2).Claim: If AB C and r is a random n-bit vector, then Pr(ABr = Cr) ½.Claim: If AB C and r is a random n-bit vector, then Pr(ABr = Cr) ½.So, if a complicated, fancy algorithm is used to compute AB in time O(n), it can be efficiently checked with only O(n2) extra work, using randomness!“Random Fingerprinting”Find a small random “fingerprint” of a large object.- the value f(z) of a polynomial at a point z- the value Cr at a random bit vector rThis fingerprint captures the essential information about the larger object: if two large objects are different, their fingerprints usually are different!Earth has huge file X that she transferred to Moon. Moon gets Y.Earth: XEarth: X Moon: YDid you get that file ok? Was Did you get that file ok? Was the transmission accurate?the transmission accurate?Uh, yeah.GaussLet (n) be the number of primes between 1 and n. I wonder how fast (n) grows? Conjecture [1790s]: ( )lim 1/ lnnnn np��=LegendreTheir estimatesx pi(x) Gauss' Li Legendre x/(log x - 1)1000 168 178 172 16910000 1229 1246 1231 1218100000 9592 9630 9588 95121000000 78498 78628 78534 7803010000000 664579 664918 665138 661459100000000 5761455 5762209 5769341 57403041000000000 50847534 50849235 50917519 5070154210000000000455052511455055614 455743004 454011971J-S HadamardTwo independent proofs of the Prime Density Theorem [1896]:( )lim 1/ lnnnn np��=De la Vallée PoussinThe Prime Density TheoremThis 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!RiemannThe Riemann Hypothesis [1859] ( ) / lnlim 0nn n nnp��-=Slightly easier to show (n)/n ≥ 1/(2 logn).Random logn bit number is a random number from 1..n(n) / n ≥ 1/2lognmeans that a random logn-bit number has at least a 1/2logn chance of being prime.Random k bit number is a random number from 1..2k(2k) / 2k ≥ 1/2kmeans that a random k-bit number has at least a 1/2k chance of being prime.Really useful factA random k-bit number has at least a 1/2k chance of being prime.So if we pick So if we pick 2k random k-bit 2k random k-bit numbersnumbers the expected number of primes on the expected number of primes on the list is the list is at least 1at least 1Picking A Random PrimeMany modern cryptosystems (e.g., RSA) include the instructions:“Pick a random n-bit prime.”How can this be done efficiently?Picking A Random Prime“Pick a random n-bit prime.”Strategy:1) Generate random n-bit numbers2) Test each one for primality [more on this later in the lecture]Picking A Random
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