Computability and Complexity Random Sources. Computability and Complexity Andrei Bulatov

Transcription

1 Coputabilit and Copleit 29- Rando Sources Coputabilit and Copleit Andrei Bulatov

2 Coputabilit and Copleit 29-2 Rando Choices We have seen several probabilistic algoriths, that is algoriths that ake soe rando choices during the coputation We have proved that those algoriths solve the corresponding probles successfull onl with soe probabilit A coon assuption for those theores is that the algoriths alwas ake a trul rando choice

3 Coputabilit and Copleit 29-3 Perfect Randoness A rando source is a device, which after pushing a button produces a potentiall infinite sequence of bits,, 2 A rando source is said to be perfect if it is fair, that is Pr [ 0] i it is independent, that is the value i of does not depend on the values of,, 2, i 2

4 Coputabilit and Copleit 29-4 Fair Sources An independent but unfair rando source can be easil converted into a perfect source Let,, be an independent source, such that 2 Pr[ i 0] p Let,, be defined such that 2 i 0 undef. if if 2i 2i 2i 2i otherwise 0 0 Let z, z, be the sequence of defined ebers of 2 Theore von Neuann z, z2, is a perfect rando source

5 Coputabilit and Copleit 29-5 Pseudorando Sources Given the difficulties in ipleenting rando sources phsicall, one a tr to find randoness in atheatical processes A pseudorando nuber generator is an algorith that given a seed, that is a short sequence of bits, produces a ver long sequence of bits that are ver hard to predict. A easure of goodness of a rando source is the copleit of the following proble Pseudorandoness Instance: A pseudorando source P Question: Using an initial segent of the output of P predict the reaining ebers of the sequence with high probabilit without knowing the seed?

6 Coputabilit and Copleit 29-6 Linear Sources Linear sources are the ost usual tpe of pseudorando sources used in the eisting software Take a large nuber, and two saller nubers a and b A seed is a nuber < 0 Then we define a bod i + i + Treating the nubers as sequences of bits we get a required pseudorando sequence Theore A linear pseudorando source can be broken in polnoial tie

7 Coputabilit and Copleit 29-7 Proof First, we find A and B such that A Bod Second, we find i + i + Define to be i. Note that i i + i ai od od i 0 i + i + +

8 Coputabilit and Copleit 29-8 Clai. If A od then, for B A we have k+ k M 0 + A + Bod M Take soe. Then A B A + A A 0 + A i i i i A i i i+ + 0od M

9 Coputabilit and Copleit 29-9 Finding A Given a linear pseudorando source,, copute,, 2 if 0 then A : 0 otherwise do 2 - find the least t and the corresponding d such that GCD d, 2,, t and divides t+ - find u for i t such that u + u + + u d i - set A u 2 + u u t t+ d set B A0 d 2 2 t t

10 Coputabilit and Copleit 29-0 Clai. The algorith coputes an A and a B such that Ak Bod k + + We show that A od k + k Fact. k If ab acod k then b cod GCD a, k k Let k', a a' GCD a, k GCD a, k Indeed, ab ac + k for soe. Therefore a ' b a' c + k' that iplies a' b a' cod k' If l is such that l a' od k' then b l a' b l a' c cod k'

11 Coputabilit and Copleit 29- Let g GCD,d. Notice that A u + u ut t u 2 + u ut t+ od da daod a Aod g Since g divides and, it also divides GCD, for Therefore a Aod GCD, Fact. If 0 is a solution of r sod then, for an k, is also a solution + k 0 GCD r, Indeed, r + k GCD r, k kr s + GCD r, k s + kr' sod k 0 where r r' GCD r, k

12 Coputabilit and Copleit 29-2 Since a is a solution of od, A is also a solution, and A + od + t log2 Clai. When calculating t, if GCD, 2,, g and g does not g divide + then GCD, 2,,, + 2 Since < we have t log2

13 Coputabilit and Copleit 29-3 Finding In general, cannot be found in polnoial tie. For eaple, if a and b, then, < 0, Given a linear pseudorando source copute A and B set : and predict that + A + B when the first incorrect prediction is ade, + A but A od Make the new guess for equal to + A + M continue predicting + A + Bod M whenever an incorrect guess is ade for i+, update M to GCD M, A i i+