Computational hardness. Feb 2 abhi shelat

Transcription

1 L Computational hardness Feb 2 abhi shelat

2 Eve Alice Bob

3 Eve Alice Bob k Gen k

4 Eve Alice Bob c=enck(mi) k Gen k

5 Eve c Alice Bob c=enck(mi) k Gen k

6 Eve c Alice c=enck(mi) Bob m=deck(c) k Gen k

7 Eve c Alice c=enck(mi) Bob m=deck(c) k Gen k

8 c Eve Test if guess m1 if not. flip coin otherwise. analyze success:

9 c Eve Test if guess m1 if not. flip coin otherwise. analyze success:

10 c Eve Test if guess m1 if not. flip coin otherwise. analyze success:

11 c Eve Test if guess m1 if not. flip coin otherwise. analyze time:

12 c Eve Test if guess m1 if not. flip coin otherwise. analyze time: recall

13 c Eve Test if guess m1 if not. flip coin otherwise. analyze time: recall generic method takes time

14 EFFICIENT COMPUTATION AND EFFICIENT ADVERSARIES

15 WHAT IS AN ALGORITHM?

16 WHAT IS AN ALGORITHM? Turing machine

17 WHAT IS AN ALGORITHM? input tape Turing machine output tape work tape

18 WHAT IS AN ALGORITHM? input tape Turing machine output tape work tape running time is T(n) if machine always halts after at most T(n) steps on input of size n.

19 WHAT IS AN ALGORITHM? input tape Turing machine output tape work tape

20 WHAT IS AN ALGORITHM? input tape Turing machine output tape computes function f(x) if... work tape

21 RANDOMIZED ALGORITHM... input tape Turing machine output tape work tape

22 RANDOMIZED ALGORITHM... input tape Turing machine output tape work tape computes function f(x) if machine always halts with output f(x) on input of x.

23 RANDOMIZED ALGORITHM... input tape Turing machine output tape work tape p.p.t. algorithm probabilistic polynomial time

24 EXAMPLES

25 EXAMPLES Problems solvable by a p.p.t. machine:

26 EXAMPLES Problems solvable by a p.p.t. machine: Problems not known to be solvable by a p.p.t. machine:

27 HOW WE MODEL ADVERSARIES Turing machine 1 machine for all inputs

28 SORTING COMPETITION Quicksort machine

29 HOW WE MODEL ADVERSARIES Turing machine 1 machine for all inputs

30 HOW WE MODEL ADVERSARIES Turing machine Turing machine 1 machine for all inputs

31 HOW WE MODEL ADVERSARIES Turing machine Turing machine 1 machine for all inputs as inputs grow machine grows

32 HOW WE MODEL ADVERSARIES Turing machine Turing machine 1 machine for all inputs as inputs grow machine grows

33 HOW WE MODEL ADVERSARIES Turing machine Turing machine 1 machine for all inputs as inputs grow machine grows

34 NON-UNIFORM P.P.T.... input tape Turing machine output tape work tape for each input len n, adversary has a special Turing machine M n of size poly(n)

35 ALGORITHMS ADVERSARIES Turing machine Turing machine p.p.t. non-uniform p.p.t.

36 KEY PROPERTY OF ENC easy hard

37 WORST-CASE ONE WAY FUNC

38 CRITIQUE OF WORST-CASE

39 NEGLIGIBLE FUNCTION

40 NEGLIGIBLE FUNCTION Definition A function ε(n) is negligible if for every c, there exists some k 0 such that for all k 0 < k, (k) 1 k. Intuitively, a negligible function is c asymptotically smaller than the inverse of any fixed polynomial

41 NEGLIGIBLE FUNCTION Definition A function ε(n) is negligible if for every c, there exists some k 0 such that for all k 0 < k, (k) 1 k. Intuitively, a negligible function is c asymptotically smaller than the inverse of any fixed polynomial

42 NEGLIGIBLE FUNCTION Definition A function ε(n) is negligible if for every c, there exists some k 0 such that for all k 0 < k, (k) 1 k. Intuitively, a negligible function is c asymptotically smaller than the inverse of any fixed polynomial

43 NON-NEGLIGIBLE? Definition A function ε(n) is negligible if for every c, there exists some k 0 such that for all k 0 < k, (k) 1 k. Intuitively, a negligible function is c asymptotically smaller than the inverse of any fixed polynomial.

44 STRONG O.W.F.

45 STRONG O.W.F.

46 STRONG O.W.F.

47 STRONG O.W.F.

48 WEAK O.W.F.

49 DIAGRAM

50 WEAK OWF -> STRONG OWF

51 EXAMPLE MULT

52 Π n =

53 FACTORING ASSUMPTION

54 IS FMULT WEAK OWF?

55 NUMBER OF PRIMES there are certainly infinitely many # of primes < x

56 THM: