Cryptography and Complexity - Hackthology [PDF]

Nov 11, 2013 - Ideally in cryptography one would like to ensure an infeasible computation's cost is greater than the rew

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Idea Transcript


" m Î M " c Î C : Pr[M = m | C = c] = Pr[M = m]

ST M n

T :N Õ N M

T(n) T

1 n n2

2n

n L Í {0, 1} *

O(n k)

−{*}

* L

0

L p(n)



n

k

0

* 0 * 0

M p(n)

M

x Î 0* y Î 0* st. | y | ≤ p( | x | )

xÎL

M

xy y Î 0* M

xÏL

xy

| 0| p ( | x | )

L = {x Î 0* | x is an encoding of an instance of }

L

P = {L Í 0* | there is a Deterministic Turing Machine, M, and a polynomial, p(n), such that T M ≤ p(n) for all n ≥ 1}

P Í NP NP Í P

1

2

1

2

2

1 1

2

2

2

h h

| 0* | | x | | 0* | | x |

2 160

160

t M (x)

T M (n) TM (n) = max{t | there exists a x Î n0 such that Pr[t M (x) = t] > 0}

t M (x) p( ⋅ ) T M (n) ≤ p(n)

L

M 2

xÎL

[M accepts x] ≥

xÏL

[M does not accept x] ≥

3 2 3

P Í BPP NP

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