Randomized Algorithms - Cryptography - Lecture Slides, Slides of Cryptography and System Security

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Probability and

Randomized Algorithms

2

Discrete Random

Variable

DEF: A discrete random variable is a set X together with an assignment of a non- negative probability Pr[ X = x ] that X takes value x ; furthermore, the sum over all possible x ε X of the probability that X takes value x must equal 1.

If X is clearly fixed from context, may

abbreviate Pr[ X = x ] to Pr[ x ] or px.

4

Independent Variables

Random variables are independent if their probabilities don’t depend on each others values: DEF: X and Y are independent if Pr[ x,y ] = Pr[ x ]Pr[ y ] for all x, y. LEMMA: Equivalently, X and Y are independent if (excluding 0-prob. y )

∀ x ∈ X , ∀ y ∈ Y , Pr[ x | y ] = Pr[ x ]

5

Baye’s Theorem

THM: If Pr[ y ] > 0 then Pr[ x|y ] = Pr[ y|x ] ⋅ Pr[ x ] / Pr[ y ]

7

Expectation

The average value taken on by a function f on probability distribution X DEF: The expectation of f is defined by: THM: COR: For n repetitions of a Binomial random variable X consider sum S which counts the number outcomes = 1. Then E ( S ) = np

E ( f ) =

x ∈ X

f ( x ) · px

E ( f + g ) = E ( f ) + E ( g )

8

Chernoff Bound

Estimates probability that sum of Binomial experiment deviate from expected sum np THM: Note: probability that sum too big falls off exponentially with n

Pr

[

S ≥ ( 1 + !) pn

]

≤ e

− ! 2 3 pn

10

Monte Carlo Algorithm

False negative allowed, but no false positives DEF: A poly-time Monte Carlo algorithm for the decision problem P is a poly-time non- deterministic Turing machine (NDTM) s.t.

Probability measured over “coin-flips” in TM or equivalently, by taking the ratio of accepting branches in NTM to total number

Defines complexity class RP “Rand-Poly” Pr[ x is accepted] :

1 2 x ∈ P = 0 x #∈ P

11

Las Vegas Algorithm

Symmetric version of Monte Carlo - no false negatives nor false positives but can “fail” DEF: A poly-time Las Vegas algorithm is a poly-time NDTM with a constant ε>0 for which Pr[fail] ≤ ε for all inputs.

Repeat algorithm to make ε arbitrarily small

Gives class ZPP “Zero-Prob-of-error-Poly”

ZPP = RP ∩ co-RP

13

Pseudo Random

Sequence

“DEF”: A pseudo random sequence is a deterministic algorithm from finite bitstrings to infinite bitstrings whose outputs cannot be distinguished from a random strings by any BPP algorithm.

14

ε-bias Detector

Given: A black box f which is known a-

priori to have some built-in bias ε in an

unknown direction.

Decide: Which direction the bias is in. n = x = output of length n from f c = number of 1’s in x return (c > n/2) // “YES” if 1 -bias, “NO” if 0-bias

Pr[output is correct] > 3/4 therefore

this problem is in BPP so ε-bias

sequences are not pseudorandom. 2 ( 1 2 − !)! 2