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Topics

• Section 6.1 – 6.6

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Original author of the slides:Vadim Bulitko

University of Alberta

http://www.cs.ualberta.ca/~bulitko/W04

Modified by T. Andrew Yang (yang@uhcl.edu)

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Counting• A random process

The set of outcomes is known, but the specific outcome is not predictable.

– Outcomes

– Sample spaceThe set of all possible outcomes of a random process (or

experiment).

– EventsAn event is a subset of a sample space.

• Examples– coins– dice

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Counting and Probability• The chance that a given event will occur• The ratio of the number of outcomes in an exhaustive set of

equally likely outcomes that produce a given event to the total number of possible outcomes

• Outcomes are assumed to be equally likely.• E: an event.• S: the sample space.• N(E): the number of outcomes in E• N(S): the total number of outcomes in S.• P(E): the probability that E will occur.

• Then P(E) =

• Examples:– Coins– Dice– Tournament

)(

)(

SN

EN

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Permutations, combinations, etc.

Attributes Ordered Unordered

Reps

No Reps

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Multiplication Rule

• p.208: Theorem 6.2.1k steps (s1, s2, …, sk) of an operation

n1 ways in s1

n2 ways in s2

…

Then, the entire operation can be performed in n1n2…nk ways.

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Multiplication Rule (cont.)

• Example: – How many different PINs are possible? p.308

(repetition is allowed)

– How if repetition is not allowed? p.309 permutations (p.313)

– Password-based authentication

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Passwords-based Authentication

• A dictionary attack is the guessing of a password by repeated trial and error.

• The dictionary may be a set of strings in random order, or a set of strings in decreasing order of probability of selection.

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Passwords-based Authentication

• Countering dictionary attack

– The goal: To maximize the time needed to guess the password

– Anderson’s Formula:

P: The probability that an attacker guesses a password in

a specified period of time

G: The number of guesses that can be tested in one time unit

T: The number of time units during which guessing occurs

N: The number of possible passwords

N

TG P

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Passwords-based Authentication

• An example:– Let S be the length of the password.

– Let A be the number of characters in the alphabet from which the characters of the password are drawn.

Then N = AS.

– Let E be the number of characters exchanged when logging in.

– Let R be the number of bytes per minute that can be sent over a communication link.

– Let G be the number of guesses per minute. Then G = R / E.

– If the attack extends over M months, T = 30 x 24 x 60 x M.

– Let P be the probability that the attack would succeed.

Then N

TG P

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Passwords-based Authentication

• Analysis of the Anderson Formula: – The goal is to maximize the time (T) needed for the attacker to

guess the password.

– That is, to decrease the chance that the attack may succeed (P).

• Approaches:– To increase N, the set of possible passwords

– To decrease the time allowed to guess the passwords, that is, to reduce T

– To decrease G

N

TG P

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Possibility Trees

• Used to count the number of outcomes

• Can be used to illustrate the multiplication rule (e.g., toss a coin for three times)

• Useful when the multiplication rule is difficult or impossible to apply

• Examples– Possibilities for tournament play: p.306– Election of officers: p.311

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Permutations

• A permutation of a set of objects is an ordering of the objects in a row.

• Example:S = {a, b}Permutations: ab, baOrder matters! Distinct objects!

• FormulaGiven n objects (n >= 1), the number of permutations is

n!

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r-permutations

• An ordered selection of r elements out of n elements• Still, order matters and no repetition

P(n,r) = n(n-1)(n-2)…(n-r+1)

=

• Exercises: P(5,3), P(7,3), P(3,3)

• Example 6.2.11 (p.317)

• Q16 on p.319

)!(

!

rn

n

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Set Operations & Counting

• The addition rule (p.321)

Example 6.3.1: number of passwords

Note: distinct, mutually disjoint sets

• To make sets disjoint:intersection, symmetric difference

• Inclusion/exclusion, difference rules– Example 6.3.6

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Combinations

• Order is not important (i.e., sets)

• c.f., Order is important in permutations

• So, the different combinations can be considered as subsets of a given set

• Example 6.4.2 (p.335)S = {0,1,2,3}

Q: How many unordered selections of two elements can be made from S?

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r-combinations• An r-combination of a set of n elements is a

subset of r of the n elements

• n choose r: the number of r-combinations that can be chosen from a set of n elementsNote: Order is not important, No repetition of elements

• p.364: computing binomial coefficients

• Formula

• Example 6.4.10: p.344

)!(!

!

rnr

n

r

n

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r-permutations vs r-combinations

• Share: no repetitions, distinct elements• Difference:

– Permutations: unordered– Combinations: ordered

• Figure 6.4.1: p.336

• !*),( rr

nrnP

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Be aware of double-counting!• A false solution: p.346

• Another example: M={a,b}, F={c,d,e}. Form 2-person teams, but one of them must be a woman.

• Questions to ask:– Am I counting everything?– Am I counting anything twice?

• Multiplication rule– Am I looking at everything at the possibility tree?– Does every outcome appear on a branch of tree?

• Addition rule:– Does every outcome appear in some subset of the diagram?– Are the subsets disjoint?

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r-permutations with repetition

• r-permutations without repetitions: order mattersP(n,r)=n(n-1)(n-2)…(n-r+1)

• What if we allow to put elements back?• How many ways can we choose r

elements from n types of elements?– Order matters– Repetitions are allowed

• Formula?

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Permutations of a set with repeated elements

• Theorem 6.4.2: p.345

• Example 6.4.11

• Which to use? c.f.: nk

• Q1: How many different bit strings can 4 bits hold?

• Q2: What are the total number of transpositions for the 4-bit bit string 0110b? That is, how many 4-bit bit strings contain exactly 2 1’s? 0110, 0101, 1010, 1001

See example 6.4.10 (p.344)

• Exercise: Try the same with 5 bits

!!...!!

!

......

321

121

3

21

2

1

1

k

k

k

nnnn

n

n

nnnn

n

nnn

n

nn

n

n

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Special case

•When k = 2, permutations with repeated elements is reduced to r-combinations. True? False?

!!...!!

!

......

321

121

3

21

2

1

1

k

k

k

nnnn

n

n

nnnn

n

nnn

n

nn

n

n

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r-combinations with repetition

• What if we allow repetitions?• Choose r elements out of n but allow

repetitions (e.g., put the elements back after drawing them)

• Order is not important• The underlying construct is multiset• Theorem 6.5.1: p.351

• Examples

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Summary

Attributes Ordered Unordered

Reps

No Repsr-permutations r-combinations

• Question: How about

!!...!!

!

......

321

121

3

21

2

1

1

k

k

k

nnnn

n

n

nnnn

n

nnn

n

nn

n

n

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Questions?