Topics

1. Topics • Section 6.1 – 6.6

2. 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)

3. Counting • A random process The set of outcomes is known, but the specific outcome is not predictable. • Outcomes • Sample space The set of all possible outcomes of a random process (or experiment). • Events An event is a subset of a sample space. • Examples • coins • dice

4. 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

5. Permutations, combinations, etc.

6. Multiplication Rule • p.208: Theorem 6.2.1 k 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.

7. 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

8. 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.

9. 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

10. 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

11. 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

12. 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

13. Permutations • A permutation of a set of objects is an ordering of the objects in a row. • Example: S = {a, b} Permutations: ab, ba Order matters! Distinct objects! • Formula Given n objects (n >= 1), the number of permutations is n!

14. 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

15. 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

16. 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?

17. 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 elements Note: Order is not important, No repetition of elements • p.364: computing binomial coefficients • Formula • Example 6.4.10: p.344

18. r-permutations vs r-combinations • Share: no repetitions, distinct elements • Difference: • Permutations: unordered • Combinations: ordered • Figure 6.4.1: p.336

19. 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?

20. r-permutations with repetition • r-permutations without repetitions: order matters P(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?

21. 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

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

23. 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

24. Summary • Question: How about

25. Questions?