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Optimizing Password Composition Policies Jeremiah Blocki Saranga Komanduri Ariel Procaccia Or Sheffet To appear at EC 2013

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Password Composition Policy password Password Composition Policy 2

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How Do Users Respond? Password1 3

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Problem: Predictable Passwords 1. password abc qwerty 6. monkey 7. letmein 8. dragon …. 25. password1 4

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Predictable Responses 1. password abc qwerty 6. monkey 7. letmein 8. dragon …. 25. password1 5

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Previous Work Initial password composition policies designed without empirical data [BDP, 2006]. User’s respond to password composition policies in predictable ways [KSKMBCCE, 2011] Trivial password choices vary widely across contexts [BX, 2012]. No theoretical models of password composition policies. 6

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Our Contributions We initiate an algorithmic study of password composition policies. Theoretical Model Security Goal Policy Structure User Model 7

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Outline User Model Policy Structure Goal Algorithms and Reductions Experiments 8

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Rankings Model User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty Each User: Passwords P ordered by preference. n = 7 (number of users). 9

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Rankings Model: Example 1 User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty Allowed PasswordsAll Passwords 10

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Rankings Model: Example 1 Pr[ | A] = 3/7 Pr[letmein | A] = 2/7 Pr[ | A]=Pr[12345 | A]=1/7 User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty 11

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Rankings Model: Example 2 User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty Allowed PasswordsAll Passwords 12

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Warm-up Fact: Let A’ A then for any w A’ Pr[w|A] ≤ Pr[w|A’] User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty Initially one person uses letmein as their password. letmein 13

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Warm-up Fact: Let A’ A then for any w A’ Pr[w|A] ≤ Pr[w|A’] User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty Every user who used letmein before is still using the same password. 14

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Outline User Model Policy Structure – Positive Rules – Negative Rules – Singleton Rules Goal Algorithms and Reductions Experiments 15

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Positive Rules 16

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Positive Rules - Example Rules R 1,…,R m P R 1 = {w | Length(w) 14}. Active Rules: S {1,…,m}. A {1} = {w | Length(w) 14}. 17

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Negative Rules 18

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Negative Rules - Example Rules R 1,…,R m P R 1 = {w | Length(w) < 8}. Active Rules: S {1,…,m}. A {1} = P - {w | Length(w) < 8}. 19

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Singleton Rules Rule R w = {w} for each w P. Can allow/ban any individual password. Special Case of Positive Rules/Negative Rules. 20

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Outline User Model Policy Structure Goal Algorithms and Reductions Experiments 21

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Online Attack password 22 Guess Limit: k-strikes policy p(k, A) – probability of a successful untargeted attack given A.

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The Wrong Goal 23

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p(k,A) - Example p(1,A) = Pr[111111] = 3/7 p(2,A) = p(1,A) + Pr[letmein] = 5/7 p(3,A) = p(2,A) + Pr[123456]= 6/7 User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword12345password letmein12345password abc baseballPassw0rd letmeinPassw0rd passwordPassw0rdabc123baseballiloveyou ………………… qwerty1 qwerty 24

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Goal: Optimize p(k,A) Goal: Find a password composition policy S {1,…,m} which minimizes p(k,A S ) for some k. p(k, A) – Fraction of accounts an adversary can crack with k guesses per account given policy A. p(1, A): minimum entropy of the password distribution resulting from policy A. 25

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Even Better Goal? Universal Approximation: Find a password composition policy S {1,…,m} such that p(k,A S ) ≤ c p(k,A S’ ) for some constant c and every k, S’ {1,…,m}. Thm: Universal approximation is unachievable in general. 26

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Outline User Model Policy Structure Goal Algorithms and Reductions Experiments 27

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Results Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard This Talk: k=1 n 1/3 -approx is NP-Hard Parameters: n, m, |P| 28

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Negative Rules are Hard! Theorem: Unless P = NP no polynomial time algorithm can even approximate p(1,A S ) to a factor of n 1/3- in the negative rules setting. 29

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Reduction Maximum Independent Set: g vertices e edges Theorem [Hastad 1996]: NP-Hard to distinguish the following two cases (1) any independent set has size at most K = g or (2) the maximum independent set has size g 1- . 30

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Reduction (Preference Lists) Preference Lists: Type 1 W1W1 …W1W1 W2W2 …W2W2 ……… WKWK …WKWK B1B1 …BgBg ……… Observation: Unless we ban W 1,…,W K we have p(1,A S ) ≥ g/n 31

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Reduction (Preference Lists) Preference Lists: Type 2 (for each edge e = {u,v}) (u,v,1)…(u,v,g) (v,u,1)…(v,u,g) X…X ……… Observation: If for any edge e = {u,v} we ban (u,v,1),…,(u,v,g) and (v,u,1),…,(v,u,g) then p(1,A S ) ≥ g/n. 32

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Reduction (Preference Lists) Preference Lists: Type 3 (for each vertex v, i j [K]) (v,i,j,1)…(v,i,j,g) (v,j,i,1)…(v,j,i,g) X…X ……… Observation: If we ban (v,i,j,1),…,(v,i,j,g) and (v,j,i,1),…,(v,j,i,g) then p(1,A S ) ≥ g/n. 33

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Reduction (Rules) R u,1 R v,2 R w,3 R x,4 K=4 Preference Lists: Type 1 W1W1 …W1W1 W2W2 …W2W2 ……… WKWK …WKWK B1B1 …BgBg ……… s t 34

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Reduction (Rules) R u,1 R v,2 R w,3 R x,4 K=4 Preference Lists: Type 2 (edge e = {u,x}) (u,x,1)…(u,x,g) (x,u,1)…(x,u,g) X…X ……… s t p(1,A S ) ≥ g/n 35

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Reduction (Rules) R u,1 R v,2 R w,3 R x,4 K=4 Preference Lists: Type 2 (edge e = {u,s}) (u,s,1)…(u,s,g) (s,u,1)…(s,u,g) X…X ……… s t 36

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Reduction (Rules) R u,1 R v,2 R w,3 R x,4 K=4 Preference Lists: Type 2 (edge e = {s,t}) (s,t,1)…(s,t,g) (t,s,1)…(t,s,g) X…X ……… s t 37

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Reduction (Rules) R u,1 R v,2 R w,3 R w,4 K=5 Preference Lists: Type 1 W1W1 …W1W1 W2W2 …W2W2 ……… WKWK …WKWK B1B1 …BgBg ……… s t p(1,A S ) ≥ g/n 38

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Reduction (Rules) R u,1 R v,2 R w,3 R w,4 K=5 Preference Lists: Type 1 W1W1 …W1W1 W2W2 …W2W2 ……… WKWK …WKWK B1B1 …BgBg ……… s t R v,5 39

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Reduction (Rules) R u,1 R v,2 R w,3 R w,4 K=5 s t R v,5 Preference Lists: Type 3 (for each vertex u, i j [K]) (v,2,5,1)…(v,2,5,g) (v,5,2,1)…(v,5,2,g) X…X ……… p(1,A S ) ≥ g/n 40

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Reduction (Rules) R u,1 R v,2 R w,3 R w,4 K=4 s t Preference Lists: Type 3 (w, i=4, j=2) (w,4,2,1)…(w,4,2,g) (w,2,4,1)…(w,2,4,g) X…X ……… 41

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Reduction (Rules) ObservationConclusion Unless we ban W 1,…,W K we have p(1,A S ) ≥ g/n. For each i, we must ban R u,i for some vertex u. If for any edge e = {u,v} we ban (u,v,1),…, (u,v,g) and (v,u,1),…,(v,u,g) then p(1,A S ) ≥ g/n. For any edge e = {u,v}, and i,j [K] we cannot ban both R u,i and R v,j. If we ban (v,i,j,1),…,(v,i,j,g) and (v,j,i,1),…,(v,j,i,g) then p(1,A S ) ≥ g/n. For i j we cannot ban both R u,i and R u,j. Impossible unless there is an independent set of size K! 42

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Reduction Independent Set of Size K?max S [m] p(1,A S ) Yes1/n Nog/n where n = O(g 3 ) 43

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Results Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard This Talk: k=1 P Parameters: n, m, |P| 44

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Key Difference: Positive vs. Negative Let S w = {i | w R i } (all rules R i that contain w). Negative Rules: Ban w - activate any rule in S w. Positive Rules: Ban w - deactivate all rules in S w. 45

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Positive Rules Fact: Let S* {1,…m} denote the optimal solution, and let S S* then either (1) p(1,A S ) = p(1,A S* ), or (S is optimal) (2) S-S w S*, where Pr[w|A S ] = p(1,A S ). All rules R i that contain the most popular word in A S. 46

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Positive Rules Fact: Let S* {1,…m} denote the optimal solution, and let S S* then either (1) p(1,A S ) = p(1,A S* ), or (S is optimal) (2) S-S w S*, where Pr[w|A S ] = p(1,A S ). Proof: Suppose for contradiction that w A S*, and observe that. Therefore,. Contradiction! 47

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Positive Rules Algorithm Iterative Elimination: Initialize: S 0 = {1,…,m} Repeat: (Ban w - current most popular password) S i+1 = S i – S w Claim: One of the S i ’s must be the optimal solution! 48

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Results Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard This Talk: k=1 Question: What if we don’t have access to the full preference lists of each user? What if we don’t want to run in time n? Parameters: n, m 49

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Results Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard This Talk: k=1 Sampling Algorithm: ε-approximation with probability 1-δ Parameters: m, 1/ε, 1/δ 50

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Sampling Algorithm Sample: q(A) returns w with probability P[w|A]. Idea: Run iterative elimination. In each round use sampling to estimate the probability of the most popular word. 51

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Sampling Lemma Lemma: Let s=100 log (m/ )/ 2 denote the number of samples in each round, and let BAD i denote the event that in iteration i, there exists a password w s.t. (e.g., our probability estimate off by /2). Then Pr[ i.BAD i ]≤ # times w sampled 52

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Sampling Lemma Partition P into buckets. … … B0B0 B1B1 BiBi w Contains at mot 2 i+1 / such passwords. 53

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Sampling Lemma Partition P into buckets. s=100 log (m/ )/ 2 … … B0B0 B1B1 BiBi Chernoff Bounds: Contains at most 2 i+1 / such passwords. w 54

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Sampling Lemma Partition P into buckets. s=100 log (m/ )/ 2 … … B0B0 B1B1 BiBi Contains at most 2 i+1 / passwords. Union Bound: 55

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Sampling Lemma Partition P into buckets. s=100 log (m/ )/ 2 … … B0B0 B1B1 BiBi Union Bound (buckets): 56

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Sampling Lemma Partition P into buckets. s=100 log (m/ )/ 2 … … B0B0 B1B1 BiBi Union Bound (rounds): 57

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Sampling Algorithm 58

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Results Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard This Talk: k=1 59

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Rankings Model - Large k Theorem: It is NP-Hard to optimize p(k,A) in the rankings model when k is a parameter. Theorem: In the normalized probabilities model there is an efficient algorithm to optimize p(k,A) for any k. 60

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Reduction Vertex Cover Edge {u,v} UV {u,v} …… g vertices e edges Question: Is there a vertex cover of size t? k = g+e-t-1 n=2e preference lists e+g passwords Uncover {u,v} by banning u or v u v x 61

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Reduction Vertex Cover n vertices m edges Question: Is there a vertex cover C of size t? k = m+n-t-1 u v Edge e = {u,v} UV ee …… p(k,A) < 1 p(k+1,A) = 1 x Set A = P - C 62

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Outline User Model Policy Structure Goal Algorithms and Reductions Experiments – RockYou Dataset – Rules – Results 63

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RockYou Dataset 64

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0.5 1 Normalized Probabilities RockYou: initial distribution over P. 0.5 letmein (0.1) P A 1 letmein (0.2) A 65

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Normalized Probabilities Rankings Model Constant kLarge k Singleton RulesPNP-Hard APX-Hard (UGC) Positive RulesPNP-Hard Negative Rulesn 1/3 -approx is NP-HardNP-Hard Normalized Probabilities Model Constant kLarge k Singleton RulesPP Positive RulesPNP-Hard Negative RulesNP-Hard 66

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Rankings vs Normalized Probabilities Let x,y A 1 A 2.Suppose that Pr[x|A 1 ]>Pr[y|A 1 ]. Is Pr[x|A 2 ]>Pr[y|A 2 ]? Rankings Model (x = password, y = ): User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword123456passwordletmein ………………… qwerty1 qwerty 67

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Rankings vs Normalized Probabilities Let x,y A 1 A 2.Suppose that Pr[x|A 1 ]>Pr[y|A 1 ]. Is Pr[x|A 2 ]>Pr[y|A 2 ]? Rankings Model (x = password, y = ): No! User 1User 2User 3User 4User 5User 6User 7 password123456letmeinpassword123456passwordletmein ………………… qwerty1 qwerty 68

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Rankings vs Normalized Probabilities Let x,y A 1 A 2.Suppose that Pr[x|A 1 ]>Pr[y|A 1 ]. Is Pr[x|A 2 ]>Pr[y|A 2 ]? Normalized Probabilities: Yes! 69

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Base Line Results 71

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Results 72

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Discussion Optimal solution was better under negative rules. However, sampled solutions were much better with positive rules. Interesting Directions: – Additional Rules? – Is the Normalized Probabilities Model reasonable? – General experiment in preference list model? 73

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Open Questions Efficient approximation algorithm in negative rules setting with normalized probabilities assumption? Adversary with limited background knowledge about the user (e.g., age, gender, birthday). 74

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