Aaron Johnson with Joan Feigenbaum Paul Syverson A Model of Onion Routing with Provable Anonymity Financial Cryptography ’07 2/12/07 Aaron Johnson with Joan Feigenbaum Paul Syverson

Overview Formally model onion routing using input/output automata Characterize the situations that provide anonymity 1

Anonymous Communication Mix Networks (1981) Dining cryptographers (1988) Onion routing (1999) Anonymous buses (2002) 2

Anonymous Communication Mix Networks (1981) Dining cryptographers (1988) Onion routing (1999) Anonymous buses (2002) 2

Onion Routing Practical design with low latency and overhead Open source implementation (http://tor.eff.org) Over 800 volunteer routers Estimated 200,000 users 3

Anonymous Communication Deployed Analyzed Mix Networks Dining cryptographers Onion routing Anonymous buses 4

Related work A Formal Treatment of Onion Routing Jan Camenisch and Anna Lysyanskaya CRYPTO 2005 A formalization of anonymity and onion routing S. Mauw, J. Verschuren, and E.P. de Vink ESORICS 2004 I/O Automaton Models and Proofs for Shared-Key Communication Systems Nancy Lynch CSFW 1999 5

Overview Formally model onion routing using input/output automata Characterize the situations that provide anonymity 6

Overview Formally model onion routing using input/output automata Simplified onion-routing protocol Non-cryptographic analysis Characterize the situations that provide anonymity 6

Overview Formally model onion routing using input/output automata Simplified onion-routing protocol Non-cryptographic analysis Characterize the situations that provide anonymity Send a message, receive a message, communicate with a destination Possibilistic anonymity 6

How Onion Routing Works 1 2 u d 3 5 User u running client Internet destination d 4 Routers running servers 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d 7

How Onion Routing Works {{{m}3}4}1 1 2 u d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u d 3 5 {{m}3}4 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u d 3 5 {m}3 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u m d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u d m’ 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u d 3 5 4 {m’}3 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u {{m’}3}4 d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 {{{m’}3}4}1 u d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged. Stream is closed. 7

How Onion Routing Works 1 2 u d 3 5 4 u creates 3-hop circuit through routers u opens a stream in the circuit to d Data is exchanged. Stream is closed. Circuit is changed every few minutes. 7

How Onion Routing Works 1 2 u d 3 5 4 8

How Onion Routing Works 1 2 u d 3 5 4 8

How Onion Routing Works 1 2 u d 3 5 4 Main theorem: Adversary can only determine parts of a circuit it controls or is next to. 8

How Onion Routing Works 1 2 u d 3 5 4 u 1 2 Main theorem: Adversary can only determine parts of a circuit it controls or is next to. 8

Anonymous Communication Sender anonymity: Adversary can’t determine the sender of a given message Receiver anonymity: Adversary can’t determine the receiver of a given message Unlinkability: Adversary can’t determine who talks to whom 9

Adversaries Passive & Global Active & Local 10

Adversaries Passive & Global Active & Local 10

Adversaries Passive & Global Active & Local 10

Adversaries Passive & Global Active & Local 10

Model Constructed with I/O automata Simplified onion-routing protocol Models asynchrony Relies on abstract properties of cryptosystem Simplified onion-routing protocol No key distribution No circuit teardowns No separate destinations No streams No stream cipher Each user constructs a circuit to one destination Circuit identifiers 11

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Automata Protocol u v w 12

Creating a Circuit u 1 2 3 13

Creating a Circuit [0,{CREATE}1] u 1 2 3 CREATE/CREATED 13

Creating a Circuit u 1 2 3 [0,CREATED] CREATE/CREATED 13

Creating a Circuit u 1 2 3 CREATE/CREATED 13

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [0,{[EXTEND,2,{CREATE}2]}1] u 1 2 3 CREATE/CREATED EXTEND/EXTENDED 14

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l1,{CREATE}2] u 1 2 3 CREATE/CREATED EXTEND/EXTENDED 14

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l1,CREATED] 14

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED 14

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [0,{{[EXTEND,3,{CREATE}3]}2}1] u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l1,{[EXTEND,3,{CREATE}3]}2] u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l2,{CREATE}3] u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l2,CREATED] CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [l1,{EXTENDED}2] CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Creating a Circuit u 1 2 3 CREATE/CREATED EXTEND/EXTENDED [Repeat with layer of encryption] 15

Input/Ouput Automata States Actions Every state has enabled actions Input, ouput, internal Actions transition between states Every state has enabled actions Input actions are always enabled Alternating state/action sequence is an execution In fair executions actions enabled infinitely often occur infinitely often In cryptographic executions no encrypted control messages are sent before they are received unless the sender possesses the key 16

I/O Automata Model Automata Notation User Server Fully-connected network of FIFO Channels Adversary replaces some servers with arbitrary automata Notation U is the set of users R is the set of routers N = U  R is the set of all agents A  N is the adversary K is the keyspace l is the (fixed) circuit length k(u,c,i) denotes the ith key used by user u on circuit c 17

User automaton 18

User automaton 18

User automaton 18

User automaton 18

User automaton 18

User automaton 18

User automaton 18

Server automaton 19

Server automaton 19

Server automaton 19

Server automaton 19

Server automaton 19

Server automaton 19

Server automaton 19

Server automaton 19

Anonymity Definition (configuration): A configuration is a function URl mapping each user to his circuit. 20

Anonymity Definition (configuration): A configuration is a function URl mapping each user to his circuit. Definition (indistinguishability): Executions  and  are indistinguishable to adversary A when his actions in  are the same as in  after possibly applying the following: : A permutation on the keys not held by A. : A permutation on the messages encrypted by a key not held by A. 20

Anonymity Definition (anonymity): User u performs action  anonymously in configuration C with respect to adversary A if, for every execution of C in which u performs , there exists an execution that is indistinguishable to A in which u does not perform . 21

Anonymity Definition (anonymity): User u performs action  anonymously in configuration C with respect to adversary A if, for every execution of C in which u performs , there exists an execution that is indistinguishable to A in which u does not perform . Definition (unlinkability): User u is unlinkable to d in configuration C with respect to adversary A if, for every fair, cryptographic execution of C in which u talk to d, there exists a fair, cryptographic execution that is indistinguishable to A in which u does not talk to d. 21

Theorem: Let C and D be configurations for which there exists a permutation : UU such that Ci(u) = Di((u)) if Ci(u) or Di((u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution  of C there exists an indistinguishable, fair, cryptographic execution  of D. The converse also holds. 22

Theorem: Let C and D be configurations for which there exists a permutation : UU such that Ci(u) = Di((u)) if Ci(u) or Di((u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution  of C there exists an indistinguishable, fair, cryptographic execution  of D. The converse also holds. C u 1 2 v 3 5 4 22

Theorem: Let C and D be configurations for which there exists a permutation : UU such that Ci(u) = Di((u)) if Ci(u) or Di((u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution  of C there exists an indistinguishable, fair, cryptographic execution  of D. The converse also holds. C D u 1 2 v 3 5 2 3 4 22

Theorem: Let C and D be configurations for which there exists a permutation : UU such that Ci(u) = Di((u)) if Ci(u) or Di((u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution  of C there exists an indistinguishable fair, cryptographic execution  of D. The converse also holds. C D u 1 2 v 2 5 2 v 3 5 u 4 2 3 4 22

Theorem: Let C and D be configurations for which there exists a permutation : UU such that Ci(u) = Di((u)) if Ci(u) or Di((u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution  of C there exists an indistinguishable fair, cryptographic execution  of D. The converse also holds. C D u u 1 2 1 2 v v 3 3 5 5 4 4 22

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys. 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys.  is an execution of D:  is fair:  is cryptographic:  is indistinguishable: 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys.  is an execution of D: Only actions by u, v, C1(u), and C1(v) have been added. These actions are modified so that they remain valid.  is fair:  is cryptographic:  is indistinguishable: 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys.  is an execution of D: Only actions by u, v, C1(u), and C1(v) have been added. These actions are modified so that they remain valid.  is fair: No new actions have been added. Router enabling is invariant under user permutations. Users only communicate with first router.  is cryptographic:  is indistinguishable: 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys.  is an execution of D: Only actions by u, v, C1(u), and C1(v) have been added. These actions are modified so that they remain valid.  is fair: No new actions have been added. Router enabling is invariant under user permutations. Users only communicate with first router.  is cryptographic: Key permutations are applied to the entire sequence, and the original sequence was cryptographic.  is indistinguishable: 23

Lemma: Let u, v be two distinct users such that neither they nor the first routers in their circuits are compromised in configuration C. Let D be identical to C except the circuits of users u and v are switched. For any fair, cryptographic execution  of C there exists a fair, cryptographic execution  of D that is indistinguishable to A. Proof: To construct : 1. Replace any message sent or received between u (v) and C1(u) (C1(v)) in  with a message sent or received between v (u) and C1(u) (C1(v)). 2. Let the permutation  send u to v and v to u and other users to themselves. Apply  to the encryption keys.  is an execution of D: Only actions by u, v, C1(u), and C1(v) have been added. These actions are modified so that they remain valid.  is fair: No new actions have been added. Router enabling is invariant under user permutations. Users only communicate with first router.  is cryptographic: Key permutations are applied to the entire sequence, and the original sequence was cryptographic.  is indistinguishable:The permutation needed to make  look like  to A is just the reverse of the key permutation used to create . 23

Unlinkability Corollary: A user is unlinkable to its destination when: 24

Unlinkability Corollary: A user is unlinkable to its destination when: 4? The last router is unknown. u 3 2 5? 24

Unlinkability Corollary: A user is unlinkable to its destination when: 4? The last router is unknown. u 3 2 5? OR The user is unknown and another unknown user has an unknown destination. 2 1 4 2? 5 4? 5? 24

Unlinkability Corollary: A user is unlinkable to its destination when: 4? The last router is unknown. u 3 2 5? OR The user is unknown and another unknown user has an unknown destination. 2 1 4 2? 5 4? 5? OR The user is unknown and another unknown user has a different destination. 2 1 4 5 2 1 24

Model Robustness Only single encryption still works Can remove circuit identifiers Can include stream ciphers May allow users to create multiple circuits 25

Future Work Construct better models of time Exhibit a cryptosystem with the desired properties Incorporate probabilistic behavior by users 26