Modeling Security Protocols in CSP ISA 763- Fall 2007 Chapter 2 and 3 of Ryan and Schneider Thanks to Professor Catuscia Palamidessi’s notes

Example: The Yahalom Protocol The protocol Message 1: a -> b : a.na Message 2: b -> s : b.{a.na.nb}ServerKey(b) Message 3: s -> a : {b.kab.na.nb}ServerKey(a) {a.kab}ServerKey(b) Message 4: a -> b : {a.kab}ServerKey(b) .{nb}kab Objectives: Authentication of the participants Kab should remain secret Secrecy also on na, nb,

Three Views of the Participants The Initiator (=A) A b: a.na A j : {b.kab.na.nb}ServerKey({a} . {a.kab}ServerKey(b) A b: {a.kab}ServerKey(b) .{nb}kab The Responder (=B) B a : a.na B j : {a.na.nb}ServerKey(b) B  a: {a.kab}ServerKey(b) .{nb}kab The Server Jeeves (= J) J b: b. {a.na.nb}ServerKey(b) J a : {b.kab.na.nb}ServerKey(Ka). {a.kab}ServerKey(b)

Objectives Develop CSP programs for all three roles In each role: Take the process through the protocol In each role: Alternate between sending and receiving Each process will Generate keys Generate nonces Perform encryption and decryption

Encoding - I Encryption function: encrypt(k,m) Decryption function: dcrypt(k,m) Keys: serverKey(b) shared between server and b Other cannot understand it!

Encoding - II A CSP process that understands only messages of the kind kab □ kab∈ Key [ recieve.a.B.({a.kab}serverKey(b).{nB}.kab)  P(B,a,nB, kab) ] One that aborts on an incorrect message □ kab∈Key [recieve.a.B.({a.kab}serverKey(B).{nB} kab)  P(B,a,nB, kab) ] □ AbortRun(B) Above can be written in the a? x format of CSP.

Initiator – the initial version Initiator(a,na) = env?b: Agent g send.a.b.a.na g receive.J.a{b.kab.na.nb}ServerKey(a)..{a.kab}kb. kab∈Key g send.a.b.{a.kab}kab nb∈Nonce g Session(a,b,kab,na,nb) m∈T

Responder – the initial version Responder(b,nb) = receive.a.b.a.na g send.b.J.b.{a.na.nb}ServerKey(b) g kab∈Key receive.a.b.{a.kab}ServerKey(b).{a.nb}kab a∈Agent g Session(b,a,kab,nb,na) na∈Nonce

The Server Jeeves Server(J,Kab ) = kab∈Key {a.kab}ServerKey(b) receive.b.J.b{a.nanb}ServerKey(b) g send.J.a.{b.kab.nanb}ServerKey(a) kab∈Key {a.kab}ServerKey(b) a,b∈Agent na,nb∈Nonce Server(J) = ||| Server(J,Kab) kab∈KeyServer

Users To model disjoint sets of nonces, we define two sets Nonce_Ia and Nonce_Ib Define User’s Anne and Bob as Useranne and UserBob where: Usera = ||| Initiator(a,n) ||| ||| Responder(b,n) n∈ Nonce_Ia n∈ Nonce_Ib

Data Types CSP operates over abstractly specified data types, and not their (bit-pattern) representations Use notations like Encrypt.k.(Sq.<a,na>) Syntax: fact := Encrypt.fact.fact | Hash.fact.fact|Sq.fact* | Key.k|Nonce.n|Text.t Abbreviations: Write {k}m for Encrypt.k.m and g(|m|) for Hash.g.m, k for Key.k, n for Nonce.n, t for text.t and <m1,…mn> for Sq.<m1,…mn>, m1.m2 for <m1,m2>. Note: Sq stands for sequence.

The Intruder Process The intruder is modeled as a process that communicates over channels with Alice and Bob [non-deterministically= at will] The intruder builds its knowledge-base of facts by learning the facts that come over the channels and logical reasoning Intruder(X) = intruder that knows X – a collection of predicate instances (knowledge-base) Intruder process: Intruder(X) ≡ learn?m:messageIntruder(Close(XU{m}) □ say!m:message∩X  Intruder(X) Close (X) ≡ deductive closure of X

Proof Rules for Intruder Deductions {k,m} ⊢ encrypt(k,m) {encrypt(k,m), k-1} ⊢ m Sq.{…, x,…} ⊢ x {x1,…xn} ⊢ Sq< x1,…xn> Can form the deductive closure with these rules and others from Predicate Logic.

Constructing the Network Reliable agents: Alice, Bob, server Jeeves Intruder: Yves Network Connections: Send.a.b.m : An event inside a reliable agent matched up with learn.m [learn = channel name] event of the intruder Receive.a.b.m: An event inside a reliable agent matched up with say.m [say=channel name] event of the intruder Send and learn channels are connected (by renaming) to the external take channel Receive and say channels are connected (by renaming) to the fake channel of the intruder

Network Connections Jeeves Anne Bob Yves leak send receive send fake.x.Bob take.Anne.y learn say Yves leak

Network Connections Sender and receiver fields of the intruder are missing in the intruder because conveys no useful information to the intruder So learn.m is mapped to all take.a.b.m of all legitimate choices of a and b So say.m is mapped to all fake.a.b.m of all legitimate choices of a and b – pretending to be from b Internally, agents and map take and fake to send and receive. So send, receive, learn and say are internal names and take and fake are external channel names.

Network Connections ||| Jeeves [[fake,take/receive,send]] Agent(anne) [[fake,take/receive,send]] ||| Agent(bob) [[fake,take/receive,send]] Jeeves [[fake,take/receive,send]] || Yves[[take.x.y,fake.x.y/learn,say|x,y∈AgentsU{Jeeves}]] {|take,fake|} The dotted lines in the picture represents renaming

A Sample Run - 1 Externally Agent Which sees Intruder Take.A.C.m A Send. A.C.M Learn.Sq<A,Na> {Sq<A,Na>} ⊢ Na {A,Na,Nc} ⊢ Sq<A,Na,Nc> {Sq<A,Na,Nc>, ServerKey(C)} ⊢ {Sq<A,Na,Nc>} ServerKey(C) {C, {Sq<A,Na,Nc>}ServerKey(C)}} ⊢Sq.<C,{Sq<A,Na,Nc>}ServerKey(C)}}>

A Sample Run -2 Agent Intruder {<A,K>} ServerKey(C)> Externally Agent Which Sees Intruder fake.C.J.m J receive.C.J.m Say. Sq.<C,{Sq<A,Na,Nc>}ServerKey(C)}> take.J.A.m send.J.A.m Say. Sq.<{Sq<C,K,Na,Nc>}ServerKey(A)}, {<A,K>} ServerKey(C)> fake.J.A.m A receive.J.A.m learn.Sq.<{Sq<C,K,Na,Nc>}ServerKey(A)}, take.A.C.m learn.<{<A,K>} ServerKey(C),{Nc}K>

Direct Communication Need to allow direct communication between agents without Intruder interference Make take and fake alternatives to the original communication: RenAgent(A)≡ Agent(A)[[fake,com,take,com / receive,receive,send,send]] Com provides an alternative channel for honest agents to communicate.

Improved Network Connections Jeeves send receive Anne Bob Com.Anne.Bob send receive send receive fake.x.Bob take.Anne.y learn say Yves leak

Specifying Protocol Goals Chapter 3 of Ryan and Schneider

Protocol Security Objectives Participants modeled as participant processes. Attackers modeled as Intruder processes. What objectives should the protocol satisfy in the presence of the intruder? Secrecy: Item m is intended to be a secret until the end of the protocol. Authenticity: When completed, some guarantees are needed about the participation of some parties the protocol is apparently run with. Non-repudiation: Evidence of the involvement of the other party Anonymity: Protecting the identity of agents with respect to particular events

Secrecy At some point in the protocol a signal Claim_secret.M will be inserted into the protocol by means of a Signal = a special channel to make secrecy claims Signals are not facts. Signal ∩ Fact =Ø Information appearing on the Signal channels are claims expressing the claimants conditions. Can specify secrecy claims (such as trace properties) without this channel, but is more complex.

Secrecy A Intr B Protocol run Claim_Secret.m

Secrecy of the Yahalom Protocol Expand Yahalom to Yahalom’ by inserting secrecy claims Secrecy claims: signal.Cliam_Secret.a.b.s Says: at this point in a’ s run apparently with b, the protocol guarantees that Yves cannot obtain s.

The Initiator with Signals Initiator(a,na ) = env?b: Agent g send.a.b.a.na g receive.J.a{b.kab.na.nb}ServerKey(a).m kab∈Key g send.a.b.m.{nb}kab nb∈Nonce g signal.Cliam_Secret.a.b.kab m∈T g Session(a,b,kab,na,nb)

Responder – the initial version Responder(b,nb ) = receive.a.b.a.na g send.b.J.b .{a.na.nb}ServerKey(b) g kab∈Key receive.a.b.{a. kab}ServerKey(b).{nb}kab na∈Nonce  signal.Claim_Secret.a.b.kab a∈Agent gSession(b,a,kab,na,nb) )

The Leak Channel In order to check if Kab appears on the say channel, we can check that by copying its messages into a leak channel. The trace Secret can be expressed as the trace property: Secretab(tr)≡ s:signal Claim_Secreta.b.s. in tr  ¬ (leak.s. in tr)

Where to insert the signal? A process should signal Claim-Secret.*.s as soon as sees s – called strong secrecy Sometimes a secret created within a protocol run and used only if the protocol terminates without problems. For example: session keys – useless unless the protocol runs to completion Then Signal.Claim-Secret.*.s should be placed at the end of the protocol.

Authentication

Authentication in CSP Insert two signals to the protocol Running.a.b in a’s protocol: a is executing a run apparently with b Commit.b.a in b’s protocol: b completed a protocol run apparently with a

Authentication A Intr B Run with A Commit with B

Authentication in CSP Authentication is achieved if Running.a.b always precedes Commit.b.a in all traces of the system Weaker or stronger forms of authentication can be achieved by varying the parameters of these signals and their constraints

Authentication in Yahalom The Yahalom Protocol wants to authenticate both parties to each other Requires two separate enhancements to the protocol We will analyze the two authentication properties separately

Authenticating the Initiator - 1 Initiator’(a,na ) = env?b: Agent g send.a.b.a.na g (receive.J.a{b. kab.na.nb}ServerKey(a) .m kab ∈ Key g signal.Running_Initiator.a.b.na.nb.kab nb∈ Nonce g send.a.b.m.{nb}kab m∈ T g Session(a,b,kab,na,nb) ) Responder’(b,nb ) = (receive.a.b.a.na g send.b.J.b .{a.na.nb}ServerKey(b) kab∈Key g receive.a.b.{a. kab}ServerKey(b) .{nb}kab nb e Nonce g signal. Commit_Responder.b.a.na.nb.kab m∈T g Session(b,a,kab,na,nb) )

Authenticating the Initiator - 2 Initiatora Responderb Server a.na b.{a.na.nb}ServerKey(b) {b.kab.na.nb}ServerKey(a) {a.kab}ServerKey(b) Run_Init.a.b.na.nb.kab {a.kab}ServerKey(b) .{nb}kab Com_Resp.b.a.na.nb.kab

Authenticating the Initiator - 3 Property to be verified: signal. Running_Initiator.a.b.na.nb.kab precedes signal.Commit_Responder.b.a.na.nb.kab in all the Traces(System) Again, this property can be verified automatically by checking all traces

Authenticating the Responder -1 Initiator’(a,na ) = env?b: Agent g send.a.b.a.na (receive.J.a{b. kab.na.nb}ServerKey(a) .m kab∈Key g send.a.b.m.{nb}kab nb∈Nonce g signal.Commit_Initiator.a.b.na.nb.kab m∈T g Session(a,b,kab,na,nb) ) Responder’(b,nb ) = (receive.a.b.a.na g send.b.J.b .{a.na.nb}ServerKey(b) kab∈Key g signal. Running_Responder.b.a.na.nb nb∈Nonce g receive.a.b.{a. kab}ServerKey(b) .{nb}kab m∈T g Session(b,a,kab,na,nb) )

Authenticating the Responder -2 Initiatora Responderb Server a.na Run_Resp.b.a.na.nb. b.{a.na.nb}ServerKey(b) {b.kab.na.nb}ServerKey(a) {a.kab}ServerKey(b) {a.kab}ServerKey(b) .{nb}kab Run_Init.a.b.na.nb.kab

Authenticating the Responder-3 The property to be verified: signal. Running_Responder.b.a.na.nb precedes signal.Commit_Initiator.a.b.na.nb.kab in all possible Traces(System) Again, this property can be verified automatically by checking the traces

Forms of Authentication Objectives From Sections 4 and 5 of Gavin Lowe’s paper on “ a hierarchy of authentication specifications” Recap of specifications: Agreeement(B-role,A-role,ds)(B) where B-role: Role of agent being authenticated A-role: Role of the recipient of authentication ds: data items agreed upon for authentication B: identity of agent being autheticated

Authentication Agreements Agreeement(B-role,A-role,{d1,d2})(B) ≡ XXXXX….. signal.Running.B-role.B?A?d’1?d’2  XXXXX…. signal.Commit.A-role.A.B.d’1.d’2  stop Objective: In all traces, signal.Running precedes signal.Commit. Can be stated as a refinement statement Agreeement(B-role,A-role,{d1,d2})(B) ⊆T SYSTEM \ (Σ - aAgreement)

Recall Trace Semantics Recall: P ⊆T Q if trace(Q) ⊆ trace(P) P \ S = removing all events in S from P So Agreeement(B-role,A-role,{d1,d2})(B) ⊆T SYSTEM \ (Σ - aAgreement) is valid if all symbols that do not belong to the alphabet of Agreement are removed from system traces, then the remaining system traces must be a subset of that of Agreeement(B-role,A-role,{d1,d2})(B) . Written as SYSTEM sat Agreeement(B-role,A-role,{d1,d2})(B) or SYSTEM meets Agreeement(B-role,A-role,{d1,d2})(B)

Generalizing Authentication Agents agreeing on some data, not others: Agreeement(B-role,A-role,{d1})(B) ≡ signal.Running.B-role.B?A?d’1?d”2  signal.Commit.A-role.A.B.d’1.?d’2  stop Commitment based on d1 but not d2. Lemma: ds2 ⊆ ds1 implies SYSTEM sat Agreeement(B-role,A-role,ds1)(B)  SYSTEM sat Agreeement(B-role,A-role,ds2)(B)

Non-injective agreements Allow A to commit on every run of B NI-Agreement(B-role,A-role,{d1,d2})(B) ≡ signal.Running.B-role.B?A?d’1?d’2  RUN({signal.Commit.A-role.A.B.d’1.?d’2})  stop One instance of Running by B-role, A-role runs as many runs as possible ! Similarly, NI-Agreement(B-role,A-role,{d1,d2})(B)n allows the protocol to be run at most n times, but allows any number of runs of B to match with a single run of A

More properties of agreements Lemma: ds2 ⊆ ds1 implies SYSTEM sat NI-agreeement(B-role,A-role,ds1)(B)  SYSTEM sat NI-agreeement(B-role,A-role,ds2)(B) Lemma: If SYSTEM sat Agreeement(B-role,A-role,ds1)(B) then SYSTEM sat NI-agreeement(B-role,A-role,ds2)(B)

A weaker version of non-injectivity Agents agreeing on some data, not others: Agreeement(B-role,A-role,{d1})(B)n ≡ [signal.Running.B-role.B?A?d’1?d”2  signal.Commit.A-role.A.B.d’1.?d’2  stop]n Agreement completed in n runs, but each commit is matched by only one run.

Weak agreements No guarantee on B’s role and data WeakAgreement(B-role,A-role,{d1,d2})(B) ≡ signal.Running.?B-role.!B?A?ds’  RUN({signal.Commit.A-role.A.B.ds|ds∈Data*})  stop A completes play A-role an arbitrary number of runs if B runs at least once playing some role using some data values

Weak vs. Non-injective Lemma: If SYSTEM sat WeakAgreement(B-role,A-role,{d1,d2})(B)n specify a system satisfying a week agreement in n rounds Lemma: If SYSTEM sat NI-agreeement(B-role,A-role,ds2)(B)m then SYSTEM sat WeakAgreement(B-role,A-role,{d1,d2})(B)m+n

Aliveness A further weakening: Aliveness(A-role)(B) ≡ A receives no guarantee that B was running the same protocol with A Only that B was running the protocol with someone Aliveness(A-role)(B) ≡ signal.Running?B-role!B?C?ds’  RUN({signal.Commit.A-role.A.B.ds | ds∈Data* /\ A∈Agent })  Stop Agreement: SYSTEM sat Aliveness(a-role)(B)

Aliveness vs. Weak-agreement Lemma: If SYSTEM sat WeakAgreement(B-role,A-role,{d1,d2})(B)n then SYSTEM sat Aliveness(a-role)(B)n

Agreeing on the Protocol Use an extra field for the protocol ID Aliveness(ProtId,A-role)(B) ≡ signal.Running.protId?B-role!B?C?ds’  RUN({signal.Commit.protId.A-role.A.B.ds| ds∈Data* /\ A∈Agent })  Stop Running different protocols Aliveness(?ProtId’,A-role)(B) ≡

Recentness Two approaches Using nonces Using time Need another field in agreements: Agreement(Resp-Role, Init-role, {Na,Nb})(Bob)m Assumptions: A and B invented Nonces Na and Nb not too distant in the past Nonces are new; i.e. have not been used in the recent past

Timed Authentication Need to authenticate within a time window AuthTime. Each agent has a built-in timeout MaxRunTime. Requires a discrete clock Add timeouts to un-timed versions of authentication objectives

Clocks Tock(n): performs at most n tocks Tock(n) ≡ If n= 0 then SKIP else (tockTock(n-1)□ SKIP), TSKIP ≡ tockTSKIP □ SKIP AddTime(P,MaxRunTime) ≡ tockAddTime(P,MaxRunTime) □ ( (P ||| Tock(MaxRunTime) ) Δ tockTSKIP) Q Δ P initially acts like Q but on receiving interrupt act like R

Timed Authentication TimedAgreeement(B-role,A-role,ds,AuthTime)(B) ≡ AddTime(Agreement(B-role,A-role,ds)(B),AuthTime) The process allows at most AuthTime tocks to occur between the Running and Commit events – i.e. during the execution Agreement(B-role,A-Role,ds). Specification: SYSTEM sat TimedAgreeement(B-role,A-role,ds,AuthTime)(B)n Need to redefine: Pn ≡ P |||{tock} P |||{tock} ……. |||{tock} P Other un-timed specifications can be enriched to have timed versions

Properties of timed Authentication If ds’ ⊆ ds and SYSTEM sat TimedAgreeement(B-role,A-role,ds,AuthTime)(B)n, then SYSTEM sat TimedAgreeement(B-role,A-role,ds’,AuthTime)(B)n Similar results hold for other timed versions of TimedNonInjectiveAgreement TimedWeakAgreement TimedAliveness

Monotonicity of Timed Specs If TimedSpec(_) stands for any of the specified timed authentication specifications and t < t’, and SYSTEM sat TimedSpec(t) then SYSTEM sat TimedSpec(t’)

Un-timed vs. Timed Specs Suppose TimedSpec(_) is the timed version of Spec(_) and SYSTEM sat TimedSpec(t,_) then, SYSTEM sat Spec(_)

Non-repudiation

Non-repudiation Objectives Provide the parties of an interaction with evidence so that later they cannot deny having participated Participants and trust: May not trust each-other. Different from previous models Can derive facts Attackers are different: Could live in the media Participants could be attackers.

Signals and Arguments of Evidence evidence.a.m : agent a signals receipt of evidence of message m Agenta(S)≡ □{b∈Agent, m∈S} send.a.b.m -> Agenti(S) □ receive.b.a?.?m -> Agenta(close├(S U {m})) □ □ {m∈S} evidence.a.m -> Agenti(S) System≡(||| {b∈Agent }Agenti(IKa) || Medium(IKm)) Close├ (S): Deductive closure of knowledge Argument: evidence.a..m in tr  b sent m’

Sending and Agent Knowledge a sent m ≡  M:fact; b:Agent send.a.b.m’/\ (m’contains m) Axioms for contains m contains m m’ contains m  m”.m’ contains m m’ contains m  m’.m” contains m m’ contains m  k(m’) contains m

The Zhou-Gollmann Protocol -1 Message 1 a -> b : {fNRO .b.l.c}Ska // send an encrypted msg C with run l, recipient b Message 2 b -> a : {fNRR .a.l.c}Skb //send signed record with sender a, run l and ciphertext c Message 3 a -> j : {fSUB .b.l.k}Ska // send encryption key k to the server j Message 4 b <-> j : {fCON .a.b.l.k}Skj // use ftp-get to retrieve a signed record of a, b, l and k Message 5 a <-> j : {fCON .a.b.l.k}Skj m= plaintext message k=encryption key c = k(m) the ciphertext a=originator, b=recipient, j=trusted server fNRO , fNRR= protocol steps L= protocol run ID Ska=a’s private signature key

The Zhou-Gollmann Protocol -2 Non-Repudiation of Receipt: a can prove that b has got the message by presenting {fNRR .a.l.c}Skb // message was received by b and {fCON .a.b.l.k}Skj // key k was deposited In terms of the evidence signal (NRR(tr)): evidence.a.{fNRR.a.l.c}SKb in tr  b sent .{fNRR.a.l.c} evidence.a.{fCON.a.b.l.k}SKj in tr  receive.a.j.{fSUB.b.l.c}SKa in tr

The Zhou-Gollmann Protocol -2 Non-Repudiation of Origin: b can prove that a has sent the message by presenting {fNRO .b.l.c}Ska // a sent the msg C and {fCON .a.b.l.k}Skj // key k was deposited In terms of the evidence signal (NRO(tr)): evidence.b.{fNRO.b.l.c}SKa in tr  a sent .{fNRO.a.l.c} evidence.b.{fCON.a.b.l.c}SKj in tr  a sent.fSUB.b.l.k

The Zhou-Gollmann in CSP -1 evidence.a evidence.b a b ftp.a ftp.b send.*.b send.*.a J receive.*.b receive.*.a receive.*.J send.*.J medium

Zhou-Gollmann in CSP - 2 □{b∈Agent, m∈S} send.a.b.m -> Agenti(S) Agenta(S) ≡ □{b∈Agent, m∈S} send.a.b.m -> Agenti(S) □ receive.a.b?m -> Agenta(close(S U {m})) □ ftp.a.Jeeves?m -> Agenta(close(S U {m})) □ {m∈S} evidence.a.m -> Agenti(S) --------------------------------------------------------------------------------------------- Server(S) ≡ receive.a.Jeeves?. {fSUB .b.l.k}Ska -> Server(S U {fCON .a.b.l.k}Skj) □{b∈Agent, m∈S} ftp.a.Jeeves.m -> Server(S)

Zhou-Gollmann in CSP - 3 Network=(|||a∈Agent Agenta(IKa) ||{|ftp|} Server {|send|}U||{|receive|} Medium(Ø) ) The correctness requirements System sat NRO(tr) and System sat NRR(tr)

Anonymity in CSP

Definitions There are many definitions of anonymity. Ref papers on: http://wiki.uni.lu/secan-lab/Anonymity-Papers.html Most definitions are probabilistic and very dependent on the application The Basic Issue: protect the identity of an agent associated with a given event or message

CSP Formulation Preventing the identity of agents associated with a message The message itself need not be hidden So, an event is modeled to have two parts: a.x a: agent (originator) x: content A protocol or a transaction must covey x without revealing a Also other selected agents must not be able to deduce that a issued a.x

The Attacker Model A group of agents that can observe and deduce information The medium may or may not be reliable Usually the medium is assumed to be reliable

Users and Observers The principle of anonymity: a data item that could have originated from one agent could have originated from a collection of agents in a given set. That is said to preserve anonymity! Anonusers = the set of users whose identities should be masked A = {a.x | a∈Anonusers} A protocol P described will provide anonymity to a set of users A if any permutation pA of A is in the acceptable set of traces of P

Other Parameters Set of Observable Events: B (say) Necessary condition for anonymity: A∩B=∅ Events that not in A or B have to be abstracted out. ABSC(P) = events to be abstracted out from the protocol P. Hence C = Σ \ (A U B) is abstracted out! Means: hidden, masked or renamed Condition for anonymity is: pA (ABSC(P))=ABSC(P) C Protocol P A B

The Dining Cryptographer’s Problem Reference: Section 3.5 Modeling and analysis of security protocols, P. Ryan and S. Snyder The dining cryptographer’s problem: unconditional sender and recipient un-traceability, Journal of Cryptology, 1-1998. The problem: A collection of cryptographers share a meal. At the end of the meal, they are secretly informed if one of them have to pay. If not, the host would pickup the tab. The cryptographers would like to know which alternative occurs without revealing which cryptographer pays, if it is one of them – without revealing which cryptographer has to pay!

Dining Cryptographer’s Algorithm Each cryptographer tosses a coin visible to herself and her right hand neighbor So each cryptographer see two coins; say self and right. Each cryptographer gets one of two messages, paying or not paying from the host = master Each cryptographer makes an announcement agree or disagree using the following algorithm If not paying say agree iff self=right The paying cryptographer says the opposite If #(disagree) is even then the host pays

The Dining Cryptographers out.0 looki,j,x: Cryptographer i reads value x from coin j Crypt(0) look.0.0 look.0.2 pays.0 notPays.0 Coin(0) Coin(2) The Master look1.0 look.2.2 notPayy.1 notPays.2 pays.1 pays.2 Crypt(1) Crypt(2) out.2 Out.1 look.1.1 look.2.1 Coin(1)

The Master as a CSP Process Master≡(Πi:Cryptnames pays.i →notPays.((i+1)mod 3) → notPays.((i+2) mod 3) →Stop) Π (notPays.0 → notPays.1 → notPays.2 → Stop)) Non-deterministically chooses to pay cryptographer i or not to pay either – implying host pays

Cryptographer as a CSP process Crypt(i) ≡notPays.i→look.i.i.?x →look.i.(i+1 mod 3).?y →(if (x=y) then (out.i.agree →Stop) else (out.i.disagree →Stop)) □(pays.i→look.i.i.?x →(if (x=y) then (out.i.disagree →Stop) else (out.i.agree →Stop)) If asked to pay and (x=y) say agree and else disagree If asked not to pay and (x=y) say disagree and else agree look.i.j.x ≡cryptographer i reading value x for coin j

The Coin as a CSP Process Coin(i)≡ Heads(i) Π Tails(i) where Heads(i)≡ (look.i.i.heads→Heads(i) ) □ ( look(i-1 mod 3).i.heads →Heads(i) ) Tails(i)≡ (look.i.i.tails→Tails(i) ) □ ( look(i-1 mod 3).i.tails→Tails(i) )

Constructing the System Crypts ≡ Crypt(0)|||Crypt(1)|||Crypt(2) Coins ≡ Coin(0)|||Coin(1)|||Coin(2) Meal≡ ((Crypts ‖ Coins) ‖ Master) {|look|} {|pays|}U{|notPays|}

Anonymity w.r.t. a Table Observer If anonymity is desired with respect to an observer of Meal (i.e. someone at the table) Such an observer should only see values agree and disagree announced on channel out notPays and look should be abstracted. Hence A = { pays.i | 0 < i < 2} = {| pays |} B = { out.i | 0 < i < 2} = {| out |} C = Σ \ (A U B) - all other events should be abstracted.

Anonymity w.r.t. a Table Observer System: Meal \ { |look, notPays |} Anonymity requirement: pA(Meal\{|look,notPays|})=Meal\{|look,notPays|} What other events are available? : {|out|} So what does a trace consist of? Only of out.i.agree or out.i. disagree events for 0<i<2: What does any permutation on {0,1,2} do? Satisfying the anonymity requirement!

Anonymity w.r.t. a Table Observer Allowing more visible events: Can allow look.i.j.x events to be visible but not their value: Let flook(look.i.j.x) = look.i.j – i.e. drop the value Erasing/hiding the data value! New anonymity requirement: pA(flook(Meal)\{|notPays|})=flook(Meal)\{|notPays|} Is it satisfied?

Anonymity of Cryptographer 0-I Cryptographer (i+1) mod 3 and (i+2) mod 3 wants to remain anonymous from cryptographer i. Take i=0 (others are similar) Events indistinguishable by cryptographer i: A= {pays.1, pays.2} Events visible to cryptographer 0: B = {pays.0, notPays.0} U {|look.0|} U {|out|} C = Σ \ (A U B)

Anonymity of Cryptographer 0-II Only one permutation pA that satisfy pA(Meal\C) = Meal\C maps pays.1 ↔ pays.2 Is tr∈ Meal\C ↔ tr ∈ pA(Meal\C) hold?

Anonymity of Cryptographer 0-III Consider the case where Cryptographer 0 has access to the value of coin 2, i.e. look.2.2.x (i.e. By eavesdropping on the transmission). Then A= {pays.1, pays.2} B = {pays.0, notPays.0} U {|out|} U {|look.0,look.2.2|} C = Σ \ (A U B) Consider the trace: tr =<pays.2, notpays.0, look.0.0.heads, look.0.1.heads, look.2.2.heads, out.2.disagree > Then tr∈ Meal\C but tr∉ pA(Meal\C). Violating anonymity!

Analysis of a Violation Heads on all three coins are observed: Hence out.2.disagree is consistent with pays.2 However not consistent with pays.2 So if pays.2 is replaced with pays.1, the rest of the trace is not consistent with that change That is, the rest of the trace contains sufficient information to distinguish pays.1 from pays.2 – violating anonymity!

Correcting a Violation Suppose look.2.2 is not available Results in the trace tr’ =<pays.2, notpays.0, look.0.0.heads, look.0.1.heads, out.2.disagree > Then tr∈Meal\C and tr∈pA(Meal\C) Regaining anonymity! Conclusion: if the information about Coin(2) is not available, then anonymity is regained!