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Some Properties (Lemmata)

Some Properties (Lemmata). In set notation: parts(parts(H)) = parts(H) analz(analz(H)) = analz(H) synt(synt(H)) = synt(H) parts(analz(H)) = analz(parts(H)) parts(synt(H)) = parts(H)  synt(H) analz(synt(H)) = analz(H)  synt(H). Example: Needham-Schroeder Symmetric Key.

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Some Properties (Lemmata)

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  1. Some Properties (Lemmata) • In set notation: • parts(parts(H)) = parts(H) • analz(analz(H)) = analz(H) • synt(synt(H)) = synt(H) • parts(analz(H)) = analz(parts(H)) • parts(synt(H)) = parts(H)  synt(H) • analz(synt(H)) = analz(H)  synt(H)

  2. Example: Needham-Schroeder Symmetric Key • Needham Schroeder Symmetric Key Protocol: • 1: A  S : A, B, NA • 2: S  A : {NA , B, KAB, {KAB, A} KBS }KAS • 3: A  B : { KAB, A} KBS • 4: B  A : {NB} KAB5: A  B : {NB ,NB} KAB • Modification in the last line (step 5) : {NB ,NB} instead of {NB + 1}

  3. Step3: Needham-Schroeder Theory

  4. Step3: Needham-Schroeder Theory • Needham-Schroeder Traces rules for protocol steps attacker

  5. Line 1 Informally 1: A  S : A, B, NA • evs is an admissible trace, e.g. evs  nsch-sym, • A  B • NA is a fresh nonce, e.g. Nonce(NA)  used(evs), • A and B are not servers (friends or spy) • then: Says(A, S, [Agent(A), Agent(B), Nonce(NA)]) # evs  nsch-sym

  6. Step 3: Axiomatization of Line 1 has to be defined appropriately

  7. Line 2 Informally 2: S  A : {NA , B, KAB, {KAB, A} KBS }KAS • evs is an admissible trace, • KAB is a fresh session key, • Gets(S, [Agent(A), Agent(B), Nonce(NA)] )  evs, • then: Says(S, A, crypt( KAS, [Nonce(NA), B, KAB , crypt(KBS, [KAB, Agent(A)]) ] ) ) # evs 2 nsch-sym

  8. Step 3: Axiomatization of Line 2

  9. Line 3 Informally 3: A  B : { KAB, A} KBS • evs is an admissible trace, • Says(A, S, [Agent(A), Agent(B), Nonce(NA)]) 2 evs • Gets(A, crypt( KAS, [Nonce(NA), B, KAB , F ]))  evs, • then: Says(A, B, F) # evs  nsch-sym • F stands for the part of the message that cannot be decrypted by A.

  10. Step 3: Axiomatization of Line 3

  11. Line 4 Informally 4: B  A : {NB} KAB • evs is an admissible trace, • NB is a fresh nonce, • Gets(B, crypt(KBS, [KAB, Agent(A)]) )  evs, • then: Says(B, A, crypt(KAB, Nonce(NB)))# evs  nsch-sym

  12. Step 3: Axiomatization of Line 4

  13. Line 5 Informally 5: A  B : {NB ,NB} KAB • evs is an admissible trace, • Says(A, S, [Agent(A), Agent(B), Nonce(NA)] 2 evs • Says(A, B, F) 2 evs, • Gets(A, crypt( KAS, [Nonce(NA), B, KAB, F ]) )  evs, • Gets(A, crypt(KAB, Nonce(NB))) 2 evs • then: Says(A, B, crypt(KAB, [Nonce(NB), Nonce(NB)]) # evs  nsch-sym

  14. Step 3: Axiomatization of Line 5

  15. Step3 : Loosing Keys • "Oops" – rule: Participants might loose keys. • Keys they got during a protocol run. • Allows to analyse what happens in these cases. • Does everything break down? • In the example: Loss of KAB after the protocol run.

  16. Step3: Modeling of Attacks • "Fake" – rule: (spy can send messages at anytime) • if evs  nsch-sym, • X  synth(analz(spies(evs))), • then: Says(Spy, B, X) # evs  nsch-sym • spies(evs) : The set of messages the spy sees (knows).

  17. Step 3: Oops Rule • Binding (the loss of key) to the current run by the two nonces!

  18. Step3: Generic Protocol Events (Attacker) What does this mean? What the spy sees (knows)

  19. Step3: Knowledge of Participants • What (arbitrary) agents learn during execution of a protocol • main interest: knowledge of the spy The spy shares knowledge with bad agents.

  20. Step3: Knowledge of Participants

  21. Step3: Knowledge of Participants

  22. Step3: Initial Knowledge

  23. Step4: Properties of Traces: Confidentiality Informally • Confidentiality: • if evs  nsch-sym,Says(S, A, crypt( KAS, [Nonce(NA), B, KAB , crypt(KBS, [KAB, Agent(A)]) ] ) ) 2 evs, • A and B are not bad agents (Bad agents share theirknowledge with the spy.), • : Notes(spy, [Nonce(NA), Nonce(NB), KAB) 2 evs for some NB (The key was not lost by an oops event after the says event from above.) • then: : Key(KAB) 2 analz(spies(evs))

  24. Step4: Properties of Traces

  25. Step4: Properties of Traces • Authentification • if evs  nsch-sym, • Gets(A, crypt( KAS, [Nonce(NA), B, KAB , F ])) 2 evs, • Says(A,B,F) 2 evs, • Gets(A, crypt(KAB(Nonce(NB))) 2 evs, • : Notes(spy, Nonce(NA), Nonce(NB), Key(KAB)) 2 evs(No loss of keys) • A and B are not bad agents, • then Says(B, A, crypt(KAB, Nonce(NB))) 2 evs • From Gets to Says

  26. Step4: Properties of Traces

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