src/HOL/Auth/Yahalom.ML
author paulson
Tue Dec 16 15:17:26 1997 +0100 (1997-12-16)
changeset 4422 21238c9d363e
parent 4238 679a233fb206
child 4449 df30e75f670f
permissions -rw-r--r--
Simplified proofs using rewrites for f``A where f is injective
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(*  Title:      HOL/Auth/Yahalom
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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Inductive relation "yahalom" for the Yahalom protocol.
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From page 257 of
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  Burrows, Abadi and Needham.  A Logic of Authentication.
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  Proc. Royal Soc. 426 (1989)
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*)
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open Yahalom;
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proof_timing:=true;
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HOL_quantifiers := false;
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Pretty.setdepth 25;
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(*A "possibility property": there are traces that reach the end*)
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goal thy 
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 "!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
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\        ==> EX X NB K. EX evs: yahalom.     \
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\               Says A B {|X, Crypt K (Nonce NB)|} : set evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS 
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          yahalom.YM4) 2);
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by possibility_tac;
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result();
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(**** Inductive proofs about yahalom ****)
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(*Nobody sends themselves messages*)
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goal thy "!!evs. evs: yahalom ==> ALL A X. Says A A X ~: set evs";
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by (etac yahalom.induct 1);
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by (Auto_tac());
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qed_spec_mp "not_Says_to_self";
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Addsimps [not_Says_to_self];
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AddSEs   [not_Says_to_self RSN (2, rev_notE)];
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(** For reasoning about the encrypted portion of messages **)
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(*Lets us treat YM4 using a similar argument as for the Fake case.*)
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goal thy "!!evs. Says S A {|Crypt (shrK A) Y, X|} : set evs ==> \
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\                X : analz (spies evs)";
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by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1);
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qed "YM4_analz_spies";
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bind_thm ("YM4_parts_spies",
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          YM4_analz_spies RS (impOfSubs analz_subset_parts));
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(*Relates to both YM4 and Oops*)
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goal thy "!!evs. Says S A {|Crypt (shrK A) {|B,K,NA,NB|}, X|} : set evs ==> \
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\                K : parts (spies evs)";
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by (blast_tac (claset() addSEs partsEs
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                        addSDs [Says_imp_spies RS parts.Inj]) 1);
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qed "YM4_Key_parts_spies";
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(*For proving the easier theorems about X ~: parts (spies evs).*)
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fun parts_spies_tac i = 
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    forward_tac [YM4_Key_parts_spies] (i+6) THEN
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    forward_tac [YM4_parts_spies] (i+5)     THEN
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    prove_simple_subgoals_tac  i;
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(*Induction for regularity theorems.  If induction formula has the form
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   X ~: analz (spies evs) --> ... then it shortens the proof by discarding
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   needless information about analz (insert X (spies evs))  *)
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fun parts_induct_tac i = 
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    etac yahalom.induct i
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    THEN 
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    REPEAT (FIRSTGOAL analz_mono_contra_tac)
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    THEN  parts_spies_tac i;
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(** Theorems of the form X ~: parts (spies evs) imply that NOBODY
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    sends messages containing X! **)
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(*Spy never sees another agent's shared key! (unless it's bad at start)*)
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goal thy 
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 "!!evs. evs : yahalom ==> (Key (shrK A) : parts (spies evs)) = (A : bad)";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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by (ALLGOALS Blast_tac);
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qed "Spy_see_shrK";
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Addsimps [Spy_see_shrK];
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goal thy 
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 "!!evs. evs : yahalom ==> (Key (shrK A) : analz (spies evs)) = (A : bad)";
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by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset()));
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qed "Spy_analz_shrK";
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Addsimps [Spy_analz_shrK];
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goal thy  "!!A. [| Key (shrK A) : parts (spies evs);       \
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\                  evs : yahalom |] ==> A:bad";
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by (blast_tac (claset() addDs [Spy_see_shrK]) 1);
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qed "Spy_see_shrK_D";
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bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
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AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
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(*Nobody can have used non-existent keys!  Needed to apply analz_insert_Key*)
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goal thy "!!evs. evs : yahalom ==>          \
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\         Key K ~: used evs --> K ~: keysFor (parts (spies evs))";
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by (parts_induct_tac 1);
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(*Fake*)
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by (best_tac
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      (claset() addSDs [impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
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                addIs  [impOfSubs analz_subset_parts]
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                addDs  [impOfSubs (analz_subset_parts RS keysFor_mono)]
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                addss  (simpset())) 1);
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(*YM2-4: Because Key K is not fresh, etc.*)
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by (REPEAT (blast_tac (claset() addSEs spies_partsEs) 1));
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qed_spec_mp "new_keys_not_used";
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bind_thm ("new_keys_not_analzd",
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          [analz_subset_parts RS keysFor_mono,
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           new_keys_not_used] MRS contra_subsetD);
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Addsimps [new_keys_not_used, new_keys_not_analzd];
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(*Describes the form of K when the Server sends this message.  Useful for
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  Oops as well as main secrecy property.*)
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goal thy 
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 "!!evs. [| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} \
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\             : set evs;                                                   \
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\           evs : yahalom |]                                          \
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\        ==> K ~: range shrK";
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by (etac rev_mp 1);
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by (etac yahalom.induct 1);
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by (ALLGOALS Asm_simp_tac);
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by (Blast_tac 1);
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qed "Says_Server_message_form";
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(*For proofs involving analz.*)
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val analz_spies_tac = 
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    forward_tac [YM4_analz_spies] 6 THEN
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    forward_tac [Says_Server_message_form] 7 THEN
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    assume_tac 7 THEN REPEAT ((etac exE ORELSE' hyp_subst_tac) 7);
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(****
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 The following is to prove theorems of the form
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  Key K : analz (insert (Key KAB) (spies evs)) ==>
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  Key K : analz (spies evs)
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 A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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goal thy  
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 "!!evs. evs : yahalom ==>                                 \
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\  ALL K KK. KK <= Compl (range shrK) -->                       \
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\            (Key K : analz (Key``KK Un (spies evs))) = \
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\            (K : KK | Key K : analz (spies evs))";
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by (etac yahalom.induct 1);
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by analz_spies_tac;
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by (REPEAT_FIRST (resolve_tac [allI, impI]));
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by (REPEAT_FIRST (rtac analz_image_freshK_lemma));
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by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
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(*Fake*) 
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by (spy_analz_tac 1);
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qed_spec_mp "analz_image_freshK";
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goal thy
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 "!!evs. [| evs : yahalom;  KAB ~: range shrK |] ==>             \
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\        Key K : analz (insert (Key KAB) (spies evs)) =       \
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\        (K = KAB | Key K : analz (spies evs))";
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by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
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qed "analz_insert_freshK";
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(*** The Key K uniquely identifies the Server's  message. **)
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goal thy 
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 "!!evs. evs : yahalom ==>                                     \
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\      EX A' B' na' nb' X'. ALL A B na nb X.                        \
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\          Says Server A                                            \
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\           {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|}        \
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\          : set evs --> A=A' & B=B' & na=na' & nb=nb' & X=X'";
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by (etac yahalom.induct 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib])));
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by (ALLGOALS Clarify_tac);
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by (ex_strip_tac 2);
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by (Blast_tac 2);
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(*Remaining case: YM3*)
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by (expand_case_tac "K = ?y" 1);
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by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
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(*...we assume X is a recent message and handle this case by contradiction*)
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by (blast_tac (claset() addSEs spies_partsEs
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                        delrules [conjI]    (*no split-up to 4 subgoals*)) 1);
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val lemma = result();
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goal thy 
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"!!evs. [| Says Server A                                                 \
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\            {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} : set evs; \
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\          Says Server A'                                                \
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\            {|Crypt (shrK A') {|Agent B', Key K, na', nb'|}, X'|} : set evs; \
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\          evs : yahalom |]                                    \
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\       ==> A=A' & B=B' & na=na' & nb=nb'";
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by (prove_unique_tac lemma 1);
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qed "unique_session_keys";
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(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
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goal thy 
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 "!!evs. [| A ~: bad;  B ~: bad;  evs : yahalom |]         \
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\        ==> Says Server A                                        \
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\              {|Crypt (shrK A) {|Agent B, Key K, na, nb|},       \
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\                Crypt (shrK B) {|Agent A, Key K|}|}              \
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\             : set evs -->                                       \
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\            Says A Spy {|na, nb, Key K|} ~: set evs -->          \
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\            Key K ~: analz (spies evs)";
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by (etac yahalom.induct 1);
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by analz_spies_tac;
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by (ALLGOALS
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    (asm_simp_tac 
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     (simpset() addsimps (expand_ifs@pushes)
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	        addsimps [analz_insert_eq, analz_insert_freshK])));
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(*Oops*)
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by (blast_tac (claset() addDs [unique_session_keys]) 3);
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(*YM3*)
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by (blast_tac (claset() delrules [impCE]
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                        addSEs spies_partsEs
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                        addIs [impOfSubs analz_subset_parts]) 2);
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(*Fake*) 
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by (spy_analz_tac 1);
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val lemma = result() RS mp RS mp RSN(2,rev_notE);
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(*Final version*)
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goal thy 
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 "!!evs. [| Says Server A                                         \
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\              {|Crypt (shrK A) {|Agent B, Key K, na, nb|},       \
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\                Crypt (shrK B) {|Agent A, Key K|}|}              \
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\             : set evs;                                          \
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\           Says A Spy {|na, nb, Key K|} ~: set evs;              \
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\           A ~: bad;  B ~: bad;  evs : yahalom |]         \
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\        ==> Key K ~: analz (spies evs)";
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by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
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by (blast_tac (claset() addSEs [lemma]) 1);
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qed "Spy_not_see_encrypted_key";
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(** Security Guarantee for A upon receiving YM3 **)
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(*If the encrypted message appears then it originated with the Server*)
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goal thy
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 "!!evs. [| Crypt (shrK A) {|Agent B, Key K, na, nb|} : parts (spies evs); \
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\           A ~: bad;  evs : yahalom |]                          \
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\         ==> Says Server A                                            \
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\              {|Crypt (shrK A) {|Agent B, Key K, na, nb|},            \
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\                Crypt (shrK B) {|Agent A, Key K|}|}                   \
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\             : set evs";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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qed "A_trusts_YM3";
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(** Security Guarantees for B upon receiving YM4 **)
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(*B knows, by the first part of A's message, that the Server distributed 
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  the key for A and B.  But this part says nothing about nonces.*)
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goal thy 
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 "!!evs. [| Crypt (shrK B) {|Agent A, Key K|} : parts (spies evs);   \
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\           B ~: bad;  evs : yahalom |]                                \
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\        ==> EX NA NB. Says Server A                                    \
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\                        {|Crypt (shrK A) {|Agent B, Key K,             \
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\                                           Nonce NA, Nonce NB|},       \
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\                          Crypt (shrK B) {|Agent A, Key K|}|}          \
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\                       : set evs";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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(*YM3*)
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by (Blast_tac 1);
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qed "B_trusts_YM4_shrK";
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(*B knows, by the second part of A's message, that the Server distributed 
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  the key quoting nonce NB.  This part says nothing about agent names. 
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  Secrecy of NB is crucial.*)
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goal thy 
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 "!!evs. evs : yahalom                                             \
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\        ==> Nonce NB ~: analz (spies evs) -->                  \
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\            Crypt K (Nonce NB) : parts (spies evs) -->         \
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\            (EX A B NA. Says Server A                             \
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\                        {|Crypt (shrK A) {|Agent B, Key K,        \
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\                                  Nonce NA, Nonce NB|},           \
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\                          Crypt (shrK B) {|Agent A, Key K|}|}     \
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\                       : set evs)";
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by (parts_induct_tac 1);
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by (ALLGOALS Clarify_tac);
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(*YM3 & Fake*)
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by (Blast_tac 2);
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by (Fake_parts_insert_tac 1);
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(*YM4*)
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(*A is uncompromised because NB is secure*)
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by (not_bad_tac "A" 1);
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(*A's certificate guarantees the existence of the Server message*)
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by (blast_tac (claset() addDs [Says_imp_spies RS parts.Inj RS parts.Fst RS
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			       A_trusts_YM3]) 1);
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bind_thm ("B_trusts_YM4_newK", result() RS mp RSN (2, rev_mp));
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(**** Towards proving secrecy of Nonce NB ****)
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paulson@3444
   315
(** Lemmas about the predicate KeyWithNonce **)
paulson@3444
   316
paulson@3444
   317
goalw thy [KeyWithNonce_def]
paulson@3444
   318
 "!!evs. Says Server A                                              \
paulson@3444
   319
\            {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|} \
nipkow@3465
   320
\          : set evs ==> KeyWithNonce K NB evs";
paulson@3444
   321
by (Blast_tac 1);
paulson@3444
   322
qed "KeyWithNonceI";
paulson@3444
   323
paulson@3444
   324
goalw thy [KeyWithNonce_def]
paulson@3444
   325
   "KeyWithNonce K NB (Says S A X # evs) =                                    \
paulson@3444
   326
\    (Server = S &                                                            \
paulson@3444
   327
\     (EX B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|}) \
paulson@3444
   328
\    | KeyWithNonce K NB evs)";
paulson@3444
   329
by (Simp_tac 1);
paulson@3444
   330
by (Blast_tac 1);
paulson@3444
   331
qed "KeyWithNonce_Says";
paulson@3444
   332
Addsimps [KeyWithNonce_Says];
paulson@3444
   333
paulson@3464
   334
(*A fresh key cannot be associated with any nonce 
paulson@3464
   335
  (with respect to a given trace). *)
paulson@3444
   336
goalw thy [KeyWithNonce_def]
paulson@3444
   337
 "!!evs. Key K ~: used evs ==> ~ KeyWithNonce K NB evs";
wenzelm@4091
   338
by (blast_tac (claset() addSEs spies_partsEs) 1);
paulson@3444
   339
qed "fresh_not_KeyWithNonce";
paulson@3444
   340
paulson@3444
   341
(*The Server message associates K with NB' and therefore not with any 
paulson@3444
   342
  other nonce NB.*)
paulson@3444
   343
goalw thy [KeyWithNonce_def]
paulson@3444
   344
 "!!evs. [| Says Server A                                                \
paulson@3444
   345
\                {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB'|}, X|} \
paulson@3466
   346
\             : set evs;                                                 \
paulson@3519
   347
\           NB ~= NB';  evs : yahalom |]                            \
paulson@3444
   348
\        ==> ~ KeyWithNonce K NB evs";
wenzelm@4091
   349
by (blast_tac (claset() addDs [unique_session_keys]) 1);
paulson@3444
   350
qed "Says_Server_KeyWithNonce";
paulson@3444
   351
paulson@3444
   352
paulson@3444
   353
(*The only nonces that can be found with the help of session keys are
paulson@3444
   354
  those distributed as nonce NB by the Server.  The form of the theorem
paulson@3444
   355
  recalls analz_image_freshK, but it is much more complicated.*)
paulson@3444
   356
paulson@3444
   357
paulson@3444
   358
(*As with analz_image_freshK, we take some pains to express the property
paulson@3444
   359
  as a logical equivalence so that the simplifier can apply it.*)
paulson@3444
   360
goal thy  
paulson@3444
   361
 "!!evs. P --> (X : analz (G Un H)) --> (X : analz H)  ==> \
paulson@3444
   362
\        P --> (X : analz (G Un H)) = (X : analz H)";
wenzelm@4091
   363
by (blast_tac (claset() addIs [impOfSubs analz_mono]) 1);
paulson@3961
   364
val Nonce_secrecy_lemma = result();
paulson@2133
   365
paulson@2133
   366
goal thy 
paulson@3519
   367
 "!!evs. evs : yahalom ==>                                         \
paulson@3444
   368
\        (ALL KK. KK <= Compl (range shrK) -->                          \
paulson@3444
   369
\             (ALL K: KK. ~ KeyWithNonce K NB evs)   -->                \
paulson@3683
   370
\             (Nonce NB : analz (Key``KK Un (spies evs))) =     \
paulson@3683
   371
\             (Nonce NB : analz (spies evs)))";
paulson@3444
   372
by (etac yahalom.induct 1);
paulson@3683
   373
by analz_spies_tac;
paulson@3444
   374
by (REPEAT_FIRST (resolve_tac [impI RS allI]));
paulson@3961
   375
by (REPEAT_FIRST (rtac Nonce_secrecy_lemma));
paulson@3444
   376
(*For Oops, simplification proves NBa~=NB.  By Says_Server_KeyWithNonce,
paulson@3444
   377
  we get (~ KeyWithNonce K NB evsa); then simplification can apply the
paulson@3444
   378
  induction hypothesis with KK = {K}.*)
paulson@3961
   379
by (ALLGOALS  (*12 seconds*)
paulson@3444
   380
    (asm_simp_tac 
paulson@3961
   381
     (analz_image_freshK_ss 
paulson@3961
   382
       addsimps expand_ifs
paulson@3961
   383
       addsimps [all_conj_distrib, analz_image_freshK,
paulson@3961
   384
		 KeyWithNonce_Says, fresh_not_KeyWithNonce, 
paulson@3961
   385
		 imp_disj_not1,		     (*Moves NBa~=NB to the front*)
paulson@3961
   386
		 Says_Server_KeyWithNonce])));
paulson@3444
   387
(*Fake*) 
paulson@3444
   388
by (spy_analz_tac 1);
paulson@4422
   389
(*YM4*)  (** LEVEL 6 **)
paulson@3683
   390
by (not_bad_tac "A" 1);
paulson@3683
   391
by (dtac (Says_imp_spies RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1
paulson@3444
   392
    THEN REPEAT (assume_tac 1));
wenzelm@4091
   393
by (blast_tac (claset() addIs [KeyWithNonceI]) 1);
paulson@3444
   394
qed_spec_mp "Nonce_secrecy";
paulson@3444
   395
paulson@3444
   396
paulson@3444
   397
(*Version required below: if NB can be decrypted using a session key then it
paulson@3444
   398
  was distributed with that key.  The more general form above is required
paulson@3444
   399
  for the induction to carry through.*)
paulson@3444
   400
goal thy 
paulson@3444
   401
 "!!evs. [| Says Server A                                                 \
paulson@3444
   402
\            {|Crypt (shrK A) {|Agent B, Key KAB, na, Nonce NB'|}, X|}    \
paulson@3466
   403
\           : set evs;                                                    \
paulson@3519
   404
\           NB ~= NB';  KAB ~: range shrK;  evs : yahalom |]         \
paulson@3683
   405
\        ==> (Nonce NB : analz (insert (Key KAB) (spies evs))) =  \
paulson@3683
   406
\            (Nonce NB : analz (spies evs))";
paulson@3444
   407
by (asm_simp_tac (analz_image_freshK_ss addsimps 
paulson@3444
   408
		  [Nonce_secrecy, Says_Server_KeyWithNonce]) 1);
paulson@3444
   409
qed "single_Nonce_secrecy";
paulson@3444
   410
paulson@3444
   411
paulson@3444
   412
(*** The Nonce NB uniquely identifies B's message. ***)
paulson@3444
   413
paulson@3444
   414
goal thy 
paulson@3519
   415
 "!!evs. evs : yahalom ==>                                            \
paulson@3444
   416
\   EX NA' A' B'. ALL NA A B.                                              \
paulson@3683
   417
\      Crypt (shrK B) {|Agent A, Nonce NA, nb|} : parts(spies evs) \
paulson@3683
   418
\      --> B ~: bad --> NA = NA' & A = A' & B = B'";
paulson@3519
   419
by (parts_induct_tac 1);
paulson@3121
   420
(*Fake*)
paulson@3121
   421
by (REPEAT (etac (exI RSN (2,exE)) 1)   (*stripping EXs makes proof faster*)
paulson@3121
   422
    THEN Fake_parts_insert_tac 1);
wenzelm@4091
   423
by (asm_simp_tac (simpset() addsimps [all_conj_distrib]) 1); 
paulson@2133
   424
(*YM2: creation of new Nonce.  Move assertion into global context*)
paulson@3501
   425
by (expand_case_tac "nb = ?y" 1);
paulson@2516
   426
by (REPEAT (resolve_tac [exI, conjI, impI, refl] 1));
wenzelm@4091
   427
by (blast_tac (claset() addSEs spies_partsEs) 1);
paulson@2133
   428
val lemma = result();
paulson@2133
   429
paulson@2110
   430
goal thy 
paulson@3683
   431
 "!!evs.[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} : parts (spies evs); \
paulson@3683
   432
\          Crypt (shrK B') {|Agent A', Nonce NA', nb|} : parts (spies evs); \
paulson@3683
   433
\          evs : yahalom;  B ~: bad;  B' ~: bad |]  \
paulson@2133
   434
\        ==> NA' = NA & A' = A & B' = B";
paulson@2451
   435
by (prove_unique_tac lemma 1);
paulson@2133
   436
qed "unique_NB";
paulson@2133
   437
paulson@2133
   438
paulson@3444
   439
(*Variant useful for proving secrecy of NB: the Says... form allows 
paulson@3683
   440
  not_bad_tac to remove the assumption B' ~: bad.*)
paulson@2133
   441
goal thy 
paulson@3501
   442
 "!!evs.[| Says C D   {|X,  Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}    \
paulson@3683
   443
\            : set evs;          B ~: bad;                               \
paulson@3501
   444
\          Says C' D' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|} \
paulson@3466
   445
\            : set evs;                                                   \
paulson@3683
   446
\          nb ~: analz (spies evs);  evs : yahalom |]        \
paulson@2133
   447
\        ==> NA' = NA & A' = A & B' = B";
paulson@3683
   448
by (not_bad_tac "B'" 1);
wenzelm@4091
   449
by (blast_tac (claset() addSDs [Says_imp_spies RS parts.Inj]
paulson@4238
   450
                        addSEs [MPair_parts]
paulson@4238
   451
                        addDs  [unique_NB]) 1);
paulson@2133
   452
qed "Says_unique_NB";
paulson@2133
   453
paulson@3444
   454
paulson@3444
   455
(** A nonce value is never used both as NA and as NB **)
paulson@3121
   456
paulson@2133
   457
goal thy 
paulson@3683
   458
 "!!evs. [| B ~: bad;  evs : yahalom  |]            \
paulson@3683
   459
\ ==> Nonce NB ~: analz (spies evs) -->           \
paulson@3683
   460
\     Crypt (shrK B') {|Agent A', Nonce NB, nb'|} : parts(spies evs) --> \
paulson@3683
   461
\     Crypt (shrK B)  {|Agent A, Nonce NA, Nonce NB|} ~: parts(spies evs)";
paulson@3519
   462
by (parts_induct_tac 1);
paulson@3121
   463
by (Fake_parts_insert_tac 1);
wenzelm@4091
   464
by (blast_tac (claset() addDs [Says_imp_spies RS analz.Inj]
paulson@4238
   465
                        addSIs [parts_insertI]
paulson@4238
   466
                        addSEs partsEs) 1);
paulson@3464
   467
bind_thm ("no_nonce_YM1_YM2", result() RS mp RSN (2,rev_mp) RSN (2,rev_notE));
paulson@2133
   468
paulson@3464
   469
(*The Server sends YM3 only in response to YM2.*)
paulson@2133
   470
goal thy 
paulson@3466
   471
 "!!evs. [| Says Server A                                                \
paulson@3466
   472
\            {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} : set evs;     \
paulson@3519
   473
\           evs : yahalom |]                                             \
paulson@2133
   474
\        ==> EX B'. Says B' Server                                       \
paulson@2284
   475
\                      {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |} \
nipkow@3465
   476
\                   : set evs";
paulson@2133
   477
by (etac rev_mp 1);
paulson@2133
   478
by (etac yahalom.induct 1);
paulson@2133
   479
by (ALLGOALS Asm_simp_tac);
paulson@3121
   480
by (ALLGOALS Blast_tac);
paulson@2133
   481
qed "Says_Server_imp_YM2";
paulson@2133
   482
paulson@2133
   483
paulson@3519
   484
(*A vital theorem for B, that nonce NB remains secure from the Spy.*)
paulson@2133
   485
goal thy 
paulson@3683
   486
 "!!evs. [| A ~: bad;  B ~: bad;  evs : yahalom |]  \
paulson@2133
   487
\ ==> Says B Server                                                    \
paulson@2284
   488
\          {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
paulson@3466
   489
\     : set evs -->                                                    \
paulson@3466
   490
\     (ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs) -->     \
paulson@3683
   491
\     Nonce NB ~: analz (spies evs)";
paulson@2133
   492
by (etac yahalom.induct 1);
paulson@3683
   493
by analz_spies_tac;
paulson@2133
   494
by (ALLGOALS
paulson@2133
   495
    (asm_simp_tac 
wenzelm@4091
   496
     (simpset() addsimps (expand_ifs@pushes)
paulson@4238
   497
	        addsimps [analz_insert_eq, analz_insert_freshK])));
paulson@3450
   498
(*Prove YM3 by showing that no NB can also be an NA*)
wenzelm@4091
   499
by (blast_tac (claset() addDs [Says_imp_spies RS parts.Inj]
paulson@4238
   500
	                addSEs [MPair_parts]
paulson@4238
   501
		        addDs  [no_nonce_YM1_YM2, Says_unique_NB]) 4
paulson@3450
   502
    THEN flexflex_tac);
paulson@3444
   503
(*YM2: similar freshness reasoning*) 
wenzelm@4091
   504
by (blast_tac (claset() addSEs partsEs
paulson@4238
   505
		        addDs  [Says_imp_spies RS analz.Inj,
paulson@4238
   506
				impOfSubs analz_subset_parts]) 3);
paulson@3450
   507
(*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*)
wenzelm@4091
   508
by (blast_tac (claset() addSIs [parts_insertI]
paulson@4238
   509
                        addSEs spies_partsEs) 2);
paulson@2377
   510
(*Fake*)
paulson@2377
   511
by (spy_analz_tac 1);
paulson@3444
   512
(** LEVEL 7: YM4 and Oops remain **)
paulson@3708
   513
by (ALLGOALS Clarify_tac);
paulson@3444
   514
(*YM4: key K is visible to Spy, contradicting session key secrecy theorem*) 
paulson@3683
   515
by (not_bad_tac "Aa" 1);
paulson@3683
   516
by (dtac (Says_imp_spies RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1);
paulson@2133
   517
by (forward_tac [Says_Server_message_form] 3);
paulson@2133
   518
by (forward_tac [Says_Server_imp_YM2] 4);
paulson@3121
   519
by (REPEAT_FIRST (eresolve_tac [asm_rl, bexE, exE, disjE]));
paulson@3519
   520
(*  use Says_unique_NB to identify message components: Aa=A, Ba=B, NAa=NA *)
wenzelm@4091
   521
by (blast_tac (claset() addDs [Says_unique_NB, Spy_not_see_encrypted_key,
paulson@4238
   522
			       impOfSubs Fake_analz_insert]) 1);
paulson@3444
   523
(** LEVEL 14 **)
paulson@3444
   524
(*Oops case: if the nonce is betrayed now, show that the Oops event is 
paulson@3444
   525
  covered by the quantified Oops assumption.*)
wenzelm@4091
   526
by (full_simp_tac (simpset() addsimps [all_conj_distrib]) 1);
paulson@2133
   527
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1 THEN etac exE 1);
paulson@2133
   528
by (expand_case_tac "NB = NBa" 1);
paulson@3444
   529
(*If NB=NBa then all other components of the Oops message agree*)
wenzelm@4091
   530
by (blast_tac (claset() addDs [Says_unique_NB]) 1 THEN flexflex_tac);
paulson@3444
   531
(*case NB ~= NBa*)
wenzelm@4091
   532
by (asm_simp_tac (simpset() addsimps [single_Nonce_secrecy]) 1);
wenzelm@4091
   533
by (blast_tac (claset() addSEs [MPair_parts]
paulson@4238
   534
		        addDs  [Says_imp_spies RS parts.Inj, 
paulson@4238
   535
			        no_nonce_YM1_YM2 (*to prove NB~=NAa*) ]) 1);
paulson@3444
   536
bind_thm ("Spy_not_see_NB", result() RSN(2,rev_mp) RSN(2,rev_mp));
paulson@2133
   537
paulson@2001
   538
paulson@3464
   539
(*B's session key guarantee from YM4.  The two certificates contribute to a
paulson@3464
   540
  single conclusion about the Server's message.  Note that the "Says A Spy"
paulson@3464
   541
  assumption must quantify over ALL POSSIBLE keys instead of our particular K.
paulson@3464
   542
  If this run is broken and the spy substitutes a certificate containing an
paulson@3464
   543
  old key, B has no means of telling.*)
paulson@2001
   544
goal thy 
paulson@3444
   545
 "!!evs. [| Says B Server                                                   \
paulson@3444
   546
\             {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}   \
paulson@3466
   547
\             : set evs;                                                    \
paulson@3444
   548
\           Says A' B {|Crypt (shrK B) {|Agent A, Key K|},                  \
paulson@3466
   549
\                       Crypt K (Nonce NB)|} : set evs;                     \
paulson@3466
   550
\           ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs;         \
paulson@3683
   551
\           A ~: bad;  B ~: bad;  evs : yahalom |]       \
paulson@3444
   552
\         ==> Says Server A                                                 \
paulson@3444
   553
\                     {|Crypt (shrK A) {|Agent B, Key K,                    \
paulson@3444
   554
\                               Nonce NA, Nonce NB|},                       \
paulson@3444
   555
\                       Crypt (shrK B) {|Agent A, Key K|}|}                 \
nipkow@3465
   556
\               : set evs";
paulson@2133
   557
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1));
paulson@3683
   558
by (etac (Says_imp_spies RS parts.Inj RS MPair_parts) 1 THEN
paulson@2133
   559
    dtac B_trusts_YM4_shrK 1);
paulson@2170
   560
by (dtac B_trusts_YM4_newK 3);
paulson@2110
   561
by (REPEAT_FIRST (eresolve_tac [asm_rl, exE]));
paulson@2133
   562
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1);
paulson@2170
   563
by (dtac unique_session_keys 1 THEN REPEAT (assume_tac 1));
wenzelm@4091
   564
by (blast_tac (claset() addDs [Says_unique_NB]) 1);
paulson@2322
   565
qed "B_trusts_YM4";
paulson@3444
   566
paulson@3444
   567
paulson@3444
   568
paulson@3444
   569
(*** Authenticating B to A ***)
paulson@3444
   570
paulson@3444
   571
(*The encryption in message YM2 tells us it cannot be faked.*)
paulson@3444
   572
goal thy 
paulson@3519
   573
 "!!evs. evs : yahalom                                            \
paulson@3683
   574
\  ==> Crypt (shrK B) {|Agent A, Nonce NA, nb|} : parts (spies evs) --> \
paulson@3683
   575
\      B ~: bad -->                                              \
paulson@3466
   576
\      Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}  \
nipkow@3465
   577
\         : set evs";
paulson@3519
   578
by (parts_induct_tac 1);
paulson@3444
   579
by (Fake_parts_insert_tac 1);
paulson@3444
   580
bind_thm ("B_Said_YM2", result() RSN (2, rev_mp) RS mp);
paulson@3444
   581
paulson@3444
   582
(*If the server sends YM3 then B sent YM2*)
paulson@3444
   583
goal thy 
paulson@3519
   584
 "!!evs. evs : yahalom                                                      \
paulson@3444
   585
\  ==> Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|} \
paulson@3466
   586
\         : set evs -->                                                     \
paulson@3683
   587
\      B ~: bad -->                                                        \
paulson@3466
   588
\      Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}  \
nipkow@3465
   589
\                 : set evs";
paulson@3444
   590
by (etac yahalom.induct 1);
paulson@3444
   591
by (ALLGOALS Asm_simp_tac);
paulson@3444
   592
(*YM4*)
paulson@3444
   593
by (Blast_tac 2);
paulson@3444
   594
(*YM3*)
wenzelm@4091
   595
by (best_tac (claset() addSDs [B_Said_YM2, Says_imp_spies RS parts.Inj]
paulson@4238
   596
		       addSEs [MPair_parts]) 1);
paulson@3444
   597
val lemma = result() RSN (2, rev_mp) RS mp |> standard;
paulson@3444
   598
paulson@3444
   599
(*If A receives YM3 then B has used nonce NA (and therefore is alive)*)
paulson@3444
   600
goal thy
paulson@3444
   601
 "!!evs. [| Says S A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|} \
paulson@3466
   602
\             : set evs;                                                    \
paulson@3683
   603
\           A ~: bad;  B ~: bad;  evs : yahalom |]                        \
paulson@3444
   604
\   ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
nipkow@3465
   605
\         : set evs";
wenzelm@4091
   606
by (blast_tac (claset() addSDs [A_trusts_YM3, lemma]
paulson@4238
   607
		        addEs spies_partsEs) 1);
paulson@3444
   608
qed "YM3_auth_B_to_A";
paulson@3444
   609
paulson@3444
   610
paulson@3444
   611
(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)
paulson@3444
   612
paulson@3444
   613
(*Assuming the session key is secure, if both certificates are present then
paulson@3444
   614
  A has said NB.  We can't be sure about the rest of A's message, but only
paulson@3444
   615
  NB matters for freshness.*)  
paulson@3444
   616
goal thy 
paulson@3519
   617
 "!!evs. evs : yahalom                                             \
paulson@3683
   618
\        ==> Key K ~: analz (spies evs) -->                     \
paulson@3683
   619
\            Crypt K (Nonce NB) : parts (spies evs) -->         \
paulson@3683
   620
\            Crypt (shrK B) {|Agent A, Key K|} : parts (spies evs) --> \
paulson@3683
   621
\            B ~: bad -->                                         \
paulson@3683
   622
\            (EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs)";
paulson@3519
   623
by (parts_induct_tac 1);
paulson@3444
   624
(*Fake*)
paulson@3444
   625
by (Fake_parts_insert_tac 1);
paulson@3444
   626
(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
paulson@4238
   627
by (fast_tac (claset() addSDs [Crypt_imp_keysFor] addss (simpset())) 1); 
paulson@3444
   628
(*YM4: was Crypt K (Nonce NB) the very last message?  If not, use ind. hyp.*)
wenzelm@4091
   629
by (asm_simp_tac (simpset() addsimps [ex_disj_distrib]) 1);
paulson@3444
   630
(*yes: apply unicity of session keys*)
paulson@3683
   631
by (not_bad_tac "Aa" 1);
wenzelm@4091
   632
by (blast_tac (claset() addSEs [MPair_parts]
paulson@4238
   633
                        addSDs [A_trusts_YM3, B_trusts_YM4_shrK]
paulson@4238
   634
		        addDs  [Says_imp_spies RS parts.Inj,
paulson@4238
   635
				unique_session_keys]) 1);
paulson@3444
   636
val lemma = normalize_thm [RSspec, RSmp] (result()) |> standard;
paulson@3444
   637
paulson@3444
   638
(*If B receives YM4 then A has used nonce NB (and therefore is alive).
paulson@3444
   639
  Moreover, A associates K with NB (thus is talking about the same run).
paulson@3444
   640
  Other premises guarantee secrecy of K.*)
paulson@3444
   641
goal thy 
paulson@3444
   642
 "!!evs. [| Says B Server                                                   \
paulson@3444
   643
\             {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}   \
paulson@3466
   644
\             : set evs;                                                    \
paulson@3466
   645
\           Says A' B {|Crypt (shrK B) {|Agent A, Key K|},                  \
paulson@3466
   646
\                       Crypt K (Nonce NB)|} : set evs;                     \
paulson@3466
   647
\           (ALL NA k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs);    \
paulson@3683
   648
\           A ~: bad;  B ~: bad;  evs : yahalom |]       \
nipkow@3465
   649
\        ==> EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs";
paulson@3444
   650
by (dtac B_trusts_YM4 1);
paulson@3444
   651
by (REPEAT_FIRST (eresolve_tac [asm_rl, spec]));
paulson@3683
   652
by (etac (Says_imp_spies RS parts.Inj RS MPair_parts) 1);
paulson@3444
   653
by (rtac lemma 1);
paulson@3444
   654
by (rtac Spy_not_see_encrypted_key 2);
paulson@3444
   655
by (REPEAT_FIRST assume_tac);
wenzelm@4091
   656
by (blast_tac (claset() addSEs [MPair_parts]
paulson@4238
   657
	       	        addDs [Says_imp_spies RS parts.Inj]) 1);
paulson@3444
   658
qed_spec_mp "YM4_imp_A_Said_YM3";