NAXOS_eCK.spthy

theory NAXOS_eCK
begin

builtin: diffie-hellman, hashing

section{* NAXOS *}

/*
 * Protocol:	NAXOS
 * Modeler: 	Cas Cremers, Benedikt Schmidt
 * Date: 	January 2012
 * Source:	"Stronger Security of Authenticated Key Exchange"
 * 		LaMacchia, Lauter, Mityagin, 2007
 * Property: 	eCK security
 *
 * Status: 	working
 */

/* Protocol rules */

/* In the description in the paper, we omitted the sorts. 
 * In this description they are made explicit.
 * '$A' is equivalent to 'A:pub'
 * '~x' is equivalent to 'x:fresh'
 */

/* Generate long-term keypair */
rule generate_ltk:
   [ Fr(~lkA) ] -->
   [ !Ltk( $A, ~lkA ), !Pk( $A, 'g'^~lkA ), Out( 'g'^~lkA ) ]

/* Initiator */
/* To formulate the responder property, we also define a SidI action for
 * the first rule. For brevity, we omitted this from the description in
 * the paper because there the responder property is not specified. */

rule Init_1:
   [ Fr( ~ekI ), !Ltk( $I, ~lkI ) ]
   --[ SidI_1(~ekI,$I,$R, 'g'^h(< '1', ~ekI, ~lkI >)) ]->
   [ Init_1( ~ekI, $I, $R, ~lkI, 'g'^h(< '1', ~ekI, ~lkI >) ),
     !Ephk(~ekI),
     Out( 'g'^h(< '1', ~ekI, ~lkI >) ) ]

rule Init_2:
   [ Init_1( ~ekI, $I, $R, ~lkI , hkI), In( Y ), !Pk( $R,'g'^~lkR ) ]
   --[SidI_2( ~ekI, $I, $R, hkI, Y,
       h( < '2', Y^~lkI, ('g'^~lkR)^h(< '1', ~ekI, ~lkI>), Y^h(< '1', ~ekI, ~lkI>), $I, $R> ) ) ]->
   [ !Sessk( ~ekI, 
       h(< '2', Y^~lkI, ('g'^~lkR)^h(< '1', ~ekI, ~lkI> ), Y^h(< '1', ~ekI, ~lkI >), $I, $R>) ) ]

/* Responder */
rule Resp_1:
   [ In( X ), Fr( ~ekR ), !Ltk($R, ~lkR), !Pk($I, 'g'^~lkI) ]
   --[ SidR_1( ~ekR, $I, $R, X, 'g'^h( < '1', ~ekR, ~lkR > ),
       h(< '2', ('g'^~lkI)^h(< '1', ~ekR, ~lkR >) ,X^~lkR, X^h(< '1', ~ekR, ~lkR >), $I, $R >) ) ]->
   [ Out( 'g'^h(<'1', ~ekR, ~lkR >) ),
     !Ephk(~ekR),
     !Sessk( ~ekR, 
       h(< '2', ('g'^~lkI)^h(<'1', ~ekR, ~lkR >) ,X^~lkR, X^h(<'1', ~ekR, ~lkR >), $I, $R >) ) ]

/* Key Reveals for the eCK model */
rule Sessk_reveal: 
   [ !Sessk(~tid, k) ]
   --[ SesskRev(~tid) ]->
   [ Out(k) ]

rule Ltk_reveal:
   [ !Ltk($A, lkA) ]
   --[ LtkRev($A) ]->
   [ Out(lkA) ]

rule Ephk_reveal:
   [ !Ephk(~ekA) ]
   --[ EphkRev(~ekA) ]->
   [ Out(~ekA) ]


/* Security properties */

/*
lemma key_agreement_reachable:
  "not (Ex #i1 #i2 ekI ekR I R k hkI hkR.
          SidI_2(ekI, I, R, hkI, hkR, k) @ i1 & SidR_1(ekR, I, R, hkI, hkR, k) @ i2)"
*/

/* An attack is valid in eCK if the session key of the test session is deduced and
   the test session is clean.
*/
lemma eCK_initiator_key:
  "not (Ex #i1 #i2 ekI I R k hkI hkR.
            SidI_2(ekI, I, R, hkI, hkR, k) @ i1 & K( k ) @ i2

            /* Not both longterm-key-reveal _and_ ephemeral-key-reveal
	     * for test thread. */
            & not(Ex #i3 #i4. LtkRev( I ) @ i3 & EphkRev( ekI ) @ i4)

            /* No session-key-reveal of test thread. */
            & not(Ex #i3. SesskRev( ekI ) @ i3 )

            /* No session-key-reveal for matching session. */
            & not(Ex #i3 #i4 ekR kpartner.
                   SidR_1( ekR,I,R,hkI,hkR,kpartner ) @i3
		   & SesskRev( ekR ) @ i4 )

            /* Not both long-term-key-reveal and ephemeral-key-reveal
	     * for matching session */
            & not(Ex #i3 #i4 #i5 ekR kpartner.
                  SidR_1( ekR,I,R,hkI,hkR,kpartner ) @i3
		  & LtkRev( R ) @ i4
		  & EphkRev( ekR ) @ i5 )

	    /* Longterm-key-reveal of partner only if there is a
	     * matching session. */
	    /* (We model eCK-wpfs, for eCK-pfs, add i1 < i3 to conclusion) */
            & (All #i3. LtkRev( R ) @ i3 ==>
                  (* (i1 < i3) | *)
	          (Ex #i4 ekR kpartner.
                      SidR_1( ekR,I,R,hkI,hkR,kpartner ) @i4)))"


/* An attack is valid in eCK if the session key of the test session is deduced and
   the test session is clean.
*/
lemma eCK_responder_key:
  "not (Ex #i1 #i2 ekR I R k hkI hkR.
            SidR_1(ekR, I, R, hkI, hkR, k) @ i1 & K( k ) @ i2

            /* Not longterm-key-reveal _and_ ephemeral-key-reveal of actor . */
            & not(Ex #i3 #i4. LtkRev( R ) @ i3 & EphkRev( ekR ) @ i4)

            /* Not session-key-reveal of test thread. */
            & not(Ex #i3. SesskRev( ekR ) @ i3 )

            /* Not session-key-reveal of partner thread. Note that we use SidI_2 here.
	       A session key reveal can only happen after SidI_2 is logged anyways.
	    */
            & not(Ex #i3 #i4 ekI kpartner.
                   SidI_2( ekI,I,R,hkI,hkR,kpartner ) @i3
		   & SesskRev( ekI ) @ i4 )

            /* If there is a partner thread, then not long-term-key-reveal and ephemeral-key-reveal. */
            & not(Ex #i3 #i4 #i5 ekI.
                  SidI_1( ekI,I,R,hkI ) @i3
		  & LtkRev( I ) @ i4
		  & EphkRev( ekI ) @ i5 )

	    /* If there is no partner thread, then there is no longterm-key-reveal for
	       the intended partner.
	       (We model eCK-wpfs, for eCK-pfs, add i1 < i3 to conclusion)
	       */
            & (All #i3. LtkRev( I ) @ i3 ==>
                  (* (i1 < i3) | *)
	          (Ex #i4 ekI.
                       SidI_1( ekI,I,R,hkI ) @i4)))"

end