chapter {* Analysing DenningSacco *} (* *********************************** This file is automatically generated from the AnB file "AnB/Denning-Sacco.AnB". Backend: Open Source Fixedpoint Model Checker version 2009c ************************************ *) theory "Denning-Sacco" imports "../src/ofmc" begin section {* Protocol Model (DenningSacco) *} datatype Role = rA | rB | rs datatype Purpose = purposeNI | purposeT | purposetimestamp | purposePayload datatype Agent = honest nat | dishonest nat datatype Nonce = "ni" "nat" | "timestamp" "nat" | "payload" "Msg" "nat" | "NI" | "T" | "Payload" and Msg = Nonce "Nonce" | Agent "Agent" | Purpose "Purpose" | pair "Msg*Msg" | scrypt "Msg*Msg" | crypt "Msg*Msg" | inv "Msg" | SID "nat" | Step "nat" | authentication | secrecy (* SymKeys *) | SymKey "Msg" | "sk" "Msg" "nat" (* Functions *) datatype Fact = Iknows Msg | State "Role * (Msg list)" | Secret "Msg * Msg" | Attack "Msg" | Witness "Msg * Msg * Msg * Msg" | Request "Msg * Msg * Msg * Msg * Msg" section {* Inductive Protocol Definition (DenningSacco) *} inductive_set DenningSacco::"Fact list set" where init_0: "[ Iknows(Nonce((ni Abs_NI)))] : DenningSacco" | init_1: "[ Iknows(Agent(dishonest(i)))] : DenningSacco" | init_2: "[ State(rA, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), SID(sid)] )] : DenningSacco" | init_3: "[ State(rA, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), SID(sid)] )] : DenningSacco" | init_4: "[ Iknows(Step(0))] : DenningSacco" | init_5: "[ Iknows(Nonce((timestamp Abs_T)))] : DenningSacco" | init_6: "[ Iknows(SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)))] : DenningSacco" | init_7: "[ Iknows(SID(sid))] : DenningSacco" | init_8: "[ Iknows(Agent(honest(a)))] : DenningSacco" | init_9: "[ State(rB, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), SID(sid)] )] : DenningSacco" | init_10: "[ State(rB, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), SID(sid)] )] : DenningSacco" | init_11: "[ State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), Agent(dishonest(i)), SID(sid)] )] : DenningSacco" | init_12: "[ State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), Agent(honest(a)), SID(sid)] )] : DenningSacco" | init_13: "[ State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), Agent(dishonest(i)), SID(sid)] )] : DenningSacco" | init_14: "[ State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), Agent(honest(a)), SID(sid)] )] : DenningSacco" | rule_0: "[| t :DenningSacco; Iknows(crypt(K, M)) : (set t); Iknows(inv(K)) : (set t)|] ==> ((Iknows(M)) #t) : DenningSacco" | rule_1: "[| t :DenningSacco; Iknows(crypt(inv(K), M)) : (set t); Iknows(K) : (set t)|] ==> ((Iknows(M)) #t) : DenningSacco" | rule_2: "[| t :DenningSacco; Iknows(scrypt(K, M)) : (set t); Iknows(K) : (set t)|] ==> ((Iknows(M)) #t) : DenningSacco" | rule_3: "[| t :DenningSacco; Iknows(pair(M1, M2)) : (set t)|] ==> ((Iknows(M1)) #(Iknows(M2)) #t) : DenningSacco" | rule_4: "[| t :DenningSacco; Secret(M, Agent(honest(a))) : (set t); Iknows(M) : (set t)|] ==> ((Attack(pair(secrecy, M))) #t) : DenningSacco" | rule_5: "[| t :DenningSacco; Request(A, B, Purpose(purposePayload), M, SID(sid)) : (set t); ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_Payload . M = Nonce((payload(pair(B, A)) Abs_Payload)))|] ==> ((Attack(pair(authentication, pair(A, pair(B, M))))) #t) : DenningSacco" | rule_6: "[| t :DenningSacco; Request(A, B, Purpose(purposeKAB), M, SID(sid)) : (set t); ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_KAB . M = SymKey((sk(pair(B, A)) Abs_KAB)))|] ==> ((Attack(pair(authentication, pair(A, pair(B, M))))) #t) : DenningSacco" | rule_7: "[| t :DenningSacco; State(rA, [Agent(A), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), SID(sid)] ) : (set t)|] ==> ((State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] )) #(Iknows(pair(Agent(A), Agent(B)))) #t) : DenningSacco" | rule_8: "[| t :DenningSacco; State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), Agent(A), SID(sid)] ) : (set t); Iknows(Agent(A)) : (set t); Iknows(Agent(B)) : (set t)|] ==> ((State(rs, [Agent(honest(a)), Step(1), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), Agent(A), pair(Agent(A), Agent(B)), SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T))))))))), SID(sid)] )) #(Iknows(scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T))))))))))) #t) : DenningSacco" | rule_9: "[| t :DenningSacco; State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] ) : (set t); Iknows(scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i))))))) : (set t)|] ==> ((State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] )) #(Iknows(Agent(dishonest(i)))) #t) : DenningSacco" | rule_10: "[| t :DenningSacco; State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] ) : (set t); Iknows(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB))) : (set t); Iknows(Agent(B)) : (set t); Iknows(SymKey(KAB)) : (set t); Iknows(Nonce(T)) : (set t); Iknows(Agent(dishonest(i))) : (set t)|] ==> ((State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] )) #(Iknows(Agent(dishonest(i)))) #t) : DenningSacco" | rule_11: "[| t :DenningSacco; State(rB, [Agent(B), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), SID(sid)] ) : (set t); Iknows(scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T))))) : (set t)|] ==> ((Secret(Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), Agent(A))) #(Witness(Agent(B), Agent(A), Purpose(purposePayload), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)))) #(State(rB, [Agent(B), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T)))), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))), SID(sid)] )) #(Iknows(scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))))) #t) : DenningSacco" | rule_12: "[| t :DenningSacco; State(rB, [Agent(B), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), SID(sid)] ) : (set t); Iknows(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB))) : (set t); Iknows(Agent(A)) : (set t); Iknows(SymKey(KAB)) : (set t); Iknows(Nonce(T)) : (set t)|] ==> ((Secret(Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), Agent(A))) #(Witness(Agent(B), Agent(A), Purpose(purposePayload), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)))) #(State(rB, [Agent(B), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T)))), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))), SID(sid)] )) #(Iknows(scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))))) #t) : DenningSacco" | rule_13: "[| t :DenningSacco; State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : (set t); Iknows(scrypt(SymKey(KAB), Nonce(Payload))) : (set t)|] ==> ((Request(Agent(A), Agent(B), Purpose(purposePayload), Nonce(Payload), SID(sid))) #(State(rA, [Agent(A), Step(3), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), Nonce(Payload), scrypt(SymKey(KAB), Nonce(Payload)), SID(sid)] )) #t) : DenningSacco" | rule_14: "[| t :DenningSacco; State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : (set t); Iknows(SymKey(KAB)) : (set t); Iknows(Nonce(Payload)) : (set t)|] ==> ((Request(Agent(A), Agent(B), Purpose(purposePayload), Nonce(Payload), SID(sid))) #(State(rA, [Agent(A), Step(3), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), Nonce(Payload), scrypt(SymKey(KAB), Nonce(Payload)), SID(sid)] )) #t) : DenningSacco" section {* Fixed-point Definition (DenningSacco) *} definition "DenningSacco_fp = {m. ( ? Abs_NI0 i8 i9 i10 i11 i12 i13 i14 i15 a10 a11 a12 a13 a14 a15 a16 a17 Abs_KAB5 a18 a19 a20 Abs_KAB6 i7 a5 a6 Abs_KAB3 Abs_T2 a7 a8 Abs_KAB4 a9 Abs_Payload1 sid0 Abs_Payload0 i3 i4 i5 i6 a0 a1 a2 a3 Abs_KAB1 a4 Abs_KAB2 Abs_T1 i0 i1 i2 Abs_KAB0 Abs_T0 . (m = Iknows(Nonce((ni Abs_NI0)))) | (m = Iknows(Agent(dishonest(i0)))) | (m = State(rA, [Agent(honest(a0)), Step(0), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(dishonest(i0)), SID(sid0)] )) | (m = State(rA, [Agent(honest(a0)), Step(0), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(honest(a3)), SID(sid0)] )) | (m = Iknows(Step(0))) | (m = Iknows(Nonce((timestamp Abs_T0)))) | (m = Iknows(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)))) | (m = Iknows(SID(sid0))) | (m = Iknows(Agent(honest(a0)))) | (m = State(rB, [Agent(honest(a0)), Step(0), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(dishonest(i0)), SID(sid0)] )) | (m = State(rB, [Agent(honest(a0)), Step(0), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(honest(a3)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Agent(dishonest(i2)), Agent(dishonest(i3)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), Agent(dishonest(i1)), Agent(honest(a4)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a3)))) Abs_KAB1)), Agent(honest(a4)), Agent(dishonest(i1)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), SymKey((sk(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_KAB1)), Agent(honest(a5)), Agent(honest(a6)), SID(sid0)] )) | (m = Iknows(pair(Agent(honest(a0)), Agent(honest(a1))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(dishonest(i0))))) | (m = State(rA, [Agent(honest(a0)), Step(1), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(dishonest(i0)), pair(Agent(honest(a3)), Agent(dishonest(i1))), SID(sid0)] )) | (m = State(rA, [Agent(honest(a0)), Step(1), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(honest(a3)), pair(Agent(honest(a4)), Agent(honest(a5))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Agent(dishonest(i2)), Agent(dishonest(i3)), pair(Agent(dishonest(i4)), Agent(dishonest(i5))), SymKey((sk(pair(Agent(dishonest(i6)), Agent(dishonest(i7)))) Abs_KAB2)), Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i8)), Agent(honest(a3)))) Abs_KAB3)), pair(Agent(dishonest(i9)), pair(SymKey((sk(pair(Agent(dishonest(i10)), Agent(dishonest(i11)))) Abs_KAB4)), pair(Nonce((timestamp Abs_T1)), scrypt(SymKey((sk(pair(Agent(dishonest(i12)), Agent(honest(a4)))) Abs_KAB5)), pair(Agent(dishonest(i13)), pair(SymKey((sk(pair(Agent(dishonest(i14)), Agent(dishonest(i15)))) Abs_KAB6)), Nonce((timestamp Abs_T2))))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), Agent(dishonest(i1)), Agent(honest(a4)), pair(Agent(honest(a5)), Agent(dishonest(i2))), SymKey((sk(pair(Agent(honest(a6)), Agent(dishonest(i3)))) Abs_KAB2)), Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a7)), Agent(honest(a8)))) Abs_KAB3)), pair(Agent(dishonest(i4)), pair(SymKey((sk(pair(Agent(honest(a9)), Agent(dishonest(i5)))) Abs_KAB4)), pair(Nonce((timestamp Abs_T1)), scrypt(SymKey((sk(pair(Agent(dishonest(i6)), Agent(honest(a10)))) Abs_KAB5)), pair(Agent(honest(a11)), pair(SymKey((sk(pair(Agent(honest(a12)), Agent(dishonest(i7)))) Abs_KAB6)), Nonce((timestamp Abs_T2))))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a3)))) Abs_KAB1)), Agent(honest(a4)), Agent(dishonest(i1)), pair(Agent(dishonest(i2)), Agent(honest(a5))), SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a6)))) Abs_KAB2)), Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i4)), Agent(honest(a7)))) Abs_KAB3)), pair(Agent(honest(a8)), pair(SymKey((sk(pair(Agent(dishonest(i5)), Agent(honest(a9)))) Abs_KAB4)), pair(Nonce((timestamp Abs_T1)), scrypt(SymKey((sk(pair(Agent(honest(a10)), Agent(honest(a11)))) Abs_KAB5)), pair(Agent(dishonest(i6)), pair(SymKey((sk(pair(Agent(dishonest(i7)), Agent(honest(a12)))) Abs_KAB6)), Nonce((timestamp Abs_T2))))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), SymKey((sk(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_KAB1)), Agent(honest(a5)), Agent(honest(a6)), pair(Agent(honest(a7)), Agent(honest(a8))), SymKey((sk(pair(Agent(honest(a9)), Agent(honest(a10)))) Abs_KAB2)), Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a11)), Agent(honest(a12)))) Abs_KAB3)), pair(Agent(honest(a13)), pair(SymKey((sk(pair(Agent(honest(a14)), Agent(honest(a15)))) Abs_KAB4)), pair(Nonce((timestamp Abs_T1)), scrypt(SymKey((sk(pair(Agent(honest(a16)), Agent(honest(a17)))) Abs_KAB5)), pair(Agent(honest(a18)), pair(SymKey((sk(pair(Agent(honest(a19)), Agent(honest(a20)))) Abs_KAB6)), Nonce((timestamp Abs_T2))))))))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(honest(a2)), pair(SymKey((sk(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_KAB1)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a5)), Agent(honest(a6)))) Abs_KAB2)), pair(Agent(honest(a7)), pair(SymKey((sk(pair(Agent(honest(a8)), Agent(honest(a9)))) Abs_KAB3)), Nonce((timestamp Abs_T1))))))))))) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(honest(a1)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_KAB2)), pair(Agent(dishonest(i2)), pair(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a5)))) Abs_KAB3)), Nonce((timestamp Abs_T1))))))))))) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(honest(a2)), Agent(dishonest(i1)))) Abs_KAB1)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))) Abs_KAB2)), pair(Agent(honest(a4)), pair(SymKey((sk(pair(Agent(honest(a5)), Agent(dishonest(i3)))) Abs_KAB3)), Nonce((timestamp Abs_T1))))))))))) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i4)), Agent(honest(a1)))) Abs_KAB2)), pair(Agent(dishonest(i5)), pair(SymKey((sk(pair(Agent(dishonest(i6)), Agent(dishonest(i7)))) Abs_KAB3)), Nonce((timestamp Abs_T1))))))))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(honest(a1))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(dishonest(i0))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i3)), pair(SymKey((sk(pair(Agent(dishonest(i4)), Agent(dishonest(i5)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i4)), pair(SymKey((sk(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(honest(a0)), pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a4)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), Nonce((timestamp Abs_T0))))))) | (m = Iknows(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)))) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Nonce((timestamp Abs_T0))))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(honest(a1))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(dishonest(i0))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i3)), pair(SymKey((sk(pair(Agent(dishonest(i4)), Agent(dishonest(i5)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i4)), pair(SymKey((sk(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(honest(a0)), pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a4)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), Nonce((timestamp Abs_T0))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((timestamp Abs_T0)))))) | (m = Secret(Nonce((payload(pair(Agent(honest(a0)), Agent(dishonest(i0)))) Abs_Payload0)), Agent(dishonest(i1)))) | (m = Witness(Agent(honest(a0)), Agent(dishonest(i0)), Purpose(purposePayload), Nonce((payload(pair(Agent(honest(a1)), Agent(dishonest(i1)))) Abs_Payload0)))) | (m = State(rB, [Agent(honest(a0)), Step(1), Nonce((timestamp Abs_T0)), SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB0)), Agent(dishonest(i0)), Nonce((timestamp Abs_T1)), SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a3)))) Abs_KAB1)), scrypt(SymKey((sk(pair(Agent(honest(a4)), Agent(honest(a5)))) Abs_KAB2)), pair(Agent(dishonest(i2)), pair(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a6)))) Abs_KAB3)), Nonce((timestamp Abs_T2))))), Nonce((payload(pair(Agent(honest(a7)), Agent(dishonest(i4)))) Abs_Payload0)), scrypt(SymKey((sk(pair(Agent(dishonest(i5)), Agent(honest(a8)))) Abs_KAB4)), Nonce((payload(pair(Agent(honest(a9)), Agent(dishonest(i6)))) Abs_Payload1))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), Nonce((payload(pair(Agent(honest(a1)), Agent(dishonest(i1)))) Abs_Payload0))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(honest(a1))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(dishonest(i0))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i3)), pair(SymKey((sk(pair(Agent(dishonest(i4)), Agent(dishonest(i5)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i4)), pair(SymKey((sk(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(honest(a0)), pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a4)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), Nonce((timestamp Abs_T0))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((timestamp Abs_T0)))))) | (m = Iknows(Nonce((payload(pair(Agent(honest(a0)), Agent(dishonest(i0)))) Abs_Payload0)))) | (m = Iknows(pair(Agent(honest(a0)), Agent(honest(a1))))) | (m = Iknows(pair(Agent(honest(a0)), Agent(dishonest(i0))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(dishonest(i3)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i3)), pair(SymKey((sk(pair(Agent(dishonest(i4)), Agent(dishonest(i5)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(dishonest(i3)), Agent(honest(a0)))) Abs_KAB1)), pair(Agent(dishonest(i4)), pair(SymKey((sk(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(honest(a2)))) Abs_KAB1)), Nonce((timestamp Abs_T1)))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a1)), Agent(honest(a2)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))) Abs_KAB2)), Nonce((timestamp Abs_T1))))))))) | (m = Iknows(pair(Agent(honest(a0)), pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))) Abs_KAB0)), pair(Nonce((timestamp Abs_T0)), scrypt(SymKey((sk(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB1)), pair(Agent(dishonest(i1)), pair(SymKey((sk(pair(Agent(dishonest(i2)), Agent(honest(a4)))) Abs_KAB2)), Nonce((timestamp Abs_T1)))))))))) | (m = Iknows(pair(SymKey((sk(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_KAB0)), Nonce((timestamp Abs_T0))))) | (m = Iknows(pair(Agent(dishonest(i0)), pair(SymKey((sk(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((timestamp Abs_T0)))))) )}" section {* Checking Fixed-point (DenningSacco) *} lemma fp_attack_free: "~ (Attack m : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_0: "Iknows(Nonce((ni Abs_NI))) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_1: "Iknows(Agent(dishonest(i))) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_2: "State(rA, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_3: "State(rA, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_4: "Iknows(Step(0)) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_5: "Iknows(Nonce((timestamp Abs_T))) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_6: "Iknows(SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB))) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_7: "Iknows(SID(sid)) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_8: "Iknows(Agent(honest(a))) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_9: "State(rB, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_10: "State(rB, [Agent(honest(a)), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_11: "State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), Agent(dishonest(i)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_12: "State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(dishonest(i)), Agent(honest(a)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_13: "State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(dishonest(i)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), Agent(dishonest(i)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma init_14: "State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(honest(a)), Agent(honest(a)))) Abs_KAB)), Agent(honest(a)), Agent(honest(a)), SID(sid)] ) : DenningSacco_fp" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_0: "[| Iknows(crypt(K, M)) : DenningSacco_fp; Iknows(inv(K)) : DenningSacco_fp|] ==> (Iknows(M) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_1: "[| Iknows(crypt(inv(K), M)) : DenningSacco_fp; Iknows(K) : DenningSacco_fp|] ==> (Iknows(M) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_2: "[| Iknows(scrypt(K, M)) : DenningSacco_fp; Iknows(K) : DenningSacco_fp|] ==> (Iknows(M) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_3: "[| Iknows(pair(M1, M2)) : DenningSacco_fp|] ==> (Iknows(M1) : DenningSacco_fp) & (Iknows(M2) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_4: "[| Secret(M, Agent(honest(a))) : DenningSacco_fp; Iknows(M) : DenningSacco_fp|] ==> (Attack(pair(secrecy, M)) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_5: "[| Request(A, B, Purpose(purposePayload), M, SID(sid)) : DenningSacco_fp; ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_Payload . M = Nonce((payload(pair(B, A)) Abs_Payload)))|] ==> (Attack(pair(authentication, pair(A, pair(B, M)))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_6: "[| Request(A, B, Purpose(purposeKAB), M, SID(sid)) : DenningSacco_fp; ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_KAB . M = SymKey((sk(pair(B, A)) Abs_KAB)))|] ==> (Attack(pair(authentication, pair(A, pair(B, M)))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_7: "[| State(rA, [Agent(A), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), SID(sid)] ) : DenningSacco_fp|] ==> (State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] ) : DenningSacco_fp) & (Iknows(pair(Agent(A), Agent(B))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_8: "[| State(rs, [Agent(honest(a)), Step(0), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), Agent(A), SID(sid)] ) : DenningSacco_fp; Iknows(Agent(A)) : DenningSacco_fp; Iknows(Agent(B)) : DenningSacco_fp|] ==> (State(rs, [Agent(honest(a)), Step(1), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), Agent(A), pair(Agent(A), Agent(B)), SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T))))))))), SID(sid)] ) : DenningSacco_fp) & (Iknows(scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Nonce((timestamp Abs_T)), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey((sk(pair(Agent(A), Agent(B))) Abs_KAB)), Nonce((timestamp Abs_T)))))))))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_9: "[| State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] ) : DenningSacco_fp; Iknows(scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i))))))) : DenningSacco_fp|] ==> (State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : DenningSacco_fp) & (Iknows(Agent(dishonest(i))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_10: "[| State(rA, [Agent(A), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), SID(sid)] ) : DenningSacco_fp; Iknows(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB))) : DenningSacco_fp; Iknows(Agent(B)) : DenningSacco_fp; Iknows(SymKey(KAB)) : DenningSacco_fp; Iknows(Nonce(T)) : DenningSacco_fp; Iknows(Agent(dishonest(i))) : DenningSacco_fp|] ==> (State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : DenningSacco_fp) & (Iknows(Agent(dishonest(i))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_11: "[| State(rB, [Agent(B), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), SID(sid)] ) : DenningSacco_fp; Iknows(scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T))))) : DenningSacco_fp|] ==> (Secret(Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), Agent(A)) : DenningSacco_fp) & (Witness(Agent(B), Agent(A), Purpose(purposePayload), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))) : DenningSacco_fp) & (State(rB, [Agent(B), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T)))), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))), SID(sid)] ) : DenningSacco_fp) & (Iknows(scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_12: "[| State(rB, [Agent(B), Step(0), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), SID(sid)] ) : DenningSacco_fp; Iknows(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB))) : DenningSacco_fp; Iknows(Agent(A)) : DenningSacco_fp; Iknows(SymKey(KAB)) : DenningSacco_fp; Iknows(Nonce(T)) : DenningSacco_fp|] ==> (Secret(Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), Agent(A)) : DenningSacco_fp) & (Witness(Agent(B), Agent(A), Purpose(purposePayload), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))) : DenningSacco_fp) & (State(rB, [Agent(B), Step(1), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), Agent(A), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(B), Agent(honest(a)))) Abs_KAB)), pair(Agent(A), pair(SymKey(KAB), Nonce(T)))), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)), scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload))), SID(sid)] ) : DenningSacco_fp) & (Iknows(scrypt(SymKey(KAB), Nonce((payload(pair(Agent(B), Agent(A))) Abs_Payload)))) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_13: "[| State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : DenningSacco_fp; Iknows(scrypt(SymKey(KAB), Nonce(Payload))) : DenningSacco_fp|] ==> (Request(Agent(A), Agent(B), Purpose(purposePayload), Nonce(Payload), SID(sid)) : DenningSacco_fp) & (State(rA, [Agent(A), Step(3), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), Nonce(Payload), scrypt(SymKey(KAB), Nonce(Payload)), SID(sid)] ) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_14: "[| State(rA, [Agent(A), Step(2), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), SID(sid)] ) : DenningSacco_fp; Iknows(SymKey(KAB)) : DenningSacco_fp; Iknows(Nonce(Payload)) : DenningSacco_fp|] ==> (Request(Agent(A), Agent(B), Purpose(purposePayload), Nonce(Payload), SID(sid)) : DenningSacco_fp) & (State(rA, [Agent(A), Step(3), Nonce((timestamp Abs_T)), SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), Agent(B), pair(Agent(A), Agent(B)), Agent(dishonest(i)), Nonce(T), SymKey(KAB), scrypt(SymKey((sk(pair(Agent(A), Agent(honest(a)))) Abs_KAB)), pair(Agent(B), pair(SymKey(KAB), pair(Nonce(T), Agent(dishonest(i)))))), Nonce(Payload), scrypt(SymKey(KAB), Nonce(Payload)), SID(sid)] ) : DenningSacco_fp)" by(simp only: DenningSacco_fp_def, simp only: set2pred, simp, auto?)+ section {* Security Proof(s) (DenningSacco) *} lemma over_approx: "t : DenningSacco ==> (set t) <= DenningSacco_fp" apply(rule DenningSacco.induct, simp_all) apply(propagate_fp, cut_tac init_0, (assumption | simp)+) apply(propagate_fp, cut_tac init_1, (assumption | simp)+) apply(propagate_fp, cut_tac init_2, (assumption | simp)+) apply(propagate_fp, cut_tac init_3, (assumption | simp)+) apply(propagate_fp, cut_tac init_4, (assumption | simp)+) apply(propagate_fp, cut_tac init_5, (assumption | simp)+) apply(propagate_fp, cut_tac init_6, (assumption | simp)+) apply(propagate_fp, cut_tac init_7, (assumption | simp)+) apply(propagate_fp, cut_tac init_8, (assumption | simp)+) apply(propagate_fp, cut_tac init_9, (assumption | simp)+) apply(propagate_fp, cut_tac init_10, (assumption | simp)+) apply(propagate_fp, cut_tac init_11, (assumption | simp)+) apply(propagate_fp, cut_tac init_12, (assumption | simp)+) apply(propagate_fp, cut_tac init_13, (assumption | simp)+) apply(propagate_fp, cut_tac init_14, (assumption | simp)+) apply(propagate_fp, cut_tac rule_0, (assumption | simp)+) apply(propagate_fp, cut_tac rule_1, (assumption | simp)+) apply(propagate_fp, cut_tac rule_2, (assumption | simp)+) apply(propagate_fp, cut_tac rule_3, (assumption | simp)+) apply(propagate_fp, cut_tac rule_4, (assumption | simp)+) apply(propagate_fp, cut_tac rule_5, (assumption | simp)+) apply(propagate_fp, cut_tac rule_6, (assumption | simp)+) apply(propagate_fp, cut_tac rule_7, (assumption | simp)+) apply(propagate_fp, cut_tac rule_8, (assumption | simp)+) apply(propagate_fp, cut_tac rule_9, (assumption | simp)+) apply(propagate_fp, cut_tac rule_10, (assumption | simp)+) apply(propagate_fp, cut_tac rule_11, (assumption | simp)+) apply(propagate_fp, cut_tac rule_12, (assumption | simp)+) apply(propagate_fp, cut_tac rule_13, (assumption | simp)+) apply(propagate_fp, cut_tac rule_14, (assumption | simp)+) done end (* theory *)