(** * Example of manipulation of inductive definitions *)
(** Needham-Schroeder protocol *)
(** simplified bversion without a server for distributing public keys *)
(** ** Basic data types *)
(** - Users;
- Agents : A is Alice, B is Bob, I is the intruder
- Keys and data: keys will be asymmetric and public;
- Messages: the keys, or names of agents,
or nonces generated by one agent for another;
messages can be encoded with the public key of an agent and paired.
*)
Inductive agent : Set := A | B | I .
Inductive message : Set :=
Name : agent -> message
| Nonce : agent * agent -> message
| SK : agent -> message
| Enc : message -> agent -> message
| P : message -> message -> message.
(** ** Description of the protocol *)
(**
Three mutually inductively defined relations:
- [send Y m] when [Y] send a message [m] encodes the protocol
-- A --> B : {NA,A} pk(B)
-- B --> A : {NA,NB} pk(A)
-- A --> B : {NB} pk(B)
The intruder [I] can send any message that (s)he knows;
- [receive Z m] when [Z] receive the message m
(ie, the message was sent by someone);
- [known m] when the message [m] is known from [I], ie
it was intercepted by [I] or deducible from informations received by [I].
*)
Section Protocol_with_flaw.
(** A global parameter [X] represents the agent with which [A] initiates the protocol *)
Variable X:agent.
(** [A] and [B] follow the protocol *)
Inductive send : agent -> message -> Prop :=
init : send A (Enc (P (Nonce (A,X)) (Name A)) X)
| trans1 : forall Y d,
receive B (Enc (P (Nonce d) (Name Y)) B)
-> send B (Enc (P (Nonce d) (Nonce (B,Y))) Y)
| trans2 : forall d, receive A (Enc (P (Nonce (A,X)) (Nonce d)) A)
-> send A (Enc (Nonce d) X)
| cheat : forall m, known m -> send I m
with receive : agent -> message -> Prop :=
link : forall m Y Z, send Y m -> receive Z m
with known : message -> Prop :=
spy : forall m, receive I m -> known m
| name : forall a, known (Name a)
| nonce : forall Y, known (Nonce (I,Y))
| secret_KI : known (SK I)
| decomp_l : forall m m', known (P m m') -> known m
| decomp_r : forall m m', known (P m m') -> known m'
| compose : forall m m', known m -> known m' -> known (P m m')
| crypt : forall m a, known m -> known (Enc m a)
| decrypt : forall m a, known (Enc m a) -> known (SK a) -> known m.
End Protocol_with_flaw.
Hint Resolve init trans1 link secret_KI
decomp_l decomp_r compose cheat spy name.
(** The protocol can end with [B] receiving the message [(Enc (Nonce (B,A)) B)]
while the protocol was initiated with [I].
*)
Lemma flaw : receive I B (Enc (Nonce (B,A)) B).
apply link with I.
apply cheat.
apply crypt; auto.
apply decrypt with I; auto.
apply spy.
apply link with A.
apply trans2.
apply link with I.
apply cheat.
apply spy.
apply link with B.
apply trans1.
apply link with I.
apply cheat.
apply crypt; auto.
apply decrypt with I; auto.
apply spy.
apply link with A.
apply init.
Qed.
(** The nonce generated by [B] for [A] is made public *)
Lemma flawB : known I (Nonce (B,A)).
apply decrypt with I; auto.
apply spy.
apply link with A.
apply trans2.
apply link with I.
apply cheat.
apply spy.
apply link with B.
apply trans1.
apply link with I.
apply cheat.
apply crypt; auto.
apply decrypt with I; auto.
apply spy.
apply link with A.
apply init.
Qed.