Library NSPK

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.
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).

The nonce generated by B for A is made public

Lemma flawB : known I (Nonce (B,A)).