%
\begin{isabellebody}%
\def\isabellecontext{Message}%
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\isamarkupsection{Agents and Messages%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
All protocol specifications refer to a syntactic theory of messages. 
Datatype
\isa{agent} introduces the constant \isa{Server} (a trusted central
machine, needed for some protocols), an infinite population of
friendly agents, and the~\isa{Spy}:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{datatype}\isamarkupfalse%
\ agent\ {\isacharequal}\ Server\ {\isacharbar}\ Friend\ nat\ {\isacharbar}\ Spy%
\begin{isamarkuptext}%
Keys are just natural numbers.  Function \isa{invKey} maps a public key to
the matching private key, and vice versa:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{types}\isamarkupfalse%
\ key\ {\isacharequal}\ nat\isanewline
\isacommand{consts}\isamarkupfalse%
\ invKey\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}key\ {\isasymRightarrow}\ key{\isachardoublequoteclose}%
\isadelimproof
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\begin{isamarkuptext}%
Datatype
\isa{msg} introduces the message forms, which include agent names, nonces,
keys, compound messages, and encryptions.%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{datatype}\isamarkupfalse%
\isanewline
\ \ \ \ \ msg\ {\isacharequal}\ Agent\ \ agent\isanewline
\ \ \ \ \ \ \ \ \ {\isacharbar}\ Nonce\ \ nat\isanewline
\ \ \ \ \ \ \ \ \ {\isacharbar}\ Key\ \ \ \ key\isanewline
\ \ \ \ \ \ \ \ \ {\isacharbar}\ MPair\ \ msg\ msg\isanewline
\ \ \ \ \ \ \ \ \ {\isacharbar}\ Crypt\ \ key\ msg%
\begin{isamarkuptext}%
\noindent
The notation $\comp{X\sb 1,\ldots X\sb{n-1},X\sb n}$
abbreviates
$\isa{MPair}\,X\sb 1\,\ldots\allowbreak(\isa{MPair}\,X\sb{n-1}\,X\sb n)$.

Since datatype constructors are injective, we have the theorem
\begin{isabelle}%
Crypt\ K\ X\ {\isacharequal}\ Crypt\ K{\isacharprime}\ X{\isacharprime}\ {\isasymLongrightarrow}\ K\ {\isacharequal}\ K{\isacharprime}\ {\isasymand}\ X\ {\isacharequal}\ X{\isacharprime}%
\end{isabelle}
A ciphertext can be decrypted using only one key and
can yield only one plaintext.  In the real world, decryption with the
wrong key succeeds but yields garbage.  Our model of encryption is
realistic if encryption adds some redundancy to the plaintext, such as a
checksum, so that garbage can be detected.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsection{Modelling the Adversary%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
The spy is part of the system and must be built into the model.  He is
a malicious user who does not have to follow the protocol.  He
watches the network and uses any keys he knows to decrypt messages.
Thus he accumulates additional keys and nonces.  These he can use to
compose new messages, which he may send to anybody.  

Two functions enable us to formalize this behaviour: \isa{analz} and
\isa{synth}.  Each function maps a sets of messages to another set of
messages. The set \isa{analz\ H} formalizes what the adversary can learn
from the set of messages~$H$.  The closure properties of this set are
defined inductively.%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse%
\isanewline
\ \ analz\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}msg\ set\ {\isasymRightarrow}\ msg\ set{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{for}\ H\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}msg\ set{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{where}\isanewline
\ \ \ \ Inj\ {\isacharbrackleft}intro{\isacharcomma}simp{\isacharbrackright}\ {\isacharcolon}\ {\isachardoublequoteopen}X\ {\isasymin}\ H\ {\isasymLongrightarrow}\ X\ {\isasymin}\ analz\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ Fst{\isacharcolon}\ \ \ \ \ {\isachardoublequoteopen}{\isasymlbrace}X{\isacharcomma}Y{\isasymrbrace}\ {\isasymin}\ analz\ H\ {\isasymLongrightarrow}\ X\ {\isasymin}\ analz\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ Snd{\isacharcolon}\ \ \ \ \ {\isachardoublequoteopen}{\isasymlbrace}X{\isacharcomma}Y{\isasymrbrace}\ {\isasymin}\ analz\ H\ {\isasymLongrightarrow}\ Y\ {\isasymin}\ analz\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ Decrypt\ {\isacharbrackleft}dest{\isacharbrackright}{\isacharcolon}\ \isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}Crypt\ K\ X\ {\isasymin}\ analz\ H{\isacharsemicolon}\ Key{\isacharparenleft}invKey\ K{\isacharparenright}\ {\isasymin}\ analz\ H{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ X\ {\isasymin}\ analz\ H{\isachardoublequoteclose}%
\isadelimproof
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\begin{isamarkuptext}%
Note the \isa{Decrypt} rule: the spy can decrypt a
message encrypted with key~$K$ if he has the matching key,~$K^{-1}$. 
Properties proved by rule induction include the following:
\begin{isabelle}%
G\ {\isasymsubseteq}\ H\ {\isasymLongrightarrow}\ analz\ G\ {\isasymsubseteq}\ analz\ H\rulename{analz{\isacharunderscore}mono}\par\smallskip%
analz\ {\isacharparenleft}analz\ H{\isacharparenright}\ {\isacharequal}\ analz\ H\rulename{analz{\isacharunderscore}idem}%
\end{isabelle}

The set of fake messages that an intruder could invent
starting from~\isa{H} is \isa{synth{\isacharparenleft}analz\ H{\isacharparenright}}, where \isa{synth\ H}
formalizes what the adversary can build from the set of messages~$H$.%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse%
\isanewline
\ \ synth\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}msg\ set\ {\isasymRightarrow}\ msg\ set{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{for}\ H\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}msg\ set{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{where}\isanewline
\ \ \ \ Inj\ \ \ \ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}X\ {\isasymin}\ H\ {\isasymLongrightarrow}\ X\ {\isasymin}\ synth\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ Agent\ \ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}Agent\ agt\ {\isasymin}\ synth\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ MPair\ \ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}X\ {\isasymin}\ synth\ H{\isacharsemicolon}\ \ Y\ {\isasymin}\ synth\ H{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymlbrace}X{\isacharcomma}Y{\isasymrbrace}\ {\isasymin}\ synth\ H{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ Crypt\ \ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}X\ {\isasymin}\ synth\ H{\isacharsemicolon}\ \ Key\ K\ {\isasymin}\ H{\isasymrbrakk}\ {\isasymLongrightarrow}\ Crypt\ K\ X\ {\isasymin}\ synth\ H{\isachardoublequoteclose}%
\isadelimproof
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\endisadelimproof
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\isatagproof
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\isadelimproof
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\endisadelimproof
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\endisatagproof
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\begin{isamarkuptext}%
The set includes all agent names.  Nonces and keys are assumed to be
unguessable, so none are included beyond those already in~$H$.   Two
elements of \isa{synth\ H} can be combined, and an element can be encrypted
using a key present in~$H$.

Like \isa{analz}, this set operator is monotone and idempotent.  It also
satisfies an interesting equation involving \isa{analz}:
\begin{isabelle}%
analz\ {\isacharparenleft}synth\ H{\isacharparenright}\ {\isacharequal}\ analz\ H\ {\isasymunion}\ synth\ H\rulename{analz{\isacharunderscore}synth}%
\end{isabelle}
Rule inversion plays a major role in reasoning about \isa{synth}, through
declarations such as this one:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{inductive{\isacharunderscore}cases}\isamarkupfalse%
\ Nonce{\isacharunderscore}synth\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}Nonce\ n\ {\isasymin}\ synth\ H{\isachardoublequoteclose}%
\begin{isamarkuptext}%
\noindent
The resulting elimination rule replaces every assumption of the form
\isa{Nonce\ n\ {\isasymin}\ synth\ H} by \isa{Nonce\ n\ {\isasymin}\ H},
expressing that a nonce cannot be guessed.  

A third operator, \isa{parts}, is useful for stating correctness
properties.  The set
\isa{parts\ H} consists of the components of elements of~$H$.  This set
includes~\isa{H} and is closed under the projections from a compound
message to its immediate parts. 
Its definition resembles that of \isa{analz} except in the rule
corresponding to the constructor \isa{Crypt}: 
\begin{isabelle}%
\ \ \ \ \ Crypt\ K\ X\ {\isasymin}\ parts\ H\ {\isasymLongrightarrow}\ X\ {\isasymin}\ parts\ H%
\end{isabelle}
The body of an encrypted message is always regarded as part of it.  We can
use \isa{parts} to express general well-formedness properties of a protocol,
for example, that an uncompromised agent's private key will never be
included as a component of any message.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimproof
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\isadelimML
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\isadelimtheory
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\endisadelimtheory
\end{isabellebody}%
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