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Table 15.1 Special Z symbols Symbol f: X Y f: X Y id X dom f ran f f (|X |) f g f g X \Y X Meaning Function between X and Y Relation between X and Y The identity function on X The domain of f The range of f Image of the set X under the function f Function which takes values of the function f except on the domain of g, where it takes the values of g Functional composition where the domain of g must equal the range of f Set difference of X and Y the set of sets with elements of X as values
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system components, notation from Table 15.1 known as Z notation [32] is used, resulting in [21]:
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Shared, Public, Secret, Auth, Conf : dual : Key Key Trusted : User User (Shared, Public, Secret) partition Key Auth Conf = Key dual dual = id Key dual(|Conf|) = Conf dual(|Auth|) = Auth dual(|Shared|) = Shared dual(|Public|) = Secret dual(|Secret|) = Public
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The rst ordinary schema de nes the variable that records what keys are known by each user, and with whom they are associated. Keys keys : User (User Key)
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A set of (user, key) pairs is associated to each user, where x maps to (y, k) means that x knows key k and uses it in communications with user y. The following three schemas de ne the state space of the model by giving formal de nitions of secure channels in terms of possession of keys.
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Con dentiality channels Keys ConfChannels : User User x, y : User
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{ k: Conf \ Secret; z:
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Con dentiality channels de ne relations between pairs of users. These are ordered pairs as the channel may be in only one direction. It will be noticed that the de nition is not symmetrical with respect to x and y. The predicate states that, for a con dentiality channel to exist from x to y, there must be a key whose use includes con dentiality and is either shared or public; x must associate this key with y; y must know the dual of the key. In the model, this means that y must associate it with some user(s), but it is not of any concern to y exactly which users know the key. To see why this is reasonable, consider a public key system providing con dentiality to y. Any user z who knows the public key of y has a con dentiality channel to y, and it is important to z that it is only y who has the secret dual key; y must of course know the secret key, but it does not matter to y who knows the public key. This corresponds to the viewpoint that con dentiality is a service provided to the sender of information. Authentication channels Keys AuthChannels : User User x, y : User
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Authentication channels also de ne relations between pairs of users. The de nition of authentication channels is dual to the de nition of con dentiality channels and corresponds to the viewpoint that authentication is a service provided to the receiver of information. Symmetric channels Keys SymmChannels : { (User)}
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SymmChannels {x, y : User|x = y x, y : User
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Symmetric channels are in both directions and so are de ned as sets of two different users. They correspond to the situation where neither key is public, and in practice the key and its dual are usually equal. State Con dentialityChannels AuthenticationChannels SymmetricChannels The system state is de ned exactly by what keys are known by each user, thereby de ning what secure channels exist. Transfer State orig , dest , recip , sender : User k : Key (orig , k ) keys (sender ) keys = keys {recip keys(recip ) (orig , k )} This schema says that if a key k is sent from one user to another, then the keys known to the recipient are updated to associate the key sent with the originator. In this model, the only state changes are those which happen as a result of passing keys from one user to another. Such a key exchange may or may not result in new channels being formed. The key passes from the sender to the recipient. It may be that the users between whom communication is intended (originator and destination) are different from the sender and recipient involved in a particular exchange. This is the situation if the sender is a key server. The recipient will therefore associate the received key with the originator, who may or may not be the sender. SecureTransfer Transfer k Secret (sender , recip ) ConfChannels k Public (sender , recip ) AuthChannels k Shared (sender , recip ) ConfChannels AuthChannels orig = sender k Public Shared sender Trusted(recip ) recip = dest k Secret Shared recip Trusted(sender )