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A sender wants to send messages , chosen from a set containing different possible messages, , to a receiver . Every has to pass through the hands of a dishonest messenger . Therefore and agree on a mathematical transformation and a secret parameter, or key , that will be used to produce the authenticator , which is sent together with . The key is chosen at random from a set of elements. knows and can find all elements in the set given enough time and computer resources. wants to change without suspecting. This means that must find the new anthenticator . Since can be found for any , it is obvious that will always succeed unless contains more than one element. Here it is proved that the average probability of success for is minimized if (a) contains elements and (b) each new known pair will diminish this set of solutions by a factor of . The minimum average probability will then be .