In $5\mathrm{.}1$ Introduction we discuss the devious secretary Bob having an automatic means of generating many messages that Alice would sign, and many messages that Bob would like to send. By the birthday problem, by the time Bob has tried a total of $2^32$ messages, he will probably have found two with the same message digest. The problem is$,$ both may be of the same type, which would not do him any good. How many messages must Bob try before it is probable that he'll have messages with matching digests, and that the messages will be of opposite types?