Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Computer

In this thesis, we explore a topic within cryptography called Message Authentication Codes (MACs). A MAC is a collection of algorithms which allows A to send a message to B in such a way that B can be certain (with very high probability) that A did in fact originate the message. Let’s say A wants to send a message (string) M to B using a MAC. First A will run the “tag generation” algorithm on M to produce a string called the “tag.” Then A sends M along with the tag to B and B runs the “verification” algorithm to determine whether A was truly the author of M . The dominant method we use for creating MACs follows an approach first employed by Wegman and Carter [30]. For tag generation we first use a member randomly chosen from a set of hash functions called a “Universal Hash Function Family” to compress M into a smaller string, then we apply some cryptographic primitive to this smaller string to produce the tag. The verification algorithm repeats these steps on M and ensures that the tag generated matches the tag sent. We examine several known MACs using this paradigm, even when those MACs were not originally designed in this way. This approach often leads to simpler proofs of security for established algorithms. We then relate several new algorithms in the same spirit. The first set of algorithms extends the well-known CBC MAC to accept messages of varying lengths and still retain a high degree of proven security. And in the final chapter we look at a new Universal Hash Function Family called NH which leads to a MAC faster than any currently known.