VLSI architectures for computations in finite rings and fields

The continuing advances in very large scale integration technology permit the realization of ever more complex functions and algorithms on silicon. However, increased complexity of algorithms will have a strong impact on the efficiency of integrated solutions with respect to area requirements and data throughput rate. Thus, a well-suited description of algorithms which exploit inherent parallelism and facil¬ itates the derivation of chip architectures and building blocks becomes more important. Algorithms based on operations in finite algebraic systems are typ¬ ical examples of involved functions coming to the fore. Such algo¬ rithms play a central role, among others, in the domains of digital signal processing, in coding theory, and in cryptography. The goal of this dissertation is to give a comprehensive treatment of the four basic arithmetic operations in finite algebraic rings and fields with respect to very large scale integration. The conditions for a successful integration of algorithms based on such operations are in¬ vestigated. To evaluate the resulting architectures, metrics are defined which allow a fair comparison of different implementation techniques. For many of the operations presented, new schemes are proposed. The investigations are performed not only theoretically. Apply¬ ing the proposed architectures, many algorithms based on finite field arithmetic can be implemented very efficiently on application-specific integrated circuits. This has been verified by student designs which have been integrated, fabricated, and successfully tested. The most