  In this paper, we derive new computational techniques for residue number
systems (RNS) based Barrett algorithm (BA). The focus of the work is an
algorithm that carries out the entire computation using only modular arithmetic
without conversion to large integers via the Chinese Remainder Theorem (CRT).
It also avoids the computationally expensive scaling-rounding operation
required in the earlier work. There are two parts to this work. First, we set
up a new BA using two constants other than powers of two. Second, a RNS based
BA is described. A complete mathematical framework is described including
proofs of the various steps in the computations and the validity of results.
Third, we present a computational algorithm for RNS based BA. Fourth, the RNS
based BA is used as a basis for new RNS based algorithms for MoM and MoE. The
applications we are dealing with are in the area of cryptography.
