Group Rings Let G = g1, g2, . . . , gn be a finite group, and let k be a field. We define the group ring k[G] to be the set of sums of the form a1g1 + a2g2 + · · ·+ angn with each ai ∈ k and gi ∈ G. Addition is defined componentwise, i.e. (a1g1 + a2g2 + · · ·+ angn) + (b1g1 + b2g2 + · · ·+ bngn) = ((a1 + b1)g1 + (a2 + b2)g2 + · · ·+ (an + bn)gn). We define multiplication in the following way: (...

An improved base-f expansion method is proposed, in which the bit-length of coefficients is shorter and the number of coefficients is smaller than in Kobayashi’s expansion method. The proposed method meshes well with efficient multi-exponentiation algorithms. In addition, two efficient algorithms based on the proposed expansion method, named f-wNAF and f-SJSF, are presented which significantly ...

In this paper we describe an efficient software implementation of characteristic 2 fields making extensive use of vector instruction sets commonly found in desktop processors. Field elements are represented in a split form so performance-critical field operations can be formulated in terms of simple operations over 4-bit sets. In particular, we detail techniques for implementing field multiplic...

The nonlocal enhancement in the velocities of charge carriers to ionization is shown to outweigh the opposing effects of dead space, increasing the avalanche speed of short avalanche photodiodes (APDs) over the predictions of a conventional local model which ignores both of these effects. The trends in the measured gain–bandwidth product of two short InAlAs APDs reported in the literature suppo...

A hardware algorithm is proposed for improving the speed of the linear digitrecurrence logarithmic algorithm. The convergence rate of this logarithmic algorithm is exponential. Furthermore, the size of the lookup tables used in the algorithm is smaller than the size of the lookup tables used in the digit-recurrence algorithms. When the word length of the operand is less than or equal to 64 bits...

We generalize Abrahamse’s interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixedpoint subalgebra of H∞ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality ...