This paper considers the normalization of Miller functions for computing “point-evaluation” pairings on an elliptic curve E over a finite field Fq, where the characteristic of Fq is neither 2 nor 3. It is shown that the normalized Miller functions for computing point-evaluation pairings on G2G1 when (i) the embedding degree k is even, or (ii) 3|k and E/Fq(q ≡ (1 mod 3)) is a curve of the form Y2=X3+b. Thus, there is no need to consider the normalization for computing pairings on many pairing-friendly elliptic curves.

A new class of nonlinear filters called Vector Median Rational Hybrid Filters (VMRHF) for multispectral image processing is introduced and applied to color image filtering problems. These filters are based on Rational Functions (RF). The VMRHF filter is a two-stage filter, which exploits the features of the vector median filter and those of the rational operator. The filter output is a result of vector rational function operating on the output of three sub-functions. Two vector median (VMF) sub-filters and one center weighted vector median filter (CWVMF) are proposed to be used here due to their desirable properties, such as, edge and details preservation and accurate chromaticity estimation. Experimental results show that the new VMRHF outperforms a number of widely known nonlinear filters for multispectral image processing such as the Vector Median ilter (VMF) and Distance Directional Filters (DDf) with respect to all criteria used.

The transfer function of IIR all-pass filters is a rational function of ω. However, the optimization of such a rational function using the successive projections method, which has a wider range of application than the Remez algorithm, has not been presented. In this paper, we propose designing IIR all-pass filters using the successive projections method.