Applying uni¯ed formula while computing point addition and doubling provides immunity to Elliptic Curve Cryptography (ECC) against power analysis attacks (a type of side channel attack). One of the popular techniques providing this uni¯edness is the Binary Hu® Curves (BHC) which got attention in 2011. In this paper we are presenting highly optimized architectures to implement point multiplication (PM) on the standard NIST curves over GF ð2 163 Þ and GF ð2 233 Þ using BHC. To achieve a high throughput over area ratio,¯rst of all, we have used a simpli¯ed arithmetic and logic unit. Secondly, we have reduced the time to compute PM through Double and Add algorithm. This is achieved by increasing the frequency of operation through a 2-stage pipelined architecture. The increase in clock cycles caused by consequent pipeline hazards is controlled through optimal scheduling of computations involved in PM. The synthesis results show that our designs can work up to a frequency of 377 MHz on Xilinx Virtex 7 FPGA. Moreover, the overall throughput/area ratio achieved through the adopted approach is up to 20% higher while comparing with available state-of-the-art solutions. J CIRCUIT SYST COMP Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/22/17. For personal use only. A. R. Jafri et al. 1750178-2 J CIRCUIT SYST COMP Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/22/17. For personal use only. Towards an Optimized Architecture for Uni¯ed Binary Hu® Curves 1750178-3 J CIRCUIT SYST COMP Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/22/17. For personal use only. A. R. Jafri et al. 1750178-4 J CIRCUIT SYST COMP Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/22/17. For personal use only.

We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form d dtIn the process we use the fundamental matrix solution of y = A(t)y and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation.

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Carry-free addition and borrow-free subtraction of modified signed-digit trinary numbers with optical symbolic substitution are presented. The proposed two-step and three-step algorithms can be easily implemented by using phase-only holograms, optical content-addressable memories, a multichannel correlator, or a polarization-encoded optical shadow-casting system.