Multiple Point Compression on Elliptic Curves

The paper aims at developing new point compression algorithms which are useful in mobile communication systems where Elliptic Curve Cryptography is employed to achieve secure data storage and transmission. Compression algorithms allow elliptic curve points to be represented in the form that balances the usage of memory and computational power. The two- and three-point compression algorithms developed by Khabbazian, Gulliver and Bhargava [4] are reviewed and extended to generic cases of four and five points. The proposed methods use only basic operations (multiplication, division, etc.) and avoids square root finding. In addition, a new two-point compression method which is heavy in compression phase and light in decompression is developed

A dimensionality reduction technique for unconstrained global optimization of functions with low effective dimensionality

We investigate the unconstrained global optimization of functions with low effective dimensionality, that are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in [Wang et al., Bayesian optimization in a billion dimensions via random embeddings. JAIR, 55(1): 361--387, 2016], we study a generic Random Embeddings for Global Optimization (REGO) framework that is compatible with any global minimization algorithm. Instead of the original, potentially large-scale optimization problem, within REGO, a Gaussian random, low-dimensional problem with bound constraints is formulated and solved in a reduced space. We provide novel probabilistic bounds for the success of REGO in solving the original, low effective-dimensionality problem, which show its independence of the (potentially large) ambient dimension and its precise dependence on the dimensions of the effective and randomly embedding subspaces. These results significantly improve existing theoretical analyses by providing the exact distribution of a reduced minimizer and its Euclidean norm and by the general assumptions required on the problem. We validate our theoretical findings by extensive numerical testing of REGO with three types of global optimization solvers, illustrating the improved scalability of REGO compared to the full-dimensional application of the respective solvers.Comment: 32 pages, 10 figures, submitted to Information and Inference: a journal of the IMA, also submitted to optimization-online repositor

Optimal Design of Helical Springs of Power Law Materials

In this paper the geometric dimensions of a compressive helical spring made of power law materials are optimized to reduce the amount of material. The mechanical constraints are derived to form the geometric programming problem. Both the prime and the dual problem are examined and solved semi-analytically for a range of spring index. A numerical example is provided to validate the solution

Optimal Design of Helical Springs of Power Law Materials

In this paper the geometric dimensions of a compressive helical spring made of power law materials are optimized to reduce the amount of material. The mechanical constraints are derived to form the geometric programming problem. Both the prime and the dual problem are examined and solved semi-analytically for a range of spring index. A numerical example is provided to validate the solution