Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the points that constitute the image of a fractal, and thus enables any affine transformation of the affine basis, done by click-and-drag, to be immediately followed by the same affine t...

This work introduces a novel tool for interactive, real-time transformations of two dimensional IFS fractals. We assign barycentric coordinates (relative to an arbitrary affine basis of R) to the points that constitute the image of a fractal. The tool uses some of the nice properties of the barycentric coordinates, enabling any affine transformation of the basis, done by click-and-drag, to be i...

In this paper we analyze the use of affine coordinates for pairing computation. We observe that in many practical settings, e. g. when implementing optimal ate pairings in high security levels, affine coordinates are faster than using the best currently known formulas for projective coordinates. This observation relies on two known techniques for speeding up field inversions which we analyze in...

Given a tau-function $\tau(t)$ of the BKP hierarchy satisfying $\tau(0)=1$, we discuss relation between its BKP-affine coordinates on isotropic Sato Grassmannian and BKP-wave function. Using this result, formulate type Kac-Schwarz operators for in terms coordinates. As an example, compute affine spin single Hurwitz numbers with completed cycles, find pair $(P,Q)$ $[P,Q]=1$. By doing this, obtai...

Elliptic Curve Public Key Cryptosystems (ECPKC) are becoming increasingly popular for use in mobile appliances where bandwidth and chip area are strongly constrained. For the same level of security, ECPKC use much smaller key length than the commonly used RSA. The underlying operation of affine coordinates elliptic curve point multiplication requires modular multiplication, division/inversion a...

Edwards curves have attracted great interest for several reasons. When curve parameters are chosen properly, the addition formulas use only 10M+1S. The formulas are strongly unified, i.e., work without change for doublings; even better, they are complete, i.e., work without change for all inputs. Dedicated doubling formulas use only 3M + 4S, and dedicated tripling formulas use only 9M + 4S. Thi...

Boolean function in even number of variables is called bent if it is at the maximal possible Hamming distance from the class of all affine Boolean functions. We have proven that every isometric mapping of the set of all Boolean functions into itself that transforms bent functions into bent functions is a combination of an affine transform of coordinates and an affine shift.