For a given set of data points lying on a low-dimensional manifold embedded in a high-dimensional space, the dimensionality reduction is to recover a low-dimensional parametrization from the data set. The recently developed Hessian Eigenmaps is a mathematically rigorous method that also sets a theoretical framework for the nonlinear dimensionality reduction problem. In this paper, we develop a ...

Lie group theory states that knowledge of a m-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by m the number of equations. We apply this principle by finding some affine derivations that induces expanded Lie point symmetries of considered system. By rewriting original problem in an invariant coordinates set for these symmetries, we reduce...

We present an algorithm that speeds exponentiation on a general elliptic curve by an estimated 3.8% to 8.5% over the best known general exponentiation methods when using affine coordinates. This is achieved by eliminating a field multiplication when we compute 2P + Q from given points P , Q on the curve. We give applications to simultaneous multiple exponentiation and to the Elliptic Curve Meth...

A control system q̇ = f(q, u) is said to be trivializable if there exists local coordinates in which the system is feedback equivalent to a control system of the form q̇ = f(u). In this paper we characterize trivializable control systems and control systems for which, up to a feedback transformation, f and ∂f/∂u commute. Characterizations are given in terms of feedback invariants of the system (i...

Bilinear maps, or pairings, initially proposed in a cryptologic context for cryptanalytic purposes, proved afterward to be an amazingly flexible and useful tool for the construction of cryptosystems with unique features. Yet, they are notoriously hard to implement efficiently, so that their effective deployment requires a careful choice of parameters and algorithms. In this paper we review the ...

This paper proves that the Montgomery form elliptic curves that are cheaply transformable into short Weierstrass form by a simple change of variables (x, y) 7→ (x+α, y) (instead of a more general affine change of variables) are precisely the curves with B = 1. The points of order 2 and 4 on these curves are described, and it is observed that the x-coordinates of these points are consecutive fie...

This paper focuses on algorithms for an efficient scalar multiplication. It proposes two algorithms for computing points of the form 2kP in affine coordinates. One works for k = 2, and the other works for an arbitrary natural number k. The efficiency of these algorithms is based on a trade-off between a field inversion and several field multiplications. Montgomery trick is used to implement thi...

Let $$f_1(n), \ldots , f_k(n)$$ be polynomial functions of n. For fixed $$n\in \mathbb {N}$$ let $$S_n\subseteq the numerical semigroup generated by $$f_1(n),\ldots ,f_k(n)$$ . As n varies, we show that many invariants $$S_n$$ are eventually quasi-polynomial in n, most notably Betti numbers, but also type, genus, and size $$\Delta $$ -set. The tool use is expressibility logical system parametri...

Now let (x, y, z) be a representative of a class in E and suppose z 6= 0 then (xz , y z , 1) is the unique representative of [(x, y, z)], the class of (x, y, z), in the plane z = 1. So we can think of projective points on E with nonzero z-coordinate as corresponding bijectively to solutions (x, y) of y = x + ax + b. Now suppose z = 0, then [(x, y, z)] ∈ E implies that x = 0, so there is exactly...

For a class of nonlinear systems we investigate the problem of under what conditions there exists a coordinate transformation that yields a state affine linear system up to output injection with implicit outputs. In particular, we provide necessary and sufficient conditions for time-varying linearization up to multi-output injection. We highlight that if the conditions hold, as a consequence, i...