Positive solutions and J-focal points for two-point boundary value problems

Cone theory is applied to a class of two-point boundary value problems for ordinary differential equations. Criteria for the existence of extremal points are obtained. These criteria are in terms of the existence of nontrivial solutions that lie in a cone, and in terms of the spectral radius of an associated compact linear operator

Software implementation of binary elliptic curves: impact of the carry-less multiplier on scalar multiplication

The availability of a new carry-less multiplication instruction in the latest Intel desktop processors significantly accelerates multiplication in binary fields and hence presents the opportunity for reevaluating algorithms for binary field arithmetic and scalar multiplication over elliptic curves. We describe how to best employ this instruction in field multiplication and the effect on performance of doubling and halving operations. Alternate strategies for implementing inversion and half-trace are examined to restore most of their competitiveness relative to the new multiplier. These improvements in field arithmetic are complemented by a study on serial and parallel approaches for Koblitz and random curves, where parallelization strategies are implemented and compared. The contributions are illustrated with experimental results improving the state-of-the-art performance of halving and doubling-based scalar multiplication on NIST curves at the 112- and 192-bit security levels, and a new speed record for side-channel resistant scalar multiplication in a random curve at the 128-bit security level

BOUNDARY VALUE PROBLEMS FOR N-TH ORDER DIFFERENCE EQUATIONS

We are concerned with solutions to the difference equation Py(t - k) = f(t,y(t)) where (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Here, k and n are fixed integers with 0 (LESSTHEQ) k \u3c n, and the coefficients (alpha)(,i)(t) are defined on I + k where I is an interval of integers of the form a,b or a,(INFIN)). Py(t - k) = 0 is said to be disconjugate on J provided no nontrivial solution has n zeros on J. We will be interested in right and left (j,n - j)-disconjugacy as defined by Peterson. We obtain a partial factorization for P if P is right (j,n - j)-disconjugate and results relating right and left (j,n-j)-disconjugacy are given. An adjoint to Py(t) = 0 and disconjugacy properties for the adjoint equation are discussed. Next, the equation Ly(t) + p(t)y(t) = 0 is considered, where L is a disconjugate operator. This work is motivated by results of Elias for the corresponding differential equation. A theorem which bounds the number of certain types of zeros for solutions on an interval is obtained. Using this, sign conditions on p(t) are determined that will guarantee that Ly(t) + p(t)y(t) = 0 is right (k,n - k)-disconjugate, and a uniqueness theorem for solutions to certain types of bound- ary value problems is given. Several results give some properties of solutions to this difference equation. Furthermore, a classification of solutions is obtained based on their behavior in a neighborhood of infinity. Finally, we consider the nonlinear equation Py(t - k) = f(t,y(t)). A comparison theorem for solutions of related linear inequalities is obtained. This leads to some disconjugacy results. The Brouwer invariance of domain theorem is used to establish some results on the (j,n - j)-problem and its related variational equation. Then we show that under suitable conditions on f and certain related linear equations that the (n - 2,2)-boundary value problem has a unique solution

A focal boundary value problem for difference equations

The eigenvalue problem in difference equations, (−1)n−kΔny(t)=λ∑i=0k−1pi(t)Δiy(t), with Δty(0)=0, 0≤i≤k, Δk+iy(T+1)=0, 0≤i<n−k, is examined. Under suitable conditions on the coefficients pi, it is shown that the smallest positive eigenvalue is a decreasing function of T. As a consequence, results concerning the first focal point for the boundary value problem with λ=1 are obtained