We study the discrete logarithm problem for the multiplicative group and for elliptic curves over a finite field by using a lifting of the corresponding object to an algebraic number field and global duality. We introduce the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of the ramification a...

In this paper, at first the elemantary and basic concepts of multiplicative discrete and continous differentian and integration introduced. Then for these kinds of differentiation invariant functions the general solution of discrete and continous multiplicative differential equations will be given. Finaly a vast class of difference equations with variable coefficients and nonlinear difference e...

The discrete logarithm problem forms the basis of numerous cryptographic systems. The most eeective attack on the discrete logarithm problem in the multiplicative group of a nite eld is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller 23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a deta...

We describe a polymorphic, typed lambda calculus with substructural features. This calculus extends the first-order substructural lambda calculus αλ associated with bunched logic. A particular novelty of our new calculus is the substructural treatment of second-order variables. This is accomplished through the use of bunches of type variables in typing contexts. Both additive and multiplicative...

We introduce a method for reconstructing macroscopic models of one-dimensional stochastic processes with long-range correlations from sparsely sampled time series by combining fractional calculus and discrete-time Langevin equations. The is illustrated the ARFIMA(1,d,0) process nonlinear autoregressive toy model multiplicative noise. reconstruct daily mean temperature data recorded at Potsdam, ...