Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields.Comment: 66 pages, 1 figur

A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac Delta Function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann's "complimentary" function introduced during his research on fractional derivatives

Discrete antiderivatives for functions over F_pⁿ

In the design of cryptographic functions, the properties of their discrete derivatives have to be carefully considered, as many cryptographic attacks exploit these properties. One can therefore attempt to first construct derivatives with the desired properties and then recover the function itself. Recently Suder developed an algorithm for reconstructing a function (also called antiderivative) over the finite field F2n given its discrete derivatives in up to n linearly independent directions. Pasalic et al. also presented an algorithm for determining a function over Fpn given one of its derivatives. Both algorithms involve solving a pn×pn system of linear equations; the functions are represented as univariate polynomials over Fpn. We show that this apparently high computational complexity is not intrinsic to the problem, but rather a consequence of the representation used. We describe a simpler algorithm, with quasilinear complexity, provided we work with a different representation of the functions. Namely they are polynomials in n variables over Fp in algebraic normal form (for p>2, additionally, we need to use the falling factorial polynomial basis) and the directions of the derivatives are the canonical basis of Fpn. Algorithms for other representations (the directions of the derivatives not being the canonical basis vectors or the univariate polynomials over Fpn mentioned above) can be obtained by combining our algorithm with converting between representations. However, the complexity of these conversions is, in the worst case, exponential. As an application, we develop a method for constructing new quadratic PN (Perfect Nonlinear) functions. We use an approach similar to the one of Suder, who used antiderivatives to give an alternative formulation of the methods of Weng et al. and Yu et al. for searching for new quadratic APN (Almost Perfect Nonlinear) functions