The Divisibility of Divisor Functions

A REMARK CONCERNING m-DIVISIBILITY AND THE DISCRETE LOGARITHM IN THE DIVISOR CLASS GROUP OF CURVES

The aim of this paper is to show that the computation of the discrete logarithm in the m-torsion part of the divisor class group of a curve X over a finite field ko (with char(/co) prime to m), or over a local field k with residue field /co , can be reduced to the computation of the discrete logarithm in /co(Cm)* • For this purpose we use a variant of the (tame) Täte pairing for Abelian varieti...

The Fourier Transform of Functions of the Greatest Common Divisor

We study discrete Fourier transformations of functions of the greatest common divisor: n ∑ k=1 f((k, n)) · exp( − 2πikm/n). Euler’s totient function: φ(n) = n ∑ k=1 (k, n) · exp(−2πik/n) is an example. The greatest common divisor (k, n) = n ∑ m=1 exp(2πikm/n) · ∑ d|n cd(m) d is another result involving Ramanujan’s sum cd(m). The last equation, interestingly, can be evaluated for k in the comple...

A divisibility criterion for exceptional APN functions

We are interested in the functions from F2m to itself which are Almost Perfectly Nonlinear over infinitely many extensions of F2, namely, the exceptional APN functions. In particular, we study the case of the polynomial functions of degree 4e with e odd and we give a necessary condition on an associated multivariate polynomial for the function to be exceptional APN. We use this condition to con...

Modular Divisor Functions and Quadratic Reciprocity

Some Asymptotic Formulas on Generalized Divisor Functions

1 . Throughout this paper, we use the following notation : c•1 , c2 , . . ., X0 , X1 , . . . denote positive absolute constants. We denote the number of elements of the finite set S by BSI . We write ex =exp (x) . We denote the least prime factor of n by p(n) . We write pall n if pain but pa+1 f n . v(n) denotes the number of the distinct prime factors of n, while the number of all the prime fa...