On the Fourier Transform of the Greatest Common Divisor

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...

Discrete Ramanujan-Fourier Transform of Even Functions (mod $r$)

An arithmetical function f is said to be even (mod r) if f (n) = f ((n, r)) for all n ∈ Z + , where (n, r) is the greatest common divisor of n and r. We adopt a linear algebraic approach to show that the Discrete Fourier Transform of an even function (mod r) can be written in terms of Ramanujan's sum and may thus be referred to as the Discrete Ramanujan-Fourier Transform.

Computational Number Theory and Applications to Cryptography

• Greatest common divisor (GCD) algorithms. We begin with Euclid’s algorithm, and the extended Euclidean algorithm [2, 12]. We will then discuss variations and improvements such as Lehmer’s algorithm [14], the binary algorithms [12], generalized binary algorithms [20], and FFT-based methods. We will also discuss how to adapt GCD algorithms to compute modular inverses and to compute the Jacobi a...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

On the Computation of the GCD of 2-D Polynomials

An interesting problem of algebraic computation is the computation of the greatest common divisor (GCD) of a set of polynomials. The GCD is usually linked with the characterisation of zeros of a polynomial matrix description of a system. The problem of finding the GCD of a set of n polynomials on R[x] of a maximal degree q is a classical problem that has been considered before (Karcanias et al....