We will show that the normal CM-fields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤ 96, and the CM-fields with class number one are of degrees ≤ 104. By many authors all normal CM-fields of degrees ≤ 96 with class number one are known except for the possib...

Let N be an imaginary abelian number field. We know that hN , the relative class number of N , goes to infinity as fN , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. I...

We introduce unbiased estimators for the Shannon entropy and the class number, in the situation that we are able to take sequences of independent samples of arbitrary length. Introduction. This paper supposes that we may pick a sequence of arbitrary length of independent samples w1, w2, . . . from an infinite population. Each sample belongs to one of M classes C1, C2, . . . , CM , and the proba...

A central problem in number theory and algebraic geometry is the determination of the size of the group of rational points on the Jacobian of an algebraic curve over a finite field. This question also has applications to cryptography, since cryptographic systems based on algebraic curves generally require a Jacobian of non-smooth order in order to foil certain types of attacks. There a variety ...

With the widespread use of communication technologies, cryptosystems are therefore critical to guarantee security over open networks as the Internet. Pseudo-random number generators (PRNGs) are fundamental in cryptosystems and information hiding schemes. One of the existing chaos-based PRNGs is using chaotic iterations schemes. In prior literature, the iterate function is just the vectorial boo...

For any positive integer n, let SL(n) denotes the least positive integer k such that L(k) ≡ 0 (mod n), where L(k) denotes the Least Common Multiple of all integers from 1 to k. The main purpose of this paper is to study the properties of the Smarandache LCM function SL(n), and give an asymptotic formula for its mean value.