Explicit Bounds for Primality Testing and Related Problems

Abstract

Abstract.Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is 0(log m). This has been generalized by Lagañas. Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log m .

The average least character non-residue and further variations on a theme of Erdős

“…earlier in Section 2, shows that (#X (q)) −1 χ∈X (q) n χ = (q) + O φ(q) #X (q) (log log q) 2 log q . (3.3) To take an example of special interest, let X (q) be the set of primitive characters modulo q, and let φ * (q) = #X (q).…”

“…Under the assumption of the generalized Riemann hypothesis for Dirichlet L-functions, we know that n χ 3 log 2 q for every non-principal character χ(mod q). (As stated, this result is due to Bach [2,Theorem 3], although the first result of its kind was proved by Ankeny [1]. If q is prime, then χ is a kth power residue character for some k dividing q − 1, and the study of the maximal order of n χ goes back to Vinogradov and Linnik in the early part of the twentieth century.)…”

Improved techniques for computing the ideal class group and a system of fundamental units in number fields

“…The use of the Minkowski bound certifies the result unconditionally, but it causes the algorithm to take a time exponential in the size of . To achieve subexponentiality, many authors chose the bound of Bach [2], who proved that under GRH, Cl.ᏻ K / was generated by the classes of the prime ideals p satisfying ᏺ.p/ Ä 12.log jj/ 2 . Although asymptotically better, in practice this bound can be larger than the one described by Belabas et al [4] who stated that under GRH, the class group is generated by the classes of the prime ideals of norm bounded by B provided that…”

Improved CRT algorithm for class polynomials in genus 2

“…1 The degree of the extension where the full` -torsion is defined depends on whether Step 1 succeeded.…”