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 .

A low-memory algorithm for finding short product representations in finite groups

“…Proving such a claim is believed to be very difficult, but under the generalized Riemann hypothesis (GRH), Bach obtains the following result [4]. Unfortunately, this says nothing about short product representations in cl(O).…”

“…The computation was run in parallel on 32 cores (3.0 GHz AMD Phenom II), using the distinguished points method. 4 The second collision yielded a short product representation after evaluating the map φ a total of 1480862431620 ≈ 1.35 √ n times.…”

Applying sieving to the computation of quadratic class groups

“…Let p max = p k be the ideal in F B with the largest norm, and let p max = N (p max ). A theorem of Bach [2] tells us that in order to ensure that our factor base contains a generating system of Cl (assuming ERH) we need p max > 6 log 2 |∆| if ∆ is fundamental and p max > 12 log 2 |∆| otherwise. Since the linear algebra step in our algorithm is rather expensive, we allow the possibility of using a smaller factor base.…”

Statistical Evidence for Small Generating Sets

“…The above sketch can be filled in to give a sequential algorithm that will compute G(n) for « < yV in polynomial amortized time. (That is, assuming ERH, the total time is N(log yV)0 (1) .) The main idea is to use a sieve to factor each « < N ; this guarantees that all relevant factorizations will be available when needed.…”

“…In the course of verifying that the ERH implies G(n) < 3log2« [1], the first author computed G(n) for « < 106. Brown and Zassenhaus [4] computed G(p) for every prime p less than 106, and conjectured that with probability "almost (but not equal to) one", the first [logp\ primes will generate Z*.…”