On the period of the linear congruential and power generators

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+\epsilon(n)}$ for any monotone function $\epsilon(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-\epsilon}$ for any $\epsilon >0$. Moreover, the period is unconditionally greater than $n^{1/2+\delta}$, for some fixed $\delta>0$, for a positive proportion of $n$ in the above mentioned sets. These bounds are related to lower bounds on the multiplicative order of an integer $e$ modulo $p-1$, modulo $\lambda(pl)$, and modulo $\lambda(m)$ where $p,l$ range over the primes, $m$ ranges over the integers, and where $\lambda(n)$ is the order of the largest cyclic subgroup of $(\Z/n\Z)^\times$.Comment: 20 pages. One of the quoted results (Theorem 23 in the previous version) is stated for any unbounded monotone function psi(x), but it appears that the proof only supports the case when psi(x) is increasing rather slowly. As a workaround, we provide a modified version of Theorem 23, and change the argument in the proof of Theorem 27 (Theorem 25 in the previous version

On a nonintegrality conjecture

It is conjectured that the sum $$S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k}$$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals

Product-free sets with high density

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.Comment: 12 pages. Many minor edits, mainly to improve the expositio

A rigorous time bound for factoring integers

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Resolution of Linear Algebra for the Discrete Logarithm Problem Using GPU and Multi-core Architectures

In cryptanalysis, solving the discrete logarithm problem (DLP) is key to assessing the security of many public-key cryptosystems. The index-calculus methods, that attack the DLP in multiplicative subgroups of finite fields, require solving large sparse systems of linear equations modulo large primes. This article deals with how we can run this computation on GPU- and multi-core-based clusters, featuring InfiniBand networking. More specifically, we present the sparse linear algebra algorithms that are proposed in the literature, in particular the block Wiedemann algorithm. We discuss the parallelization of the central matrix--vector product operation from both algorithmic and practical points of view, and illustrate how our approach has contributed to the recent record-sized DLP computation in GF($2^{809}$).Comment: Euro-Par 2014 Parallel Processing, Aug 2014, Porto, Portugal. \<http://europar2014.dcc.fc.up.pt/\&gt

A hyperelliptic smoothness test, I

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe