Gradual sub-lattice reduction and a new complexity for factoring polynomials

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well

Correlated twin-photon generation in a silicon nitride loaded thin film PPLN waveguide

Photon-pair sources based on thin film lithium niobate on insulator technology have a great potential for integrated optical quantum information processing. We report on such a source of correlated twin-photon pairs generated by spontaneous parametric down conversion in a silicon nitride (SiN) rib loaded thin film periodically poled lithium niobate (LN) waveguide. The generated correlated photon pairs have a wavelength centred at 1560 nm compatible with present telecom infrastructure, a large bandwidth (21 THz) and a brightness of ∼2.5 × 105 pairs/s/mW/GHz. Using the Hanbury Brown and Twiss effect, we have also shown heralded single photon emission, achieving an autocorrelation g (2) H (0) ≃ 0.04.Antoine Henry, David Barral, Isabelle Zaquine, Andreas Boes, Arnan Mitchell, Nadia Belabas, and Kamel Bencheik

Number Fields Ramified at One Prime

Abstract. For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. We study the existence of G-p fields for fixed G and varying p. For G a finite group and p a prime, we define a G-p field to be a Galois number field K ⊂ C satisfying Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. Let KG,p denote the finite, and often empty, set of G-p fields. The sets KG,p have been studied mainly from the point of view of fixing p and varying G; see [Har94], for example. We take the opposite point of view, as we fix G and let p vary. Given a finite group G, we let PG be the sequence of primes where each prime p is listed |KG,p | times. We determine, for various groups G, the first few primes in PG and their corresponding fields. Only the primes p dividing |G | can be wildly ramified in a G-p field, and so the sequences PG which are infinite are dominated by tamely ramified fields. In Sections 1, 2, and 3, we consider the cases when G is solvable with length 1, 2, and ≥ 3 respectively, using mainly class field theory. Section 4 deals wit

Discriminants cubiques et progressions arithmétiques

Algorithmique des fonctions