Felix Fontein

PotLLL: a polynomial time version of LLL with deep insertions

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where the insertions are restricted to blocks, behave nicely in practice concerning the running time. However no proof of polynomial running time is known. In this paper PotLLL, a new variant of DeepLLL with provably polynomial running time, is presented. We compare the practical behavior of the new algorithm to classical LLL, BKZ as well as blockwise variants of DeepLLL regarding both the output quality and running time.

On the probability of generating a lattice

We study the problem of determining the probability that m vectors selected uniformly at random from the intersection of the full-rank lattice @L in R^n and the window [0,B)^n generate @L when B is chosen to be appropriately large. This problem plays an important role in the analysis of the success probability of quantum algorithms for solving the Discrete Logarithm Problem in infrastructures obtained from number fields and also for computing fundamental units of number fields. We provide the first complete and rigorous proof that 2n+1 vectors suffice to generate @L with constant probability (provided that B is chosen to be sufficiently large in terms of n and the covering radius of @L and the last n+1 vectors are sampled from a slightly larger window). Based on extensive computer simulations, we conjecture that only n+1 vectors sampled from one window suffice to generate @L with constant success probability. If this conjecture is true, then a significantly better success probability of the above quantum algorithms can be guaranteed.

Abstract Infrastructures of Unit Rank Two (abstract only)

Infrastructures of Unit Rank Two Felix Fontein, University of Zürich, felix.fontein@math.uzh.ch On our poster, we want to give information on the infrastructure of a global field of unit rank two. The infrastructure of a global field is the set of all minima of a fractional ideal, together with the neighbor relation and the baby step operations [HMPLR87, Fon08c]. In the case of unit rank one, it is both used for computation of fundamental units [Buc85a] and for cryptography [SSW96, JSS07]. One main emphasis lies on visualization, both of the set of minima together with baby steps (in the sense of J. Buchmann in [Buc85a]) and the generalized Voronŏı algorithm. The generalized Voronŏı algorithm was first described by J. Buchmann in [Buc85a, Buc85b] for number fields. In the case of purely cubic function fields, it has been introduced by Y. Lee, R. Scheidler and C. Yarrish in [LSY03]. Some screenshots of the current experimental version of our program can be seen on the second page; the shown cases are purely cubic function fields over F5. We also plan to present a live version on our laptop during the poster session. Depending on our progress, we also plan to include new results on the unit rank two case [Fon08a], which are related to the interpretation of certain unit rank one infrastructures as cyclic groups as in [Fon08b]. References [Buc85a] Johannes Buchmann. A generalization of Voronŏı’s unit algorithm. I. J. Number Theory, 20(2):177–191, 1985. [Buc85b] Johannes Buchmann. A generalization of Voronŏı’s unit algorithm. II. J. Number Theory, 20(2):192–209, 1985. [Fon08a] Felix Fontein. The infrastructure of a global field of arbitrary unit rank. In preparation. [Fon08b] Felix Fontein. Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures, 2008. To appear in Advances in Mathematics of Communications. [Fon08c] Felix Fontein. The infrastructure of a global field of unit rank one, 2008. In preparation. [HMPLR87] Y. Hellegouarch, D. L. McQuillan, and R. Paysant-Le Roux. Unités de certains sousanneaux des corps de fonctions algébriques. Acta Arith., 48(1):9–47, 1987. [JSS07] M. J. Jacobson, R. Scheidler, and A. Stein. Cryptographic protocols on real hyperelliptic curves. Adv. Math. Commun., 1(2):197–221, 2007.