A p-adic quasi-quadratic point counting algorithm

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page

On the rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A version with all figures can be obtained from http://www.physik.uni-ulm.de/theo/qc/ (File: http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any problems contact Arnd B\"acker (e-mail: [email protected]) or Roman Schubert (e-mail: [email protected]

Elliptic curve cryptography: Generation and validation of domain parameters in binary Galois Fields

Elliptic curve cryptography (ECC) is an increasingly popular method for securing many forms of data and communication via public key encryption. The algorithm utilizes key parameters, referred to as the domain parameters. These parameters must adhere to specific characteristics in order to be valid for use in the algorithm. The American National Standards Institute (ANSI), in ANSI X9.62, provides the process for generating and validating these parameters. The National Institute of Standards and Technology (NIST) has identified fifteen sets of parameters; five for prime fields, five for binary fields, and five for Koblitz curves. The parameter generation and validation processes have several key issues. The first is the fast reduction within the proper modulus. The modulus chosen is an irreducible polynomial having degree greater than 160. Choosing irreducible polynomials of a particular order is less critical since they have isomorphic properties, mathematically. However, since there are differences in performance, there are standards that determine the specific polynomials chosen. The NIST standards are also based on word lengths of 32 bits. Processor architecture, primality, and validation of irreducibility are other important characteristics. The area of ECC that is researched is the generation and validation processes, as they are specified for binary Galois Fields F (2m). The rationale for the parameters, as computed for 32 bit and 64 bit computer architectures, and the algorithms used for implementation, as specified by ANSI, NIST and others, are examined. The methods for fast reduction are also examined as a baseline for understanding these parameters. Another aspect of the research is to determine a set of parameters beyond the 571-bit length that meet the necessary criteria as determined by the standards

Computing Zeta Functions of Nondegenerate Curves

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed.Comment: 41 page