The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.

We study the behavior of D-branes at distances far shorter than the string length scale ls. We argue that short-distance phenomena are described by the IR behavior of the D-brane world-volume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupling gs this corresponds to momenta and energies far above string scale. We use 0-brane quantum mechanics to study 0-brane collisions and find structure at length scales corresponding to the eleven-dimensional Planck length (l 11 P ∼ g1/3 s ls) and to the radius of the eleventh dimension in M-theory (R11 ∼ gsls). We use 0-branes to probe non-trivial geometries and topologies at sub-stringy scales. We study the 0-brane 4-brane system, calculating the 0-brane moduli space metric, and find the bound state at threshold, which has characteristic size l11 P. We examine the blowup of an orbifold and are able to resolve the resulting S2 down to size l11 P. A 0-brane with momentum approaching 1/R11 is able to explore a larger configuration space in which the blowup is embedded. Analogous phenomena occur for small instantons. We finally turn to 1-branes and calculate the size of a bound state to be ∼ g 1/2 s ls, the 1-brane tension scale. August

Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identity-based encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be capable of providing security equivalent to 128-, 192-, or 256-bit AES keys. In this paper we examine the implications of heightened security needs for pairing-based cryptosystems. We first describe three different reasons why high-security users might have concerns about the long-term viability of these systems. However, in our view none of the risks inherent in pairing-based systems are sufficiently serious to warrant pulling them from the shelves. We next discuss two families of elliptic curves E for use in pairingbased cryptosystems. The first has the property that the pairing takes values in the prime field Fp over which the curve is defined; the second family consists of supersingular curves with embedding degree k = 2. Finally, we examine the efficiency of the Weil pairing as opposed to the Tate pairing and compare a range of choices of embedding degree k, including k = 1 and k = 24. Let E be the elliptic curve 1.

Reach set computations are of fundamental importance in control theory. We consider the reach set problem for open-loop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. The method employs a reduction of forward and backward reach set and control parameter set problem to the transcendental implicitization problem for the components of special solutions of simpler non-parametric systems. Numerous examples are computed using the redlog and qepcad packages.

Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain L-functions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular L-functions. The central question is the following: given a holomorphic newform f with integral coefficients and associated L-function Lf(s), for how many fundamental discriminants d with |d | ≤ x, does Lf(s, χd), the L-function twisted by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the L-function is associated with an elliptic curve, in light of the conjecture of Birch and Swinnerton-Dyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated L-function (1) LE(s) = for ℜs> 1. Then, as a consequence of the Taniyama-Shimura conjecture, recently solved by Wiles, Taylor, ([W], [TW]), and Breuil, Conrad, and Diamond, LE is entire and satisfies a functional equation n=1 a ∗ n n s

When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the

We construct stable operations Tn : E" ( ) \Gamma! E"(1=n) ( ) for n ? 0 in the version of elliptic cohomology where the coefficient ring E" agrees with the ring of modular forms for SL 2 (Z) which are meromorphic at 1, and Tn restricts to the n th Hecke operator Tn on E" . In the past few years, the idea of elliptic cohomology has emerged from the combined efforts of a variety of mathematicians and physicists, and it is widely expected that it will play as important a role in global analysis and topology as K--theory and bordism have in the past. At present, there is no explicit geometric description of the cohomology theories that arise in this area, although there are several promising ideas which it is hoped will eventually lead to such a description. On the other hand, there are constructions of these theories based upon cobordism theories and for many purposes these seem to be adequate, at least for problems within the realm of stable homotopy theory. In particular, in ...

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New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simplifications in some of the algorithms.