Do All Elliptic Curves of the Same Order Have the Same Difficulty of Discrete Log?

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.Comment: 26 pages, revised, to appear in Advances in Cryptology -- Asiacrypt 200

Two Suns in The Sky: Stellar Multiplicity in Exoplanet Systems

We present results of a reconnaissance for stellar companions to all 131 radial-velocity-detected candidate extrasolar planetary systems known as of July 1, 2005. CPM companions were investigated using the multi-epoch DSS images, and confirmed by matching the trigonometric parallax distances of the primaries to companion distances estimated photometrically. We also attempt to confirm or refute companions listed in the Washington Double Star Catalog, the Catalogs of Nearby Stars, in Hipparcos results, and in Duquennoy & Mayor (1991). Our findings indicate that a lower limit of 30 (23%) of the 131 exoplanet systems have stellar companions. We report new stellar companions to HD 38529 and HD 188015, and a new candidate companion to HD 169830. We confirm many previously reported stellar companions, including six stars in five systems that are recognized for the first time as companions to exoplanet hosts. We have found evidence that 20 entries in the Washington Double Star Catalog are not gravitationally bound companions. At least three, and possibly five, of the exoplanet systems reside in triple star systems. Three exoplanet systems have potentially close-in stellar companions ~ 20 AU away from the primary. Finally, two of the exoplanet systems contain white dwarf companions. This comprehensive assessment of exoplanet systems indicates that solar systems are found in a variety of stellar multiplicity environments - singles, binaries, and triples; and that planets survive the post-main-sequence evolution of companion stars.Comment: 52 pages, 7 figures, Accepted for publication in Ap

MV3: A new word based stream cipher using rapid mixing and revolving buffers

MV3 is a new word based stream cipher for encrypting long streams of data. A direct adaptation of a byte based cipher such as RC4 into a 32- or 64-bit word version will obviously need vast amounts of memory. This scaling issue necessitates a look for new components and principles, as well as mathematical analysis to justify their use. Our approach, like RC4's, is based on rapidly mixing random walks on directed graphs (that is, walks which reach a random state quickly, from any starting point). We begin with some well understood walks, and then introduce nonlinearity in their steps in order to improve security and show long term statistical correlations are negligible. To minimize the short term correlations, as well as to deter attacks using equations involving successive outputs, we provide a method for sequencing the outputs derived from the walk using three revolving buffers. The cipher is fast -- it runs at a speed of less than 5 cycles per byte on a Pentium IV processor. A word based cipher needs to output more bits per step, which exposes more correlations for attacks. Moreover we seek simplicity of construction and transparent analysis. To meet these requirements, we use a larger state and claim security corresponding to only a fraction of it. Our design is for an adequately secure word-based cipher; our very preliminary estimate puts the security close to exhaustive search for keys of size < 256 bits.Comment: 27 pages, shortened version will appear in "Topics in Cryptology - CT-RSA 2007

The Solar Neighborhood XV: Discovery of New High Proper Motion Stars with mu >= 0.4"/yr between Declinations -47 degrees and 00 degrees

We report the discovery of 152 new high proper motion systems (mu >= 0.4"/yr) in the southern sky (Declination = -47 degrees to 00 degrees) brighter than UKST plate R_{59F} =16.5 via our SuperCOSMOS-RECONS (SCR) search. This paper complements Paper XII in The Solar Neighborhood series, which covered the region from Declination = -90 degrees to -47 degrees and discussed all 147 new systems from the southernmost phase of the search. Among the total of 299 systems from both papers, there are 148 (71 in Paper XII, 77 in this paper) new systems moving faster than 0.5"/yr that are additions to the classic ``LHS'' (Luyten Half Second) sample. These constitute an 8% increase in the sample of all stellar systems with mu >= 0.5"/yr in the southern sky. As in Paper XII, distance estimates are provided for the systems reported here based upon a combination of photographic plate magnitudes and 2MASS photometry, assuming all stars are on the main sequence. Two SCR systems from the portion of the sky included in this paper are anticipated to be within 10 pc, and an additional 23 are within 25 pc. In total, the results presented in Paper XII and here for this SCR sweep of the entire southern sky include five new systems within 10 pc and 38 more between 10 and 25 pc. The largest number of nearby systems have been found in the slowest proper motion bin, 0.6"/yr > mu >= 0.4"/yr, indicating that there may be a large population of low proper motion systems very near the Sun.Comment: 36 pages, 5 figures, accepted for publication in Astronomical Journa

The Solar Neighborhood XVII: Parallax Results from the CTIOPI 0.9m Program -- Twenty New Members of the RECONS 10 Parsec Sample

Astrometric measurements for 25 red dwarf systems are presented, including the first definitive trigonometric parallaxes for 20 systems within 10 pc of the Sun, the horizon of the RECONS sample. The three nearest systems that had no previous trigonometric parallaxes (other than perhaps rough preliminary efforts) are SO 0253+1652 (3.84 +/- 0.04 pc, the 23rd nearest system), SCR 1845-6357 AB (3.85 +/- 0.02 pc, 24th), and LHS 1723 (5.32 +/- 0.04 pc, 56th). In total, seven of the systems reported here rank among the nearest 100 stellar systems. Supporting photometric and spectroscopic observations have been made to provide full characterization of the systems, including complete VRIJHK photometry and spectral types. A study of the variability of 27 targets reveals six obvious variable stars, including GJ 1207, for which we observed a flare event in the V band that caused it to brighten by 1.7 mag. Improved parallaxes for GJ 54 AB and GJ 1061, both important members of the 10 pc sample, are also reported. Definitive parallaxes for GJ 1001 A, GJ 633, and GJ 2130 ABC, all of which have been reported to be within 10 pc, indicate that they are beyond 10 pc. From the analysis of systems with (previously) high trigonometric parallax errors, we conclude that parallaxes with errors in excess of 10 mas are insufficiently reliable for inclusion in the RECONS sample. The cumulative total of new additions to the 10 pc sample since 2000 is now 34 systems -- 28 by the RECONS team and six by other groups. This total represents a net increase of 16% in the number of stellar systems reliably known to be nearer than 10 pc.Comment: 33 pages, including 3 figures and 3 table

A High Angular Resolution Survey of Massive Stars in Cygnus OB2: Results from the Hubble Space Telescope Fine Guidance Sensors

We present results of a high angular resolution survey of massive OB stars in the Cygnus OB2 association that we conducted with the Fine Guidance Sensor 1R (FGS1r) on the Hubble Space Telescope. FGS1r is able to resolve binary systems with a magnitude difference delta-V < 4 down to separations as small as 0.01 arcsec. The sample includes 58 of the brighter members of Cyg OB2, one of the closest examples of an environment containing a large number of very young and massive stars. We resolved binary companions for 12 targets and confirmed the triple nature of one other target, and we offer evidence of marginally resolved companions for two additional stars. We confirm the binary nature of 11 of these systems from complementary adaptive optics imaging observations. The overall binary frequency in our study is 22% to 26% corresponding to orbital periods ranging from 20 - 20,000 years. When combined with the known short-period spectroscopic binaries, the results supports the hypothesis that the binary fraction among massive stars is > 60%. One of the new discoveries is a companion to the hypergiant star MT 304 = Cyg OB2-12, and future measurements of orbital motion should provide mass estimates for this very luminous star.Comment: accepted for AJ, 84 pages, 61 figure

Rotation, magnetism, and metallicity of M dwarf systems

Close M-dwarf binaries and higher multiples allow the investigation of rotational evolution and mean magnetic flux unbiased from scatter in inclination angle and age since the orientation of the spin axis of the components is most likely parallel and the individual systems are coeval. Systems composed of an early (M0.0 -- M4.0) and a late (M4.0 -- M8.0) type component offer the possibility to study differences in rotation and magnetism between partially and fully convective stars. We have selected 10 of the closest dM systems to determine the rotation velocities and the mean magnetic field strengths based on spectroscopic analysis of FeH lines of Wing-Ford transitions at 1 $\mu$m observed with VLT/CRIRES. We also studied the quality of our spectroscopic model regarding atmospheric parameters including metallicity. A modified version of the Molecular Zeeman Library (MZL) was used to compute Land\'e g-factors for FeH lines. Magnetic spectral synthesis was performed with the Synmast code. We confirmed previously reported findings that less massive M-dwarfs are braked less effectively than objects of earlier types. Strong surface magnetic fields were detected in primaries of four systems (GJ 852, GJ 234, LP 717-36, GJ 3322), and in the secondary of the triple system GJ 852. We also confirm strong 2 kG magnetic field in the primary of the triple system GJ 2005. No fields could be accurately determined in rapidly rotating stars with \vsini>10 \kms. For slow and moderately rotating stars we find the surface magnetic field strength to increase with the rotational velocity \vsini which is consistent with other results from studying field stars.Comment: Accepted by MNRAS, 10 pages, 4 figures, 4 table