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

New evidence for solar cycle variations at great distances

Recent studies of solar planetary relationships are directed toward exploring how far out from the sun one could observe solar cycle variations. A positive solar Jovian relationship is suggested from a Chree superposed epoch study of the intensity of the great red spot of Jupiter over a period of about six solar cycles. The characteristic double maxima observed in the solar cycle variation is common to other observations of solar events in the photosphere, chromosphere, and corona; radio and corpuscular emissions from the sun; cosmic ray intensity and geomagnetic activity. The same method of analysis adopted for the study of luminosity changes of the planets Jupiter, Saturn, Uranus, and Neptune indicates that the fluctuations of luminosity follow the single maximum solar cycle represented by sunspot numbers. In conjunction with changes of upper atmospheric density and temperature, it is suggested that the extreme ultraviolet (EUV) emission from the sun may be connected with luminosity changes. A method of distinguishing between phenomena related to solar wind and those related to solar EUV is presented

Voyager 1 and 2 measurements of radial and latitudinal cosmic ray gradients during 1981 - 1984

The cosmic ray radial gradient was determined during 1981-84 using data from very similar detectors onboard spacecraft Voyagers 1 and 2 (radial separation approx. 6 AU, heliolatitude separation approx. 25 deg.) and from the Earth-orbiting satellite IMP 8. The principal result is that the radial gradient over this period decreased at the rate approx. 2.0%/AU between 1 and 16 AU and approx. 0.6%/AU between approx. 16 and 22 AU

The radial gradient of interplanetary radiation measured by Mariners 4 and 5

Interplanetary radiation radial gradient measured from Mariners 4 and

Non-Abelian Analogs of Lattice Rounding

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we give an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.Comment: 30 page

Longitudinal distribution of cosmic rays in the heliosphere

The longitudinal distribution of cosmic ray intensity was examined during the years 1974-1976 when the persistent high speed solar wind stream structures produced a well ordered inner heliosphere. Solar wind velocity is mapped back to the Sun and compared with cosmic ray intensity which is represented relative to the solar rotation average. Low solar wind velocity is observed to be a necessary, but not sufficient, condition for the occurrence of higher cosmic ray intensities at 1 AU. These relative enhancements cover a restricted range of heliographic longitudes and persist for several solar rotations. The observed solar wind and cosmic ray intensity relationships are consistent with a simple model suggested here in which cosmic ray modulation is very weak in the inner heliosphere, sunward of the first shock crossing on each field line and more intense in the outer heliosphere

The influence of geomagnetic activity on polar cap absorption

Cosmic noise absorption data on polar cap analyzed for geomagnetic activit