Computing the cardinality of CM elliptic curves using torsion points

Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).Comment: Revised and shortened version, including more material using discriminants of curves and division polynomial

Elliptic curves and primality proving

Projet ICSLAThis report describes the theory and implementation of the Elliptic Curve Primality Proving algorithm - ECPP - algorithm. This includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields ; the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields. We then use this theory to design a very powerful primality proving algorithm. Half of the paper is devoted to the description of its implementation. In particular, we give the currently best algorithms to speed up each part of the program. The resulting program is very fast. We can prove the primality of 100-digit numbers in less than five minutes on a SUN 3/60 workstation, and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

Kansas environmental and resource study: A Great Plains model, tasks 1-6

There are no author identified significant results in this report. Environmental and resources investigations in Kansas utilizing ERTS-1 imagery are summarized for the following areas: (1) use of feature extraction techniqued for texture context information in ERTS imagery; (2) interpretation and automatic image enhancement; (3) water use, production, and disease detection and predictions for wheat; (4) ERTS-1 agricultural statistics; (5) monitoring fresh water resources; and (6) ground pattern analysis in the Great Plains

Kansas environmental and resource study: A Great Plains model

The land use category of subimage regions over Kansas within an MSS image can be identified with an accuracy of about 70% using the textural-spectral features of the multi-images from the four MSS bands. Ground truth measurements indicate that reflectance ratios of the 545 and 655 nm wavebands provide an index of plant development and possibly physiological stress. Agricultural consultants have expressed substantial interest in work conducted on center pivot irrigation and have inquired as to how they may use ERTS-1 imagery to aid those in the irrigation field. Results of the land use mapping experiment indicate that ERTS-1 imagery has major potential in regionalization. The ways in which land is utilized within these regions may then be studied more effectively than if no adequate regionalization is available

Kansas environmental and resource study: A Great Plains model

The author has identified the following significant results. Improvement in the land use classification accuracy of ERTS-1 MSS multi-images over Kansas can be made using two distances between neighboring grey tone N-tuples instead of one distance. Much more information is contained texturally than spectrally on the Kansas image. Ground truth measurements indicate that reflectance ratios of the 545 and 655 nm wavebands provide an index of plant development and possibly physiological stress. Preliminary analysis of MSS 4 and 5 channels substantiate the ground truth interpretation. Results of the land use mapping experiment indicate that ERTS-1 imagery has major potential in regionalization. The ways in which land is utilized within these regions may then be studied more effectively than if no adequate regionalization is available. A model for estimating wheat yield per acre has been applied to acreage estimates derived from ERTS-1 imagery to project the 1973 wheat yields for a ten county area in southwest Kansas. The results are within 3% of the preharvest estimates for the same area prepared by the USDA. Visual identification of winter wheat is readily achieved by using a temporal sequence of images. Identification can be improve by stratifying the project area into subregions having more or less homogeneous agricultural practices and crop mixes

Reverse flow digital artery pedicle flap for closure of diabetic forefoot ulceration

Digital artery pedicle flap is a useful surgical technique for coverage of plantar foot defects. For diabetic forefoot ulcers that are subject to recurrence despite consistent care, this flap can provide long-term durable closure. The authors provide a case report and overview of this innovative reconstructive procedure