16,760 research outputs found
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
Simple verification of completeness of two addition formulas on twisted Edwards curves
Daniel Bernstein and Tanja Lange proved thattwo given addition formulas on twisted Edwards elliptic curvesax^2 + y^2 = 1 + dxy are complete (i.e. the sum of any two pointson a curve can be computed using one of these formulas). Inthis paper we give other simple verification of completenessof these formulas using for example Groebner bases and an ¨algorithm implemented in Magma, which is based on the fact thatcompleteness means that some systems of polynomial equationshave no solutions. This method may be also applied to verifycompleteness of additions formulas on other models of ellipticcurves
Simple verification of completeness of two addition formulas on twisted Edwards curves
Daniel Bernstein and Tanja Lange proved thattwo given addition formulas on twisted Edwards elliptic curvesax^2 + y^2 = 1 + dxy are complete (i.e. the sum of any two pointson a curve can be computed using one of these formulas). Inthis paper we give other simple verification of completenessof these formulas using for example Groebner bases and an ¨algorithm implemented in Magma, which is based on the fact thatcompleteness means that some systems of polynomial equationshave no solutions. This method may be also applied to verifycompleteness of additions formulas on other models of ellipticcurves
Efficient arithmetic on elliptic curves in characteristic 2
International audienceWe present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time and for addition of points in time , where is the cost of multiplication, the cost of squaring , and the cost of multiplication by a constant. By a study of the Kummer curves , we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with per bit where is multiplication by a constant that depends on
Scaling Theory for Steady State Plastic Flows in Amorphous Solids
Strongly correlated amorphous solids are a class of glass-formers whose
inter-particle potential admits an approximate inverse power-law form in a
relevant range of inter-particle distances. We study the steady-state plastic
flow of such systems, firstly in the athermal, quasi-static limit, and secondly
at finite temperatures and strain rates. In all cases we demonstrate the
usefulness of scaling concepts to reduce the data to universal scaling
functions where the scaling exponents are determined a-priori from the
inter-particle potential. In particular we show that the steady plastic flow at
finite temperatures with efficient heat extraction is uniquely characterized by
two scaled variables; equivalently, the steady state displays an equation of
state that relates one scaled variable to the other two. We discuss the range
of applicability of the scaling theory, and the connection to density scaling
in supercooled liquid dynamics. We explain that the description of transient
states calls for additional state variables whose identity is still far from
obvious.Comment: 9 pages, 9 figure
Stress Propagation and Arching in Static Sandpiles
We present a new approach to the modelling of stress propagation in static
granular media, focussing on the conical sandpile constructed from a point
source. We view the medium as consisting of cohesionless hard particles held up
by static frictional forces; these are subject to microscopic indeterminacy
which corresponds macroscopically to the fact that the equations of stress
continuity are incomplete -- no strain variable can be defined. We propose that
in general the continuity equations should be closed by means of a constitutive
relation (or relations) between different components of the (mesoscopically
averaged) stress tensor. The primary constitutive relation relates radial and
vertical shear and normal stresses (in two dimensions, this is all one needs).
We argue that the constitutive relation(s) should be local, and should encode
the construction history of the pile: this history determines the organization
of the grains at a mesoscopic scale, and thereby the local relationship between
stresses. To the accuracy of published experiments, the pattern of stresses
beneath a pile shows a scaling between piles of different heights (RSF scaling)
which severely limits the form the constitutive relation can take ...Comment: 38 pages, 24 Postscript figures, LATEX, minor misspellings corrected,
Journal de Physique I, Ref. Nr. 6.1125, accepte
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