96 research outputs found
A CM construction for curves of genus 2 with p-rank 1
We construct Weil numbers corresponding to genus-2 curves with -rank 1
over the finite field \F_{p^2} of elements. The corresponding curves
can be constructed using explicit CM constructions. In one of our algorithms,
the group of \F_{p^2}-valued points of the Jacobian has prime order, while
another allows for a prescribed embedding degree with respect to a subgroup of
prescribed order. The curves are defined over \F_{p^2} out of necessity: we
show that curves of -rank 1 over \F_p for large cannot be efficiently
constructed using explicit CM constructions.Comment: 19 page
Constructing pairing-friendly hyperelliptic curves using Weil restriction
A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over that become pairing-friendly over a finite extension of . Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the Cocks-Pinch and Brezing-Weng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded -value for simple, non-supersingular abelian surfaces
Generalized explicit descent and its application to curves of genus 3
We introduce a common generalization of essentially all known methods for
explicit computation of Selmer groups, which are used to bound the ranks of
abelian varieties over global fields. We also simplify and extend the proofs
relating what is computed to the cohomologically-defined Selmer groups. Selmer
group computations have been practical for many Jacobians of curves over Q of
genus up to 2 since the 1990s, but our approach is the first to be practical
for general curves of genus 3. We show that our approach succeeds on some
genus-3 examples defined by polynomials with small coefficients.Comment: 58 pages; added a few references, and updated a few other
Computing genus 2 curves from invariants on the Hilbert moduli space
AbstractWe give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required
ΠΡΠ°Π½ΠΈΡΡ ΡΠ±Π°Π»Π°Π½ΡΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠΉ ΡΡΠ΅ΠΏΠ΅Π½ΠΈ Π²Π»ΠΎΠΆΠ΅Π½ΠΈΡ Π΄Π»Ρ ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΠΈΠΈ Π½Π° Π±ΠΈΠ»ΠΈΠ½Π΅ΠΉΠ½ΡΡ ΡΠΏΠ°ΡΠΈΠ²Π°Π½ΠΈΡΡ
ΠΠ²ΠΎΠ΄ΠΈΡΡΡ ΡΠΎΡΠΌΡΠ»Π° Π΄Π»Ρ ΡΠ°ΡΡΡΡΠ° Π³ΡΠ°Π½ΠΈΡ ΡΠ±Π°Π»Π°Π½ΡΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠΉ ΡΡΠ΅ΠΏΠ΅Π½ΠΈ Π²Π»ΠΎΠΆΠ΅Π½ΠΈΡ Π³ΠΈΠΏΠ΅ΡΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΊΡΠΈΠ²ΠΎΠΉ. ΠΡΡΠΈΡΠ»Π΅Π½Ρ ΡΠ΅ΠΊΡΡΠΈΠ΅ Π³ΡΠ°Π½ΠΈΡΡ Π΄Π»Ρ ΠΊΡΠΈΠ²ΡΡ
ΡΠΎΠ΄Π° 1-3. ΠΠ»Ρ ΠΊΡΠΈΠ²ΡΡ
Ρ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠΌΠΈ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°ΠΌΠΈ Π³Π΅Π½Π΅ΡΠ°ΡΠΈΠΈ, Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠΈΠΌΠΈ Ρ-Π·Π½Π°ΡΠ΅Π½ΠΈΡΠΌΠΈ ΠΈ ΡΡΠ΅ΠΏΠ΅Π½ΡΠΌΠΈ Π²Π»ΠΎΠΆΠ΅Π½ΠΈΡ ΠΎΡ 1 Π΄ΠΎ 10 Π²ΡΡΠΈΡΠ»Π΅Π½ Π΄ΠΈΠ°ΠΏΠ°Π·ΠΎΠ½ Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ, ΠΊΠΎΡΠΎΡΠΎΠΌΡ ΠΏΡΠΈΠ½Π°Π΄Π»Π΅ΠΆΠΈΡ ΡΡΠΎΠ²Π΅Π½Ρ Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎΡΡΠΈ ΠΊΡΠΈΠ²ΠΎΠΉ
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