Non-supersingular Hyperelliptic Jacobians
نویسنده
چکیده
In his previous papers [25, 26, 28] the author proved that in characteristic 6= 2 the jacobian J(C) of a hyperelliptic curve C : y = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group Sn or the alternating group An. Here n ≥ 9 is the degree of f . The goal of this paper is to extend this result to the case when either n = 7, 8 or n = 5, 6 and char(K) > 3. 2000 Mathematics Subject Classification: Primary 14H40; Secondary 14K05.
منابع مشابه
Hyperelliptic Jacobians without Complex Multiplication in Positive Characteristic
has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] of even degree is “very big”. More precisely, if f is a polynomial of even degree n ≥ 10 and Gal(f) is either the symmetric group Sn or the alternating group An then End(J(C)) = Z. Notice that it is known [10] that in this case (and even for all integers n ≥ 5) ei...
متن کاملSpeeding Up Pairing Computations on Genus 2 Hyperelliptic Curves with Efficiently Computable Automorphisms
Pairings on the Jacobians of (hyper-)elliptic curves have received considerable attention not only as a tool to attack curve based cryptosystems but also as a building block for constructing cryptographic schemes with new and novel properties. Motivated by the work of Scott, we investigate how to use efficiently computable automorphisms to speed up pairing computations on two families of non-su...
متن کاملEfficient Pairing Computation on Genus 2 Curves in Projective Coordinates
In recent years there has been much interest in the development and the fast computation of bilinear pairings due to their practical and myriad applications in cryptography. Well known efficient examples are the Weil and Tate pairings and their variants such as the Eta and Ate pairings on the Jacobians of (hyper-)elliptic curves. In this paper, we consider the use of projective coordinates for ...
متن کاملIsogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves
We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (Z/2Z) for any hyperelliptic genus 3 curve. These algorithms provide a rational isogeny...
متن کاملNon-hyperelliptic modular Jacobians of dimension 3
We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomo...
متن کامل