Large Cyclic Subgroups of Jacobians of Hyperelliptic Curves
نویسنده
چکیده
In this paper we obtain conditions on the divisors of the group order of the Jacobian of a hyperelliptic genus 2 curve, generated by the complex multiplication method described by Weng (2003) and Gaudry et al (2005). Examples, where these conditions imply that the Jacobian has a large cyclic subgroup, are given.
منابع مشابه
Isogenies 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 کاملDecomposing Jacobians of Hyperelliptic Curves
Many interesting questions can be asked about the decomposition of Jacobians of curves. For instance, we may want to know which curves have completely decomposable Jacobians (Jacobians which are the product of g elliptic curves) [4]. We may ask about number theoretic properties of the elliptic curves that show up in the decomposition of Jacobians of curves [2]. We would also like to know how ma...
متن کاملExplicit Descent for Jacobians of Cyclic Covers of the Projective Line
We develop a general method for bounding Mordell-Weil ranks of Jacobians of arbitrary curves of the form y = f(x). As an example, we compute the Mordell-Weil ranks over Q and Q( √ −3) for a non-hyperelliptic curve of genus 8.
متن کاملThe average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
We prove that when all hyperelliptic curves of genus n ≥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational p...
متن کاملComputational Aspects of Jacobians of Hyperelliptic Curves
Nowadays, one area of research in cryptanalysis is solving the Discrete Logarithm Problem (DLP) in finite groups whose group representation is not yet exploited. For such groups, the best one can do is using a generic method to attack the DLP, the fastest of which remains the Pollard rho algorithm with r-adding walks. For the first time, we rigorously analyze the Pollard rho method with r-addin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- IACR Cryptology ePrint Archive
دوره 2007 شماره
صفحات -
تاریخ انتشار 2007