4 results sorted by ID
Possible spell-corrected query: canonical left
Some applications of higher dimensional isogenies to elliptic curves (overview of results)
Damien Robert
Foundations
We give some applications of the "embedding Lemma". The first one is a polynomial time (in $\log q$) algorithm to compute the endomorphism ring $\mathrm{End}(E)$ of an ordinary elliptic curve $E/\mathbb{F}_q$, provided we are given the factorisation of $Δ_π$. In particular, this computation can be done in quantum polynomial time.
The second application is an algorithm to compute the canonical lift of $E/\mathbb{F}_q$, $q=p^n$, (still assuming that $E$ is ordinary) to precision $m$ in...
The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives
H. Gopalakrishna Gadiyar, R. Padma
Public-key cryptography
In this brief note we connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.
A Framework for the Sound Specification of Cryptographic Tasks
Juan A. Garay, Aggelos Kiayias, Hong-Sheng Zhou
Cryptographic protocols
Nowadays it is widely accepted to formulate the security of a protocol
carrying out a given task via the ``trusted-party paradigm,'' where
the protocol execution is compared with an ideal process where the
outputs are computed by a trusted party that sees all the inputs. A
protocol is said to securely carry out a given task if running the
protocol with a realistic adversary amounts to ``emulating'' the ideal
process with the appropriate trusted party. In the Universal
Composability (UC)...
An AGM-type elliptic curve point counting algorithm in characteristic three
Trond Stølen Gustavsen, Kristian Ranestad
Foundations
Given an ordinary elliptic curve on Hesse form over a finite field of characteristic three, we give a sequence of elliptic curves which leads to an effective construction of the canonical lift, and obtain an algorithm for computing the number of points. Our methods are based on the study of an explicitly and naturally given $3$-isogeny between elliptic curves on Hesse form.
We give some applications of the "embedding Lemma". The first one is a polynomial time (in $\log q$) algorithm to compute the endomorphism ring $\mathrm{End}(E)$ of an ordinary elliptic curve $E/\mathbb{F}_q$, provided we are given the factorisation of $Δ_π$. In particular, this computation can be done in quantum polynomial time. The second application is an algorithm to compute the canonical lift of $E/\mathbb{F}_q$, $q=p^n$, (still assuming that $E$ is ordinary) to precision $m$ in...
In this brief note we connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.
Nowadays it is widely accepted to formulate the security of a protocol carrying out a given task via the ``trusted-party paradigm,'' where the protocol execution is compared with an ideal process where the outputs are computed by a trusted party that sees all the inputs. A protocol is said to securely carry out a given task if running the protocol with a realistic adversary amounts to ``emulating'' the ideal process with the appropriate trusted party. In the Universal Composability (UC)...
Given an ordinary elliptic curve on Hesse form over a finite field of characteristic three, we give a sequence of elliptic curves which leads to an effective construction of the canonical lift, and obtain an algorithm for computing the number of points. Our methods are based on the study of an explicitly and naturally given $3$-isogeny between elliptic curves on Hesse form.
