Anzeige
Erschienen in:
13.12.2019 | Original Paper
On the algebraic structure of \(E_p^{(m)}\) and applications to cryptography
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 4/2021
EinloggenAktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
In this paper we show that the \({\mathbb {Z}}/p^{m}{\mathbb {Z}}\)-module structure of the ring \(E_p^{(m)}\) is isomorphic to a \({\mathbb {Z}}/p^{m}{\mathbb {Z}}\)-submodule of the matrix ring over \({\mathbb {Z}}/p^{m}{\mathbb {Z}}\). Using this intrinsic structure of \(E_p^{(m)}\), solving a linear system over \(E_p^{(m)}\) becomes computationally equivalent to solving a linear system over \({\mathbb {Z}}/p^{m}{\mathbb {Z}}\). As an application we break the protocol based on the Diffie–Hellman decomposition problem and ElGamal decomposition problem over \(E_p^{(m)}\). Our algorithm terminates in a provable running time of \(O(m^{6})\) \({\mathbb {Z}}/p^{m}{\mathbb {Z}}\)-operations.