منابع مشابه
Isogeny volcanoes
The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This expository paper recounts the theory behind these graphs and examines several recently developed algorithms that realize substantial (often dramatic) performan...
متن کاملModular polynomials via isogeny volcanoes
We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X,Y ] and (Z/mZ)[X, Y ], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected run...
متن کاملOn Avoiding ZVP-Attacks Using Isogeny Volcanoes
The usage of elliptic curve cryptography in smart cards has been shown to be efficient although, when considering curves, one should take care about their vulnerability against the Zero-Value Point Attacks (ZVP). In this paper, we present a new procedure to find elliptic curves which are resistant against these attacks. This algorithm finds, in an efficient way, a secure curve by means of volca...
متن کاملComputing isogeny volcanoes of composite degree
Isogeny volcanoes are an interesting structure that have had several recent applications. An isogeny volcano is a connected component of a larger graph called a cordillera. In this paper, we further explore properties of how to compute volcanoes given that we have already computed one of a different degree. This allows us to compute volcanoes of composite degree more efficiently than a direct c...
متن کاملModular Isogeny Complexes
We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.
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ژورنال
عنوان ژورنال: The Open Book Series
سال: 2013
ISSN: 2329-907X,2329-9061
DOI: 10.2140/obs.2013.1.507