giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of ...

We study Tamagawa numbers of elliptic curves with torsion Z/2Z?Z/14Z over cubic fields and an n-isogeny Q, for n?{6,8,10,12,14,16,17,18,19,37,43,67,163}. Bruin Najman [3] proved that every curve a field is base change defined Q. find Q are always divisible by 142, each factor 14 coming from rational prime split multiplicative reduction type I14k, one which p=2. The only exception the 1922.e2, c...

On an elliptic curve, the degree of an isogeny corresponds essentially to the degrees of the polynomial expressions involved in its application. The multiplication–by– map [ ] has degree , therefore the complexity to directly evaluate [ ](P ) is O( ). For a small prime (= 2, 3) such that the additive binary representation provides no better performance, this represents the true cost of applicat...

In this paper, we investigate the e ciency of implementing a post-quantum key exchange protocol over isogenies (PQCrypto 2011) on ARM-powered embedded platforms. This work proposes to employ new primes to speed up constant-time nite eld arithmetic and perform isogenies quickly. Montgomery multiplication and reduction are employed to produce a speedup of 3 over the GNU Multiprecision Library. We...

Generating a supersingular elliptic curve such that nobody knows its endomorphism ring is notoriously hard task, despite several isogeny-based protocols relying on an object. A trusted setup often proposed as workaround, but aspects remain unclear. In this work, we develop the tools necessary to practically run distributed trusted-setup ceremony. Our key contribution first statistically zero-kn...

This article explores the connection between radical isogenies and modular curves. Radical are formulas designed for computation of chains fixed small degree $N$, introduced by Castryck, Decru, Vercauteren at Asiacrypt 2020. One significant advantage isogeny over other with a similar purpose is that they eliminate need to generate point order $N$ generates kernel isogeny. While were originally ...