scholarly journals Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions

2018 ◽  
pp. 329-368 ◽  
Author(s):  
Kirsten Eisenträger ◽  
Sean Hallgren ◽  
Kristin Lauter ◽  
Travis Morrison ◽  
Christophe Petit
Keyword(s):  
Endomorphism Rings ◽  
Isogeny Graphs

scholarly journals Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs

2020 ◽  
Vol 4 (1) ◽  
pp. 215-232
Author(s):  
Kirsten Eisenträger ◽  
Sean Hallgren ◽  
Chris Leonardi ◽  
Travis Morrison ◽  
Jennifer Park
Keyword(s):  
Elliptic Curves ◽  
Path Finding ◽  
Endomorphism Rings ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs

scholarly journals Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

2021 ◽  
Vol 15 (1) ◽  
pp. 454-464
Author(s):  
Guanju Xiao ◽  
Lixia Luo ◽  
Yingpu Deng
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Endomorphism Rings ◽  
Principal Ideals ◽  
Supersingular Elliptic Curves ◽  
Isogeny Graphs ◽  
Quadratic Order

Abstract Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽 p 2 , if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.

scholarly journals Endomorphism Rings Via Minimal Morphisms

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Manuel Cortés-Izurdiaga ◽  
Pedro A. Guil Asensio ◽  
D. Keskin Tütüncü ◽  
Ashish K. Srivastava
Keyword(s):  
Endomorphism Rings

On the computation of the endomorphism rings of abelian surfaces

2021 ◽  
Author(s):  
Claus Fieker ◽  
Tommy Hofmann ◽  
Sogo Pierre Sanon
Keyword(s):  
Abelian Surfaces ◽  
Endomorphism Rings

Anti-isomorphisms of the endomorphism rings of a class of free modules

1965 ◽  
Vol 159 (4) ◽  
pp. 278-284 ◽  
Author(s):  
Leonard Gewirtzman
Keyword(s):  
Endomorphism Rings ◽  
Free Modules

Isomorphisms and anti-isomorphisms of endomorphism rings of graded modules close to free ones

2009 ◽  
Vol 79 (2) ◽  
pp. 255-257 ◽  
Author(s):  
I. N. Balaba ◽  
A. V. Mikhalev
Keyword(s):  
Endomorphism Rings ◽  
Graded Modules

Multiplication complexe et équivalence élémentaire dans le langage des corps (Complex multiplication and elementary equivalence in the language of fields)

2002 ◽  
Vol 67 (2) ◽  
pp. 635-648
Author(s):  
Xavier Vidaux
Keyword(s):  
Function Fields ◽  
Complex Multiplication ◽  
Closed Field ◽  
Constant Symbol ◽  
Endomorphism Rings ◽  
Modular Invariant ◽  
Elementary Equivalence ◽  
Algebraically Closed Field

AbstractLet K and K′ be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0. non k-isomorphic, and let C and C′ be two curves with respectively K and K′ as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K′ are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras.

Self-Small Abelian Groups as Modules over Their Endomorphism Rings

2003 ◽  
Vol 31 (10) ◽  
pp. 4911-4924 ◽  
Author(s):  
Simion Breaz
Keyword(s):  
Abelian Groups ◽  
Endomorphism Rings

scholarly journals Some modules with nearly prescribed endomorphism rings

1972 ◽  
Vol 23 (2) ◽  
pp. 250-262 ◽  
Author(s):  
Sheila Brenner
Keyword(s):  
Endomorphism Rings
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