Elliptic Loops

Cycle 32th Oral Defence of the Phd Thesis

June 11, 2020
Versione stampabile

Venue: The event will take place online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)

Time: 11.00 a.m.

  • Daniele Taufer - PhD in Mathematics, University of Trento

Abstract:
Given an elliptic curve E over Fp and an integer e ≥ 1, we define a new object, called “elliptic loop”, as the set of plane projective points over Z/p^e Z lying over E, endowed with an operation inherited by the curve addition. This object is proved to be a power-associative abelian algebraic loop. Its substructures are investigated by means of other algebraic cubics defined over the same ring, which we named “shadow curve” and “layers”. When E has trace 1, a distinctive behavior is detected and employed for producing an isomorphism attack to the discrete logarithm on this family of curves. Stronger properties are derived for small values of e, which lead to an explicit description of the infinity part and to characterizing the geometry of rational |E|-torsion points.

Supervisor: Massimiliano Sala