Supersingular isogeny Diffie-Hellman (SIDH) is a proposal for a quantumresistant key exchange. The state-of-the-art implementation works entirely with Montgomery curves and basically can be divided into elliptic curve arithmetic and isogeny arithmetic. It is well known that twisted Edwards curves can provide a more efficient elliptic curve arithmetic. Therefore it was hinted by Costello and His...

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if characteristic field equals 2, and it 112, 58, 52, 3. If not supersingular, 60 both cases. also give complete classification large configurations lines.

Introduction. In our earlier papers [2,3,4,5,6], we investigated stable operations and cooperations in elliptic cohomology and its variants, relating these to known operations on rings of modular forms. The purpose of this article is to give an introduction to these stable operation algebras, in particular explaining the connections with Hecke algebras and Morava stabilizer algebras; further de...

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic proper...

We show that the Mathieu groups M22 and M11 can act on the supersingular K3 surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group M23 with three orbits on 24 letters act on a supersingular K3 surface with Artin invariant 1 in a suitable characteristic.

We investigate configurations of rational double points with the total Milnor number 21 on supersingular K3 surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular K3 surfaces.

Consider an elliptic curve E over a number field K. Suppose that has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational p. We completely classify valuations $$p^n$$ -torsion points by valuation coefficient $$p{\text {th}}$$ division polynomial. This classification corrects error in earlier work Lozano-Robledo. As application, we find minimum necessary ramifica...

In a secret sharing scheme, is distributed among several participants in such way that only any authorized subset of able to recover the secret. So far, security many schemes has been based on hardness some mathematical problems, as discrete logarithm and factorization. These problems can be solved polynomial time using Shor?s algorithm for quantum computer. this paper, we propose an efficient ...

The irreducible supersingular mod p representations of G = GL2(F ), where F is a finite extension of Qp, are the building blocks of the mod p representation theory of G. They all arise as irreducible quotients of certain universal supersingular representations. We investigate the structure of these universal modules in the case when F/Qp is totally ramified.