Elliptic curve cryptography has raised attention as it allows for having shorter keys and ciphertexts. For example, to obtain similar security levels with 2048 bit RSA key, it is necessary to use only 256 bit keys using over elliptic curve cryptography. Additionally, developments in the index calculus method for solving a discrete logarithm problem increases the sizes of the keys to keep the se...

Elliptic Curve Cryptography (ECC) can be considered an approach to public-key cryptography based on the arithmetic of elliptic curves and the Elliptic Curve Discrete Logarithm Problem (ECDLP). Regarding encryption, the best-known scheme based on ECC is the Elliptic Curve Integrated Encryption Scheme (ECIES), included in standards from ANSI, IEEE, and also ISO/IEC. In the present work, we provid...

The modular curve X1(N) parametrizes elliptic curves with a point of order N . For N ≤ 50 we obtain plane models of X1(N) that have been optimized for fast computation, and provide explicit birational maps to transform a point on our model of X1(N) to an elliptic curve. Over a finite field, these allow us to quickly construct elliptic curves containing a point of order N , and can accelerate th...

Suppose M ≡ N . Recently, there has been much interest in the classification of groups. We show that ‖π‖ > ∅. In [31], the authors address the structure of manifolds under the additional assumption that xΩ = πV . Therefore in [31], the main result was the classification of non-elliptic monoids.

This paper presents an extensive and careful study of the software implementation on workstations of the NIST-recommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation.

1 SL2(Z) and elliptic curves 2 1.1 SL2(Z) and the moduli of complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Fundamental region and a system of generators . . . . . . . . . . . . . . . . . . . . . 3 1.3 The Weierstrass ℘ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Nonsingular cubics and the invariant j . . . . . . . . . . . ...

For a class of anisotropic elliptic problems in bounded domains Ω we show that the convexity of Ω plays an important role in regularity and nonexistence results. Using recent results in [9] we improve some statements in [3]. 2000 Mathematical Subject Classification: 35J70, 46E35, 35B65.

Let p ≥ 5 be a prime number. We find all the possible subgroups G of GL2(Z/pZ) such that there exists a number field k and an elliptic curve E defined over k such that the Gal(k(E [p])/k)-module E [p] is isomorphic to the G-module (Z/pZ)2 and there exists n ∈ N such that the local-global divisibility by pn does not hold over E(k).

We describe a new version of the elliptic curve encryption schemes PSEC (Provably Secure Elliptic Curve). PSEC{3 is a public-key encryption system that uses the elliptic curve El Gamal trapdoor function and two random functions (hash functions) as well as any semantically secure symmetric encryption scheme, such as the one-time pad, or any classical block-cipher. Furthermore, we dene a new prob...