Elliptic Curve Cryptography - Introduction

Introduction

Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible. The size of the elliptic curve determines the difficulty of the problem. The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements—i.e., that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key—e.g., a 256-bit ECC public key should provide comparable security to a 3072-bit RSA public key (see key sizes below).

For current cryptographic purposes, an elliptic curve is a plane curve which consists of the points satisfying the equation

along with a distinguished point at infinity, denoted ∞. (The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.)

This set together with the group operation of the elliptic group theory form an Abelian group, with the point at infinity as identity element. The structure of the group is inherited from the divisor group of the underlying algebraic variety.

As for other popular public key cryptosystems, no mathematical proof of security has been published for ECC as of 2009. However, the U.S. National Security Agency has endorsed ECC by including schemes based on it in its Suite B set of recommended algorithms and allows their use for protecting information classified up to top secret with 384-bit keys. While the RSA patent expired in 2000, there are patents in force covering certain aspects of ECC technology, though some argue that the Federal elliptic curve digital signature standard (ECDSA; NIST FIPS 186-3) and certain practical ECC-based key exchange schemes (including ECDH) can be implemented without infringing them.

Read more about this topic:  Elliptic Curve Cryptography

Other articles related to "introduction":

John Frame (theologian) - Selected Works
... Introduction to Presuppositional Apologetics Part 2 ... Van Til The Theologian, 1976 ISBN 0-916034-02-X Medical Ethics, 1988 ISBN 0-87552-261-0 ...
Introduction To Metaphysics
... An Introduction to Metaphysics (Introduction à la Métaphysique) is a 1903 essay by Henri Bergson that explores the concept of reality ...
China Miéville - Bibliography - Nonfiction - Introductions To Fiction By Other Authors
... The Borribles An Introduction, 2001 ... Things That Never Happen An Introduction, 2002 ... Wizardry and Wild An Introduction, 2004 ...
Introduction - Music - Songs and Tracks
... Introduction", by Chicago from The Chicago Transit Authority "Introduction", by Hood from Outside Closer "Introduction", by Kajagoogoo from White Feathers "Introduction", by Mike Oldfield from Tubular Bells ...

Famous quotes containing the word introduction:

    For better or worse, stepparenting is self-conscious parenting. You’re damned if you do, and damned if you don’t.
    —Anonymous Parent. Making It as a Stepparent, by Claire Berman, introduction (1980, repr. 1986)

    We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.
    Angela Carter (1940–1992)

    My objection to Liberalism is this—that it is the introduction into the practical business of life of the highest kind—namely, politics—of philosophical ideas instead of political principles.
    Benjamin Disraeli (1804–1881)