Hyperelliptic curve cryptography
- Hyperelliptic curve cryptography
Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insomuch as the algebraic geometry construct of a hyperelliptic curve with an appropriate group law provides an Abelian group on which to do arithmetic.
The use of hyperelliptic curves in cryptography came about in 1989 from Neal Koblitz. Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA). Also for hyperelliptic curves of genus higher that 3, there are known efficient sub-exponential attacksFact|date=July 2008, in contrast to the case of elliptic curves whose best known attack is exponential.
The hyperelliptic curves used are typically of the sort where the degree of is 5 — a genus two hyperelliptic curve, or 7 — a genus three hyperelliptic curve.
Taking the idea of hyperelliptic curve cryptography to the next level, tori can be used in torus based cryptography. These systems are even more complicated (computationally) than hyperelliptic curve based cryptosystems.
External links
* Colm Ó hÉigeartaigh [http://www.computing.dcu.ie/~coheigeartaigh/crypto.html Implementation of some hyperelliptic curves algorithms] using MIRACL
Wikimedia Foundation.
2010.
Look at other dictionaries:
Hyperelliptic curve — In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form:y^2 = f(x)where f(x) is a polynomial of degree n > 4 with n distinct roots. A hyperelliptic function is a function from … Wikipedia
List of curve topics — This is a list of curve topics in mathematics. See also curve, list of curves, and list of differential geometry topics.*acnode *algebraic curve *arc *asymptote *asymptotic curve *Barbier s theorem *barycentric… … Wikipedia
Doubling-oriented Doche–Icart–Kohel curve — A Doubling oriented Doche Icart Kohel curve of equation y2 = x3 − x2 − 16x In mathematics, the doubling oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written. It is a special case of Weierstrass form and it is also… … Wikipedia
Trace Zero Cryptography — In the year 1998 Gerhard Frey firstly purposed using trace zero varieties for cryptographic purpose. These varieties are subgroups of the divisor class group on a low genus hyperelliptic curve defined over a finite field. These groups can be used … Wikipedia
Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a … Wikipedia
Neal Koblitz — Born December 24, 1948 (1948 12 24) (age 62) Nationality … Wikipedia
Centre for Applied Cryptographic Research — The Centre for Applied Cryptographic Research (CACR) is a group of industrial representatives, professors, and students at the University of Waterloo in Waterloo, Ontario, Canada who work and do research in the field of cryptography. The CACR… … Wikipedia
List of curves topics — This is a list of curve topics in mathematics. See also curve, list of curves, and list of differential geometry topics. acnode algebraic curve arc asymptote asymptotic curve Barbier s theorem barycentric[1] Bézier curve Bézout s theorem Birch… … Wikipedia
List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia
Fangguo Zhang — is an associate professor at the Department of Electronics and Communication Engineering of the School of Information Science and Technology at Sun Yat sen University in P.R. China. His main research interests include Pairings Based Cryptosystems … Wikipedia
