Twists of curves
Encyclopedia
In mathematics
an elliptic curve
E over a field
K has its quadratic twist, that is another elliptic curve which is isomorphic
to E over an algebraic closure
of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic
and quartic twists. The curve and its twists have the same j-invariant
.
over K of the form:
Given
and 
, the quadratic twist of E is the curve Ed, defined by the equation:
The two elliptic curves E and Ed are not isomorphic over K, but over the field extension
K(
).
Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
2.Twisted Edwards curve
3.Twisted tripling-oriented Doche–Icart–Kohel curve
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
E over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K has its quadratic twist, that is another elliptic curve which is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to E over an algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic
and quartic twists. The curve and its twists have the same j-invariant
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
.
Quadratic twist
Let E be an elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
over K of the form:
Given
and , the quadratic twist of E is the curve Ed, defined by the equation:
The two elliptic curves E and Ed are not isomorphic over K, but over the field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
K(
).Quartic twist
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a quartic twist, one obtain four curves: one is isomorphic to E, one is its quadratic twist, and only the other two are really new.Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
Cubic twist
Analogously to the quartic twist case, an elliptic curve over K with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.Examples
1.Twisted Hessian curvesTwisted Hessian curves
In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic. In some operations , it is close in speed to Edwards...
2.Twisted Edwards curve
Twisted Edwards curve
In algebraic geometry, the Twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein et al. and named after Harold M. Edwards...
3.Twisted tripling-oriented Doche–Icart–Kohel curve