Twists of curves

In the mathematical field of  algebraic geometry, an elliptic curve E over a field K  has an associated 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.

Quadratic twist

First assume K is a field of characteristic different from 2.
Let E be an elliptic curve over K of the form:

 

Given  not quadratic residue, the quadratic twist of  is the curve , defined by the equation:

 

or equivalently

 

The two elliptic curves  and  are not isomorphic over , but rather over the field extension .

Now assume K is of characteristic 2. Let E be an elliptic curve over K of the form:

 

Given  such that  is an irreducible polynomial over K, the quadratic twist of E is the curve Ed, defined by the equation:

 

The two elliptic curves  and  are not isomorphic over , but over the field extension .

Quadratic twist over finite fields

If  is a finite field with  elements, then for all  there exist a  such that the point  belongs to either  or .
In fact, if  is on just one of the curves, there is exactly one other  on that same curve (which can happen if the characteristic is not ).

As a consequence,  or equivalently 

where  is the trace of the Frobenius endomorphism of the curve.

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 obtains precisely 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  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

 Twisted Hessian curves
 Twisted Edwards curve
 Twisted tripling-oriented Doche–Icart–Kohel curve

References

 

 

 

Category:Elliptic curves
Category:Elliptic curve cryptography