In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one defined by the Weierstrass equation. Sometimes it is used in cryptography instead of the Weierstrass form because it can provide a defence against simple and differential power analysis style (SPA) attacks; it is possible, indeed, to use the general addition formula also for doubling a point on an elliptic curve of this form: in this way the two operations become indistinguishable from some side-channel information. The Jacobi curve also offers faster arithmetic compared to the Weierstrass curve. The Jacobi curve can be of two types: the Jacobi intersection, that is given by an intersection of two surfaces, and the Jacobi quartic.
Elliptic Curves: Basics Given an elliptic curve, it is possible to do some "operations" between its points: for example one can add two points P and Q obtaining the point P + Q that belongs to the curve; given a point P on the elliptic curve, it is possible to "double" P, that means find [2]P = P + P (the square brackets are used to indicate [n]P, the point P added n times), and also find the negation of P, that means find –P. In this way, the points of an elliptic curve forms a group. Note that the identity element of the group operation is not a point on the affine plane, it only appears in the projective coordinates: then O = (0: 1: 0) is the "point at infinity", that is the neutral element in the group law. Adding and doubling formulas are useful also to compute [n]P, the n-th multiple of a point P on an elliptic curve: this operation is considered the most in elliptic curve cryptography. An elliptic curve E, over a field K can be put in the Weierstrass form y2 = x3 + ax + b, with a, b in K. What will be of importance later are point of order 2, that is P on E such that [2]P = O and P ≠ O. If P = (p, 0) is a point on E, then it has order 2; more generally the points of order 2 correspond to the roots of the polynomial f(x) = x3 + ax + b. From now on, we will use Ea,b to denote the elliptic curve with Weierstrass form y2 = x3 + ax + b. If Ea,b is such that the cubic polynomial x3 + ax + b has three distinct roots in K and b = 0 we can write Ea,b in the Legendre normal form:
Ea,b: y2 = x(x + 1)(x + j) In this case we have three points of order two: (0, 0), (–1, 0), (–j, 0). In this case we use the notation E[j]. Note that j can be expressed in terms of a, b.
Definition: Jacobi intersection An elliptic curve in P3(K) can be represented as the intersection of two quadric surfaces:
Q : { Q 1 ( X 0 , X 1 , X 2 , X 3 ) = 0 } ∩ { Q 2 ( X 0 , X 1 , X 2 , X 3 ) = 0 } {\displaystyle Q:\{Q_{1}(X_{0},X_{1},X_{2},X_{3})=0\}\cap \{Q_{2}(X_{0},X_{1},X_{2},X_{3})=0\}}
It is possible to define the Jacobi form of an elliptic curve as the intersection of two quadrics. Let Ea,b be an elliptic curve in the Weierstrass form, we apply the following map to it:
Φ : ( x , y ) ↦ ( X , Y , Z , T ) = ( x , y , 1 , x 2 ) {\displaystyle \Phi :(x,y)\mapsto (X,Y,Z,T)=(x,y,1,x^{2})}
We see that the following system of equations holds:
S : { X 2 − T Z = 0 Y 2 − a X Z − b Z 2 − T X = 0 {\displaystyle \mathbf {S} :{\begin{cases}X^{2}-TZ=0\\Y^{2}-aXZ-bZ^{2}-TX=0\end{cases}}}
The curve E[j] corresponds to the following intersection of surfaces in P3(K):
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