Pairing Programming

Needs maintenance; refer to my thesis instead.

Minimal pairing-based cryptography requires:

  1. Arithmetic in \(\mathbb{Z}_p\): I built mine on top of the GMP library, which conveniently provides number theoretic functions such as inversion modulo a prime and the Jacobi symbol.

  2. Elliptic curve groups: mostly routines for solving \(y^2 = x^3 + a x + b\) over \(\mathbb{Z}_p\), point addition and multiplication.

  3. Bilinear pairing: Miller’s algorithm.

The above suffices for what I call Type A and B (and A1) pairings: very fast, but elements take a lot of space to represent.

Field extensions must be implemented for other pairing types.

Finding certain pairing-friendly curves requires more work. The the MNT curve construction method requires routines for finding roots modulo a given prime, testing polynomial irreducibility, computing Hilbert polynomials. These in turn depend on high precision complex floating point arithmetic and also an algorithm to solve a Pell-type equation.

Arithmetic on Elliptic Curves

Curves over fields of characteristic greater than 3 can be written in the form \(y^2 = x^3 + a x + b\). For such curves, point addition and multiplication routines are straightforward. (See Blake, Seroussi and Smart.) We give the explicit formulae here.

Suppose we wish to compute \(R = P + Q\). If \(P = O\), then \(R = Q\) and similarly \(Q = O\) is easily handled. If \(P_x = Q_x\) then either \(P_y = -Q_y\) in which case \(R = O\) (for we have \(P = -Q\)), or \(P = Q\) in which case the point doubling algorithm is invoked. Note that the point doubling algorithm assumes the input does not have order 2, i.e. the \(y\)-coordinate is nonzero. (If \(P\) has order 2, the result is \(O\).)

Point addition for \(P \ne Q\):

\[ \array { \lambda & = & \frac{Q_y - P_y}{Q_x - P_x} \\ R_x & = & \lambda^2 - P_x - Q_x \\ R_y & = & (P_x - R_x)\lambda - P_y } \]

Point doubling:

\[ \array { \lambda & = & \frac{3 P_x^2 + a}{2 P_y} \\ R_x & = & \lambda^2 - 2 P_x \\ R_y & = & (P_x - R_x)\lambda - P_y } \]

Generating Random Points

The standard method to generate a random point on an elliptic curve is to choose a random \(x\)-coordinate and solve a quadratic equation for \(y\). (If no solution exists, a new \(x\)-coordinate is chosen.) For odd characteristics, this can be done once one is able to find square roots of elements. In a finite field of prime order, square roots can be obtained via the Tonelli-Shanks algorithm. In general fields, see D.J. Bernstein’s draft of a paper entitled "Faster square roots in annoying finite fields". At the moment I just solve \(x^2 - a\) using the Cantor-Zassenhaus method (which is referred to as Legendre’s method in the aforementioned reference).

Easier ways to generate random points exist for particular cases. For example if \(p\) is a prime with \(p = 2 \bmod 3\) then the cube root of \(x \in \mathbb{F}_q\) is simply \(x^{(2p-1)/3}\). Thus if \(a = 0\) in the equation of the elliptic curve, then it is a simple matter to pick \(y\) first and then solve for \(x\).

Ben Lynn 💡