tate-bilinear-pairing 0.2.3

a library for calculating Tate bilinear pairing especially on super-singularelliptic curve E:y^2=x^3-x+1 in affine coordinates defined over a Galois Field GF(3^m)

Introduction

This package is a Python library for calculating Tate bilinear pairing, especially on super-singular elliptic curve $E:y^2=x^3-x+1$ in affine coordinates defined over a Galois Field $GF(3^m)$.

This package is also for calculating the addition of two elements in the elliptic curve group, and the addition of $k$ identical element in the elliptic curve group.

The code of this package for computing the Tate bilinear pairing follows the paper by Beuchat et al [3]. The code of this package for computing the elliptic curve group operation follows the paper by Kerins et al [2].

This package is in PURE Python, working with Python 2.7 and 3.2.

This package computes one Tate bilinear pairing within 10.1 seconds @ AMD P320 (2.1GHz).

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What is Tate bilinear pairing

Generally speaking, The Tate bilinear pairing algorithm is a transformation that takes two points on an elliptic curve and outputs a nonzero element in the extension field $GF(3^{6m})$. The state-of-the-art way of computing the Tate bilinear pairing is eta pairing, introduced by Barreto et al [4]. For more information, please refer to [1,2,3,4].

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Usage 1: calculating Tate bilinear pairing

Given two random numbers like this:

>>> import random
>>> a = random.randint(0,1000)
>>> b = random.randint(0,1000)

Computing two elements $[inf1, x1, y1]$, and $[inf2, x2, y2]$ in the elliptic curve group:

>>> from tate_bilinear_pairing import ecc
>>> g = ecc.gen()
>>> inf1, x1, y1 = ecc.scalar_mult(a, g)
>>> inf2, x2, y2 = ecc.scalar_mult(b, g)

Tate bilinear pairing is done via:

>>> from tate_bilinear_pairing import eta
>>> t = eta.pairing(x1, y1, x2, y2)

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Usage 2: calculating the addition of two elements in the elliptic curve group

Given two elements $p1=[inf1, x1, y1]$, and $p2=[inf2, x2, y2]$ in the elliptic curve group, the addition is done via:

>>> p3 = ecc.add(p1, p2)

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Usage 3: calculating the addition of $k$ identical elements

Given a non-negative integer $k$ and an group element $p1=[inf1, x1, y1]$, $k cdot p1$ is computed via:

>>> p3 = ecc.scalar_mult(k, p1)

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References

[1] I. Duursma, H.S. Lee.

“Tate pairing implementation for hyper-elliptic curves $y^2=x^p-x+d$”.

[2] T. Kerins, W.P. Marnane, E.M. Popovici, and P.S.L.M. Barreto.

“Efficient hardware for the Tate pairing calculation in characteristic

three”.

[3] J. Beuchat, N. Brisebarre, J. Detrey, E. Okamoto, M. Shirase, and T. Takagi.

“Algorithms and Arithmetic Operators for Computing the $eta_T$ Pairing

in Characteristic Three”.

[4] P.S.L.M. Barreto, S.D. Galbraith, C. O hEigeartaigh, and M. Scott,

“Efficient Pairing Computation on Supersingular Abelian Varieties”.