# tate_bilinear_pairing 0.6

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)\$. The largest order of \$G_1\$ is 911 bits.

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 3.26 seconds @ Intel Core2 L7500 CPU (1.60GHz) if the order of \$G_1\$ is 151 bits.

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].

Usage 1: calculating Tate bilinear pairing

Specify the order of G1 is of 151 bits:

```>>> from tate_bilinear_pairing import eta
>>> eta.init(151)
```

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)
```

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:

```>>> p1 = [inf1, x1, y1]
>>> p2 = [inf2, x2, y2]
```

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:

```>>> k = random.randint(0,1000)
>>> p3 = ecc.scalar_mult(k, p1)
```

References

[1] I. Duursma, H.S. Lee.
Tate pairing implementation for hyper-elliptic curves \$y^2=x^p-x+d\$. Advances in Cryptology - Proc. ASIACRYPT ’03, pp. 111-123, 2003.
[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. Cryptographic Hardware and Embedded Systems - Proc. CHES ’05, pp. 412-426, 2005.
[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. IEEE Transactions on Computers, Special Section on Special-Purpose Hardware for Cryptography and Cryptanalysis, vol. 57 no. 11 pp. 1454-1468, 2008.
[4] P.S.L.M. Barreto, S.D. Galbraith, C. O hEigeartaigh, and M. Scott,
Efficient Pairing Computation on Supersingular Abelian Varieties. Designs, Codes and Cryptography, vol. 42, no. 3, pp. 239-271, Mar. 2007.

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