Cairo BN

Contents

• Mod arithmetics
• Fq Field element arithmetics
• WIP Fq2 Field extension arithmetics
• WIP Fq6 Field extension arithmetics
• WIP Fq12 Field extension arithmetics
• G1: (x, y) = (Fq, Fq)
• G2: (x, y) = (Fq2, Fq2)
• Pairing: Line function
• WIP Pairing: Miller loop

Refrence material.

Point operations

Pairings for Beginners

Field extensions

Multiplication and Squaring on Pairing-Friendly Fields

Todo

We are probably already doing a bunch of these, but room for thought.

• Speeding scalar multiplication

K. Eisentrger, K. Lauter and P. L. Montgomery, “Fast Elliptic Curve Arithmetic and Improved Weil Pairing Evaluation”, LNCS, Springer, vol. 2612, (2003), pp. 343-354.

• Reducing the loop length in Miller's algorithm

D. Lubicz and D. Robert, “A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties”, IACR Cryptology ePrint Archive, (2013), pp. 192.

• Performing the computing over the field Fqk/d instead of the field Fqk using the twists

C. Costello, T. Lange and M. Naehrig, “Faster pairing computations on curves with high-degree twists”, In Public Key Cryptography: 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings, Springer Verlag, Paris, (2010), pp. 224-242.

• Using other variant of Miller's formula

J. Boxall, N. El Mrabet, F. Laguillaumie and P. Le Duc, “A Variant of Miller's Formula and Algorithm”, The 4th International Conference on Pairing Based Cryptography, Pairing, (2010).

• Deleting the computing for the denominator

P. S. L. M. Barreto, H. Y. Kim and M. Scott, “e_cient algorithms for pairing based cryptosystems”, CRYPTO, LNCS, Springer, Heidelberg, vol. 2442, (2002), pp. 354-369.

• Optimisations of Miller's loop

https://www.researchgate.net/publication/288646605_Optimizing_the_computing_of_pairing_with_Miller's_algorithm