Pairing in cryptography: an arithmetic point of view

J. C. Bajard; N. El Mrabet

doi:10.1117/12.733789

21 September 2007 Pairing in cryptography: an arithmetic point of view

J. C. Bajard, N. El Mrabet

Proceedings Volume 6697, Advanced Signal Processing Algorithms, Architectures, and Implementations XVII; 66970O (2007) https://doi.org/10.1117/12.733789
Event: Optical Engineering + Applications, 2007, San Diego, California, United States

Abstract

The pairing is a mathematical notion wich appeared in cryptography during the 80'. At the beginning, it was used to build attacks on cryptosystems, transferring the discrete logarithm problem on elliptic curves, to a discrete logarithm problem on finite fields, the first was the MOV³⁶ attack in 1993. Now, pairings are used to construct some cryptographic protocols: Diffie Hellman tripartite, identity based encryption, or short signature. The main two pairings usually used are the Tate and Weil pairings. They use distortions and rationnal functions, and their complexities depends of the curve and the field involved. This study deals with two particular papers: one due to N. Koblitz and A. Menezes²⁷ published in 2005, and a second one written by R Granger, D. Page and N. Smart²⁴ in 2006. These two papers compare Tate andWeil pairings, but they differ in their conclusions. We consider the different arithmetic tricks used, trying to precise each point, in a way to avoid any ambiguity. Thus, the arithmetics proposed take into account the features of the fields and the curves used. We clarify the complexity of the possible implementations. We compare the different approaches, in order to clarify the conclusions of the previous papers.

Citation Download Citation

J. C. Bajard and N. El Mrabet "Pairing in cryptography: an arithmetic point of view", Proc. SPIE 6697, Advanced Signal Processing Algorithms, Architectures, and Implementations XVII, 66970O (21 September 2007); https://doi.org/10.1117/12.733789

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