Physics, mathematics, and technology

2014 Issue №10

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About pairing on abelian varieties p-rank one and their cryptographic applications



Described and analyzed in terms of computational efficiency different types of pairings on hyperelliptic curves, especially on curves of genus 2 and p-rank one for the cryptographic applications.


1. Алешников С. И., Алешникова М. В., Горбачёв А. А. Протокол доверенного шифрования на основе модифицированного алгоритма вычисления спаривания Вейля на алгебраических кривых для облачных вычислений // Информационные технологии. 2013. № 9 (205). С. 36—39.
2. Barreto P. S. L. M., Galbraith S. et al. Efficient pairing computation on supersingular Abelian varieties // Designs, Codes and Cryptography, 2007. Vol. 42(3).
3. Freeman D., Stevenhagen D., Streng M. Abelian varieties with prescribed embedding degree //ANTS. 2008. Vol. 5011. P. 60—73.
4. Galbraith S. D., Hess F., Vercauteren F. Hyperelliptic pairing // Pairing 2007. LNCS 4575. 2007. P. 108—131.
5. Galbraith S. D., McKee J. F., Valenca P. C. Ordinary Abelian varieties having small embedding degree // J. Finite Fields and Their Applications. 2007. 13. P. 800—814.
6. Granger R., Hess F. et al. Ate pairing on hyperelliptic curves // Advance in Cryptology–EUROCRYPT’2007. LNCS 4515. 2007. P. 430—447.
7. Granger R., Page D., Smart N. P. High security pairing-based cryptography revisited.Cryptology ePrint Archive. Report 2006/059. 2006.
8. Hess F., Smart N. P., Vercauteren F. The Ate pairing revisited // IEEE Transactions on Information Theory. 2006. Vol. 52. P. 4595—4602.
9. Koblitz N., Menezes A. Pairing-based cryptography at high seccurity levels //Cryptography and Coding. LNCS 3796. 2005. P. 235—249.
10. Matsuda S., Kanayama N., Hess F. et al. Optimized version of the Ate and twisted Ate pairing // The 11th Int. Conf. on Cryptography and Coding. LNCS 4887.2007. P. 302—312.
11. O’Connor L. H., McGuire G., Naehrig M. et al. A CM construction for curves of genus 2 with p-rank 1. arXiv:0811.3434v3 [math.AG] 11 May 2010.
12. Oort F. Abelian varieties over finite fields // Higher-dimensional varieties over finite fields. Summer school in Göttingen. 2007.
13. Zhang F. Twisted Ate pairing on hyperelliptic curves and applications.Cryptology ePrint Archive, Report 2008/274, 2008