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2019 Issue №04

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Acceleration of Computations in Jacobian Hyperelliptic Curve

Pages
13-20

Abstract

In article is stated the method of acceleration of procedures of addition and doubling points Jacobian of a hyperelliptic curve in affine and projective coordinates. Corresponding modified Miller algorithms are developed. In af­fine coordinates an expense for group operation of doubling there are more than expenses for group operation of addition, therefore it is more favorable to realize consistently two additions, than doubling and addition. Estimations of efficiency of the modified algorithms are received.

Reference

1.  Sakai R., Ohgishi K., Kasahara M. Cryptosystems Based on Pairings // Procee­dings of the 2000 Symposium on Cryptography and Information Security. 2000. Р. 26—28.

2.  Joux A. A One-Round Protocol for Tripartite Diffe-Hellman // Algorithmic Number Theory Symposium. Springer-Verlag, 2000. Р. 385—394.

3.  Lange T. Formulae for Arithmetic on Genus 2 Hyperelliptic Curves // Appli­cable Algebra in Engineering, Communication and Computing. 2005. Vol. 15, iss. 5. P. 295—328.

4.  Choie Y., Lee E. Implementation of Tate Pairing on Hyperelliptic Curve of Ge­nus 2 // Information Security and Cryptology. 2004. Р. 97—111.

5.  H’Eigeartaigh C. O., Scott M. Pairing Calculation on Supersingular Genus 2 Cur­ves // Selected Areas in Cryptography. 2007. Р. 302—316.

6.  Duursma I., Gaudry P., Morain F. Speeding up the Discrete Log Computation on Curves with Automorphisms // Advances in Cryptology. 1999. Р. 103—121.

7.  Gaudry P. An Algorithm for Solving the Discrete Log Problem on Hyperellip­tic Curves // Advances in Cryptology. 2000. Р. 19—34.

8.  Fan X., Gong G. Efficient Explicit Formulae for Genus 2 Hyperelliptic Curves over Prime Fields and Their Implementations. LNCS 4876. Springer, 2007. Р. 155—172.

9.  Koblitz N., Menezes A. Pairing-based Cryptography at High Security Levels // Cryptography and Coding. LNCS 3796. Springer, 2005. Р. 235—249.

10.  Teruya T., Saito K., Kanayama N. et al. Constructing Symmetric Pairings over Supersingular Elliptic Curves with Embedding Degree Three // LNCS 8365. Sprin­ger, 2014. P. 97—112.

11.  Granger R., Hess F. Ate Pairing on Hyperelliptic Curves // Advance in Cryp­to­logy. LNCS 4515. Springer, 2007. P. 430—447.

12.  Zhang F. Twisted Ate Pairing on Hyperelliptic Curves and Applications // Cryptology ePrint Archive, Report 2008/274, 2008.

13.  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. С. 36—39.