ФОРМУЛА СЛОЖЕНИЯ ДИВИЗОРОВ С ИДЕНТИЧНЫМИ Z-КООРДИНАТАМИ В ЯКОБИАНЕ ГИПЕРЭЛЛИПТИЧЕСКОЙ КРИВОЙ ВТОРОГО РОДА НАД ПРОСТЫМИ ПОЛЯМИ
DOI:
https://doi.org/10.18372/2410-7840.12.1966Abstract
Началом применения алгебраических кривых в криптографии послужили публикации Миллера и Коблица [1, 2]. Ими было предложено воспользоваться свойством точек эллиптической кривой (ЭК), образовывать аддитивную Абелевую группу. В своей более поздней работе [3] Коблиц обосновал возможность использования более сложных кривых – гиперэллиптических (ГЭК). Для ГЭК следует рассматривать группу (якобиан) уже не самих точек кривой, а более сложных структур – дивизоров. Как отмечено в [3], ГЭК, в свою очередь, обладают целым рядом преимуществ над ЭК: являются более богатым источником Абелевых групп [3, 4] (образуют Абелевую группу, размер которой определяется произведением размера базового поля на род кривой).References
N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, 48(177), 1987, pp. 203–209.
I. V. S. Miller. Use of elliptic curves in cryptography. In H. C. Williams, editor, Advances in Cryptology - CRYPTO’85, volume 218 of LNCS, Springer, 1985, pp. 417–426.
N. Koblitz. Hyperelliptic cryptosystems. Journal of cryptology, No 1. 1989. pp.139-150.
Menezes A., Wu Y.H., Zuccherato R. An elementary introduction to hyperelliptic curves / In: Koblitz N. ed. // Algebraic aspects of cryptography. –Berlin, Heidelberg, New York: Springer–Verlag, 1998. -pp. 28–63.
Cantor D.G. Computing in Jacobian of a Hyperelliptic curve // Mathematics of Computation. –Vol. 48. -No.177. -1987. -pp.95–101.
Gaudry P., Harley R. Counting points on hyperelliptic curves over finite fields / In W. Bosma, ed. // ANTS IV. –LNCS 1838. –Berlin: Springer–Verlag, 2000. –pp.297–312.
Harley R. Fast arithmetic on genius two curves. –2000. Available at: http://cristal.inria.fr/harley/hyper/, adding.txt and doubling.c
Jacobson M. (Jr), Menezes A., Stain A. Hyperelliptic curves and cryptography // Report of Fields Institute Communications. –Vol.7. –2002. –28p.
Lange T. Efficient arithmetic on hyperelliptic curves: PhD thesis: Mathematics and Informatics. –University of Essen: Institute for experimental mathematics. – Germany: Essen, 2001. –122p.
Matsuo K., Chao J., Tsujii S. Fast genius two hyperelliptic curve cryptosystem // Technical report IEICE. –ISEC2001–31. –IEICE`2001. –2001. –8p.
Miyamoto Y., Doi H., Matsuo K., Chao J., Tsujii S. A fast addition algorithm of genius two hyperelliptic curve // In the 2002 Symposium on cryptography and information security. –SCIS`2002. Japan: IEICE, 2002. –pp.497– 502. (In Japanese)
Takahashi M. Improving Harley algorithms for jacobians of genius 2 hyperelliptic curves // In the 2002 Symposium on cryptography and information security. –SCIS`2002. Japan: IEICE, 2002. –pp.155–160. (In Japanese)
Sugizaki H., Matsuo K., Chao J., Tsujii S. An extension of Harley addition algorithm for hyperelliptic curves over finite fields of characteristic two // Technical report IEICE. –ISEC2002–09. –IEICE`2002. –2002. –8p.
Lange T. Efficient arithmetic on genius 2 hyperelliptic curves over finite fields via explicit formulae // Cryptology ePrint Archive. –Report 2002/121. –2002. –13p. Available http://eprint.iacr.org.
Lange T. Inversion–free arithmetic on genius 2 hyperelliptic curves // Cryptology ePrint Archive. –Report 2002/147. –2002. –7p. Available http://eprint.iacr.org.
Lange T. Formulae for arithmetic on genius 2 hyperelliptic curves. September 2003. Available http://www.ruhr–uni–bochum.de/itsc/tanja/preprints/expl_sub.pdf.
Lange T. Weighted coordinates on genius 2 hyperelliptic curves // Cryptology ePrint Archive. –Report 2002/153. –2002. –20p. Available http://eprint.iacr.org.
Wollinger T. Software and hardware implementation of hyperelliptic curve cryptosystems: PhD dissertation: Electronics and informatics. –Worchester Polytechnic Institute. –Germany: Bochum, 2004. –218p.
Chudnovsky D.V., Chudnovsky G.V. Sequence of number generated by addition in formal group and new primality and factorization test // Advanced in Applied Math. –№8. –1986. –pp.385–434.
Ковтун В.Ю., Збитнев С.И. Арифметические операции в якобиане гиперэллиптической кривой рода 2 в проективных координатах с уменьшенной сложностью // Восточно–Европейский журнал передовых технологий. –2004. –Вып. №½ (13). –С. 14–22.
Cohen H., Miyaji A., Ono T. Efficient elliptic curve exponentiation using mixed coordinates // Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology. –CRYPTO`98. –LNCS 1514. –Berlin: Springer–Verlag, 1998. –pp. 51–65.
N. Meloni. New point addition formul. for ECC applications. In C. Carlet and B. Sunar, editors, Arithmetic of Finite Fields (WAIFI 2007), LNCS 4547, Springer, 2007, pp. 189–201.
Downloads
Issue
Section
License
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
