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Trisection for supersingular genus $2$ curves in characteristic $2$
Comparison of scalar multiplication on real hyperelliptic curves
1. | Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada |
2. | Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada, Canada |
References:
[1] |
E. Barker, W. Barker, W. Polk and M. Smid, Recommendation for key management - part 1: General (revised), NIST Special Publication 800-57, 2007. |
[2] |
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercouteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Boca Raton, 2006.
doi: 10.1201/9781420034981. |
[3] |
W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inf. Theory, 22 (1976), 472-492.
doi: 10.1109/TIT.1976.1055638. |
[4] |
S. Erickson, M. J. Jacobson, Jr. and A. Stein, Explicit formulas for real hyperelliptic curves of genus $2$ in affine representation, Adv. Math. Commun., 5 (2011), 623-666.
doi: 10.3934/amc.2011.5.623. |
[5] |
F. Fontein, Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures, Adv. Math. Commun., 2 (2008), 293-307.
doi: 10.3934/amc.2008.2.293. |
[6] |
F. Fontein, Holes in the infrastructure of global hyperelliptic function fields,, preprint, ().
|
[7] |
E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des Nombres (Québec, PQ), de Gruyter, Berlin, 1989, 227-239. |
[8] |
S. D. Galbraith, M. Harrison and D. J. Mireles Morales, Efficient hyperelliptic curve arithmetic using balanced representation for divisors, in Algorithmic Number Theory - ANTS 2008 (Berlin), Springer, 2008, 342-356.
doi: 10.1007/978-3-540-79456-1_23. |
[9] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Cryptographic protocols on real hyperelliptic curves, Adv. Math. Commun., 1 (2007), 197-221.
doi: 10.3934/amc.2007.1.197. |
[10] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Fast arithmetic on hyperelliptic curves via continued fraction expansions, in Advances in Coding Theory and Cryptology (eds. T. Shaska, W.C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing, 2007, 201-244.
doi: 10.1142/9789812772022_0013. |
[11] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Cryptographic aspects of real hyperelliptic curves, Tatra Mountains Math. Publ., 40 (2010), 1-35.
doi: 10.2478/v10127-010-0030-9. |
[12] |
N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology, 1 (1989), 139-150.
doi: 10.1007/BF02252872. |
[13] |
T. Lange, Formulae for arithmetic on genus 2 hyperelliptic curves, Appl. Algebra Eng. Commun. Comput., 15 (2005), 295-328.
doi: 10.1007/s00200-004-0154-8. |
[14] |
D. J. Mireles Morales, An analysis of the infrastructure in real function fields, Cryptology eprint archive no. 2008/299, 2008. |
[15] |
R. Scheidler, J. A. Buchmann and H. C. Williams, A key exchange protocol using real quadratic fields, J. Cryptology, 7 (1994), 171-199.
doi: 10.1007/BF02318548. |
[16] |
R. Scheidler, A. Stein and H. C. Williams, Key-exchange in real quadratic congruence function fields, Des. Codes Crypt., 7 (1996), 153-174.
doi: 10.1007/BF00125081. |
[17] |
V. Shoup, NTL: A Library for doing Number Theory (version 5.4.2), http://www.shoup.net, 2008. |
[18] |
A. Stein, Explicit infrastructure for real quadratic function fields and real hyperelliptic curves, Glas. Mat. Ser. III, 44(64) (2009), 89-126.
doi: 10.3336/gm.44.1.05. |
show all references
References:
[1] |
E. Barker, W. Barker, W. Polk and M. Smid, Recommendation for key management - part 1: General (revised), NIST Special Publication 800-57, 2007. |
[2] |
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercouteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Boca Raton, 2006.
doi: 10.1201/9781420034981. |
[3] |
W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inf. Theory, 22 (1976), 472-492.
doi: 10.1109/TIT.1976.1055638. |
[4] |
S. Erickson, M. J. Jacobson, Jr. and A. Stein, Explicit formulas for real hyperelliptic curves of genus $2$ in affine representation, Adv. Math. Commun., 5 (2011), 623-666.
doi: 10.3934/amc.2011.5.623. |
[5] |
F. Fontein, Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures, Adv. Math. Commun., 2 (2008), 293-307.
doi: 10.3934/amc.2008.2.293. |
[6] |
F. Fontein, Holes in the infrastructure of global hyperelliptic function fields,, preprint, ().
|
[7] |
E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des Nombres (Québec, PQ), de Gruyter, Berlin, 1989, 227-239. |
[8] |
S. D. Galbraith, M. Harrison and D. J. Mireles Morales, Efficient hyperelliptic curve arithmetic using balanced representation for divisors, in Algorithmic Number Theory - ANTS 2008 (Berlin), Springer, 2008, 342-356.
doi: 10.1007/978-3-540-79456-1_23. |
[9] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Cryptographic protocols on real hyperelliptic curves, Adv. Math. Commun., 1 (2007), 197-221.
doi: 10.3934/amc.2007.1.197. |
[10] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Fast arithmetic on hyperelliptic curves via continued fraction expansions, in Advances in Coding Theory and Cryptology (eds. T. Shaska, W.C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing, 2007, 201-244.
doi: 10.1142/9789812772022_0013. |
[11] |
M. J. Jacobson, Jr., R. Scheidler and A. Stein, Cryptographic aspects of real hyperelliptic curves, Tatra Mountains Math. Publ., 40 (2010), 1-35.
doi: 10.2478/v10127-010-0030-9. |
[12] |
N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology, 1 (1989), 139-150.
doi: 10.1007/BF02252872. |
[13] |
T. Lange, Formulae for arithmetic on genus 2 hyperelliptic curves, Appl. Algebra Eng. Commun. Comput., 15 (2005), 295-328.
doi: 10.1007/s00200-004-0154-8. |
[14] |
D. J. Mireles Morales, An analysis of the infrastructure in real function fields, Cryptology eprint archive no. 2008/299, 2008. |
[15] |
R. Scheidler, J. A. Buchmann and H. C. Williams, A key exchange protocol using real quadratic fields, J. Cryptology, 7 (1994), 171-199.
doi: 10.1007/BF02318548. |
[16] |
R. Scheidler, A. Stein and H. C. Williams, Key-exchange in real quadratic congruence function fields, Des. Codes Crypt., 7 (1996), 153-174.
doi: 10.1007/BF00125081. |
[17] |
V. Shoup, NTL: A Library for doing Number Theory (version 5.4.2), http://www.shoup.net, 2008. |
[18] |
A. Stein, Explicit infrastructure for real quadratic function fields and real hyperelliptic curves, Glas. Mat. Ser. III, 44(64) (2009), 89-126.
doi: 10.3336/gm.44.1.05. |
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