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Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$
1. | CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, F-34095 Montpellier, France |
2. | Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada |
References:
[1] |
R. Avanzi, M. J. Jacobson, Jr. and R. Scheidler, Efficient divisor reduction on hyperelliptic curves, Adv. Math. Commun., 4 (2010), 261-279.
doi: 10.3934/amc.2010.4.261. |
[2] |
D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp., 48 (1987), 95-101.
doi: 10.1090/S0025-5718-1987-0866101-0. |
[3] |
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996. |
[4] |
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Boca Raton, 2006. |
[5] |
C. Costello and K. Lauter, Group law computations on jacobians of hyperelliptic curves, in Selected Areas in Cryptography, 2011, 92-117. |
[6] |
X. Ding, Acceleration of algorithm for the reduced sum of two divisors of a hyperelliptic curve, in Information Computing and Applications, Springer, 2011, 177-184.
doi: 10.1007/978-3-642-16336-4_24. |
[7] |
A. Enge and P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith., 102 (2002), 83-103.
doi: 10.4064/aa102-1-6. |
[8] |
G. Frey, Applications of arithmetical geometry to cryptographic constructions, in Proceedings of the Fifth International Conference on Finite Fields and Applications, Springer, 2001, 128-161. |
[9] |
P. Gaudry, F. Hess and N. Smart, Constructive and destructive facets of Weil descent on elliptic curves, J. Cryptology, 15 (2002), 19-46.
doi: 10.1007/s00145-001-0011-x. |
[10] |
P. Gaudry, E. Thomé, N. Thériault and C. Diem, A double large prime variation for small genus hyperelliptic index calculus, Math. Comp., 76 (2007), 475-492.
doi: 10.1090/S0025-5718-06-01900-4. |
[11] |
L. Imbert, M. J. Jacobson, Jr. and A. Schmidt, Fast ideal cubing in imaginary quadratic number and function fields, Adv. Math. Commun., 4 (2010), 237-260.
doi: 10.3934/amc.2010.4.237. |
[12] |
M. J. Jacobson, Jr., A. J. Menezes and A. Stein, Solving elliptic curve discrete logarithm problems using Weil descent, J. Ramanujan Math. Soc., 16 (2001), 231-260. |
[13] |
M. J. Jacobson, Jr. and A. J. van der Poorten, Computational aspects of NUCOMP, in Algorithmic Number Theory - ANTS-V, Springer-Verlag, Berlin, 2002, 120-133.
doi: 10.1007/3-540-45455-1_10. |
[14] |
M. J. Jacobson, Jr., R. E. Sawilla and H. C. Williams, Efficient ideal reduction in quadratic fields, Int. J. Math. Comp. Sci., 1 (2006), 83-116. |
[15] |
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, World Scientific Publishing, 2007, 201-244.
doi: 10.1142/9789812772022_0013. |
[16] |
M. J. Jacobson, Jr. and H. C. Williams, Solving the Pell Equation, Springer-Verlag, 2009. |
[17] |
D. E. Knuth, The Art of Computer Programming, Third edition, Addison-Wesley, Reading, MA, 1997. |
[18] |
N. Koblitz, Elliptic curve cryptosystems, Math. Comp., 48 (1987), 203-209.
doi: 10.1090/S0025-5718-1987-0866109-5. |
[19] |
N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology, 1 (1989), 139-150.
doi: 10.1007/BF02252872. |
[20] |
V. Miller, Use of elliptic curves in cryptography, in Advances in Cryptology - CRYPTO '85, 1986, 417-426.
doi: 10.1007/3-540-39799-X_31. |
[21] |
M. Musson, Another Look at the Gaudry, Hess and Smart Attack on the Elliptic Curve Discrete Logarithm Problem, Ph.D thesis, University of Calgary, 2011. |
[22] |
K. Nagao, Improving group law algorithms for Jacobians of hyperelliptic curves, in Algorithmic Number Theory (Leiden, 2000), Springer, Berlin, 2000, 439-447.
doi: 10.1007/10722028_28. |
[23] |
V. Shoup, NTL: A Library for doing Number Theory, Software, 2010. Available from: http://www.shoup.net/ntl. |
[24] |
E. Teske, An elliptic curve trapdoor system, J. Cryptology, 19 (2006), 115-133.
doi: 10.1007/s00145-004-0328-3. |
[25] |
M. D. Velichka, M. J. Jacobson, Jr. and A. Stein, Computing discrete logarithms on high-genus hyperelliptic curves over even characteristic finite fields, Math. Comp., to appear.
doi: 10.1090/S0025-5718-2013-02748-2. |
show all references
References:
[1] |
R. Avanzi, M. J. Jacobson, Jr. and R. Scheidler, Efficient divisor reduction on hyperelliptic curves, Adv. Math. Commun., 4 (2010), 261-279.
doi: 10.3934/amc.2010.4.261. |
[2] |
D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp., 48 (1987), 95-101.
doi: 10.1090/S0025-5718-1987-0866101-0. |
[3] |
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996. |
[4] |
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Boca Raton, 2006. |
[5] |
C. Costello and K. Lauter, Group law computations on jacobians of hyperelliptic curves, in Selected Areas in Cryptography, 2011, 92-117. |
[6] |
X. Ding, Acceleration of algorithm for the reduced sum of two divisors of a hyperelliptic curve, in Information Computing and Applications, Springer, 2011, 177-184.
doi: 10.1007/978-3-642-16336-4_24. |
[7] |
A. Enge and P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith., 102 (2002), 83-103.
doi: 10.4064/aa102-1-6. |
[8] |
G. Frey, Applications of arithmetical geometry to cryptographic constructions, in Proceedings of the Fifth International Conference on Finite Fields and Applications, Springer, 2001, 128-161. |
[9] |
P. Gaudry, F. Hess and N. Smart, Constructive and destructive facets of Weil descent on elliptic curves, J. Cryptology, 15 (2002), 19-46.
doi: 10.1007/s00145-001-0011-x. |
[10] |
P. Gaudry, E. Thomé, N. Thériault and C. Diem, A double large prime variation for small genus hyperelliptic index calculus, Math. Comp., 76 (2007), 475-492.
doi: 10.1090/S0025-5718-06-01900-4. |
[11] |
L. Imbert, M. J. Jacobson, Jr. and A. Schmidt, Fast ideal cubing in imaginary quadratic number and function fields, Adv. Math. Commun., 4 (2010), 237-260.
doi: 10.3934/amc.2010.4.237. |
[12] |
M. J. Jacobson, Jr., A. J. Menezes and A. Stein, Solving elliptic curve discrete logarithm problems using Weil descent, J. Ramanujan Math. Soc., 16 (2001), 231-260. |
[13] |
M. J. Jacobson, Jr. and A. J. van der Poorten, Computational aspects of NUCOMP, in Algorithmic Number Theory - ANTS-V, Springer-Verlag, Berlin, 2002, 120-133.
doi: 10.1007/3-540-45455-1_10. |
[14] |
M. J. Jacobson, Jr., R. E. Sawilla and H. C. Williams, Efficient ideal reduction in quadratic fields, Int. J. Math. Comp. Sci., 1 (2006), 83-116. |
[15] |
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, World Scientific Publishing, 2007, 201-244.
doi: 10.1142/9789812772022_0013. |
[16] |
M. J. Jacobson, Jr. and H. C. Williams, Solving the Pell Equation, Springer-Verlag, 2009. |
[17] |
D. E. Knuth, The Art of Computer Programming, Third edition, Addison-Wesley, Reading, MA, 1997. |
[18] |
N. Koblitz, Elliptic curve cryptosystems, Math. Comp., 48 (1987), 203-209.
doi: 10.1090/S0025-5718-1987-0866109-5. |
[19] |
N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology, 1 (1989), 139-150.
doi: 10.1007/BF02252872. |
[20] |
V. Miller, Use of elliptic curves in cryptography, in Advances in Cryptology - CRYPTO '85, 1986, 417-426.
doi: 10.1007/3-540-39799-X_31. |
[21] |
M. Musson, Another Look at the Gaudry, Hess and Smart Attack on the Elliptic Curve Discrete Logarithm Problem, Ph.D thesis, University of Calgary, 2011. |
[22] |
K. Nagao, Improving group law algorithms for Jacobians of hyperelliptic curves, in Algorithmic Number Theory (Leiden, 2000), Springer, Berlin, 2000, 439-447.
doi: 10.1007/10722028_28. |
[23] |
V. Shoup, NTL: A Library for doing Number Theory, Software, 2010. Available from: http://www.shoup.net/ntl. |
[24] |
E. Teske, An elliptic curve trapdoor system, J. Cryptology, 19 (2006), 115-133.
doi: 10.1007/s00145-004-0328-3. |
[25] |
M. D. Velichka, M. J. Jacobson, Jr. and A. Stein, Computing discrete logarithms on high-genus hyperelliptic curves over even characteristic finite fields, Math. Comp., to appear.
doi: 10.1090/S0025-5718-2013-02748-2. |
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