Cryptographic aspects of real hyperelliptic curves
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- [1] AVANZI, R.-JACOBSON, M. J., JR.-SCHEIDLER, R.: Efficient reduction of large divisors on hyperelliptic curves, Adv. Math. Communications 4 (2010), 261-279.
- [2] BERNSTEIN, D. J.-LANGE, T.: Faster addition and doubling on elliptic curves, in: Advances in Cryptology-ASIACRYPT ’07, Kuching, Malaysia, 2007 (K. Kurosawa, ed.), Lecture Notes in Comput. Sci., Vol. 4833, Springer, Berlin, 2008, pp. 29-50.
- [3] BUCHMANN, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, in: S´eminaire de Th´eorie des Nombres, Paris, 1988-89, Prog. Math., Vol. 91, 1990, pp. 27-41.
- [4] CANTOR, D. G.: Computing the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), 95-101.
- [5] CASSELS, J. W. S.-FLYNN, E. V.:, in: London Math. Soc. Lecture Note Ser., Vol. 230, Cambridge Univ. Press, Cambridge, 1996.
- [6] CHEN, Z.-STORJOHANN, A.-FLETCHER, C.:, http://www.cs.uwaterloo.ca/ z4chen/iml.html, 2007.
- [7] COHEN, H.-FREY, G.-AVANZI, R.-DOCHE, C.-LANGE, T.-NGUYEN, K.- -VERCAUTEREN, F.:, in: Discrete Math. Appl., Chapman & Hall/CRC, Boca Raton, FL, 2006.
- [8] DIFFIE, W.-HELLMAN, M. E.: New directions in cryptography, IEEE Trans. Inform. Theory 22 (1976), 644-654.
- [9] ELGAMAL, V.: A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory IT-31 (1985), 469-472.
- [10] ERICKSON, S.-HO, T.-ZEMEDKUN, S.: Explicit projective formulas for real hyperelliptic curves of genus 2, 2009 (in preparation).
- [11] ERICKSON, S.-JACOBSON,M. J., JR.-SHANG, N.-SHEN, S.-STEIN, A.: Explicit formulas for real hyperelliptic curves of genus 2 in affine representation, in: Proceedings of First InternationalWorkshop-WAIFI ’07, Madrid, 2007 (C. Carlet et al., eds.), Lecture Notes in Comput. Sci., Vol. 4547, Springer, Berlin, 2007, pp. 202-218.
- [12] ERICKSON, S.-JACOBSON, M. J. JR.-STEIN, A.: Explicit formulas for real hyperelliptic curves of genus 2 in affine representation, 2009 (in preparation).
- [13] FONTEIN, F.: Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures, Adv. Math. Commun. 2 (2008), 293-307.
- [14] ,. Ph.D. Thesis, University of Z¨urich, Z¨urich, Switzerland, 2008.
- [15] GALBRAITH, S. D.-HARRISON, M.-MIRELES MORALES, D. J.: Efficient hyperelliptic curve arithmetic using balanced representation for divisors, in: Algorithmic Number Theory-ANTS ’08, Banff, Canada, 2008 (A. van der Poorten, ed.), Lecture Notes in Comput. Sci., Vol. 5011, Springer, Berlin, 2008, pp. 342-356.
- [16] GALBRAITH, S. D.-LIN, X.-MIRELES MORALES, D. J.: Pairings on hyperelliptic curves with a real model, in: Pairing-Based Cryptography-Pairing ’08 (S. Galbraith, ed.) Egham, UK, 2008, Lecture Notes in Comput. Sci., Vol. 5209, Springer, Berlin, 2008, pp. 265-281.
- [17] GALBRAITH, S. D.-PUJOLAS, J.-RITZENTHALER, C.-SMITH, B.: Distortion maps for supersingular genus two curves, J. Math. Cryptology 3 (2009), 1-18.
- [18] GAUDRY, P.-THOM´E, E.-TH´ERIAULT, N.-DIEM, C.: A double large prime variation for small genus hyperelliptic index calculus, Math. Comp. 76 (2007), 475-492.
- [19] HAMMELL, J. F.:. Master’s Thesis, University of Calgary, Canada, 2008.
- [20] HAMMELL, J. F.-JACOBSON, M. J., JR.: Index-calculus algorithms in real quadratic function fields, 2009 (in preparation).
- [21] HANKERSON, D.-MENEZES, A.-VANSTONE, S.:. Springer, New York, 2004.
- [22] IMBERT, L.-JACOBSON, M. J., JR.-SCHMIDT, A.: Fast ideal cubing in imaginary quadratic number and function fields, Adv. Math. Communications 4 (2010), 237-260.
- [23] JACOBSON, M. J., JR.-MENEZES, A. J.-STEIN, A.: Hyperelliptic curves and cryptography, in: Selected papers from the International Conference on Number Theory, Banff, AB, Canada, 2003 (A. van der Poorten, et. al., eds.), Fields Inst. Commun., Vol. 41, Amer. Math. Soc., Providence, RI, 2004, pp. 255-282.
- [24] JACOBSON, M. J., JR.-SCHEIDLER, R.-STEIN, A.: Cryptographic protocols on real and imaginary hyperelliptic curves, Adv. Math. Commun. 1 (2007), 197-221.
- [25] , Fast arithmetic on hyperelliptic curves via continued fraction expansions, in: Advances in Coding Theory and Cryptology (T. Shaska, T. et al., eds.), Series on Coding Theory and Cryptology, Vol. 3, World Scientific, Hackensack, NJ, 2007, pp. 200-243.
- [26] JACOBSON, M. J., JR.-STEIN, A.-VELICHKA, M. D.: Computing discrete logarithms on high-genus hyperelliptic curves over even characteristic finite fields (in preparation), 2009.
- [27] JACOBSON, M. J., JR.-VAN DER POORTEN, A. J.: , Computational aspects of NUCOMP, in: Algorithmic Number Theory-ANTS-V, Sydney, Australia, 2002 (C. Fieker et al., eds.), Lecture Notes in Comput. Sci., Vol. 2369, Springer, Berlin, 2002, pp. 120-133.
- [28] KOBLITZ, N.: Elliptic curve cryptosystems, Math. Comp. 48 (1987), 203-209.
- [29] , Hyperelliptic cryptosystems, J. Cryptology 1 (1989), 139-150.
- [30] LANGE, T.: Formulae for arithmetic on genus 2 hyperelliptic curves, Appl. Algebra Engrg. Comm. Comput. 15 (2005), 295-328.
- [31] MENEZES, A. J.-WU, Y.-H.-ZUCCHERATO, R. J.: An elementary introduction to hyperelliptic curves, in: Algebraic Aspects of Cryptography, Algorithms Comput. Math., Vol. 3, Springer, Berlin, 1998, pp. 155-178.
- [32] MILLER, V.: Use of elliptic curves in cryptography, in: Advances in Cryptology- -CRYPTO ’85, Santa Barbara, California, 1985, Lecture Notes in Comput. Sci., Vol. 218, Springer, Berlin, 1986, pp. 417-426.
- [33] MIRELES MORALES, D. J.: An analysis of the infrastructure in real function fields, Eprint archive no. 2008/299, 2008.
- [34] M¨ULLER,V.-STEIN, A.-THIEL, C.: Computing discrete logarithms in real quadratic congruence function fields of large genus, Math. Comp. 68 (1999), 807-822.
- [35] PAULUS, S.-R¨UCK, H.-G.: Real and imaginary quadratic representations of hyperelliptic function fields, Math. Comp. 68 (1999), 1233-1241.
- [36] POHLIG, S. C.-HELLMAN, M. E.: An improved algorithm for computing logarithms over GF(p) and it’s cryptographic significance, IEEE Trans. Inf. Theory 24 (1978), 106-110.
- [37] SCHEIDLER, R.: Cryptography in quadratic function fields, Des. Codes Cryptogr. 22 (2001), 239-264.
- [38] SCHEIDLER, R.-BUCHMANN, J. A.-WILLIAMS, H. C.: A key exchange protocol using real quadratic fields, J. Cryptology 7 (1994), 171-199.
- [39] SCHEIDLER, R.-STEIN, A.-WILLIAMS, H. C.: Key-exchange in real quadratic congruence function fields, Des. Codes Cryptogr. 7 (1996), 153-174.
- [40] SHANKS, D.: The infrastructure of a real quadratic field and its applications, in: Proceedings of Number Theory Conf., Univ. Colorado, Boulder, Colorado, 1972, pp. 217-224.
- [41] SHOUP, V.:, 2008, http://www.shoup.net.
- [42] STEIN, A.: Equivalences between elliptic curves and real quadratic congruence function fields, J. Th´eorie Nombr. Bordeaux 9 (1997), 75-95.
- [43] , Sharp upper bounds for arithmetics in hyperelliptic function fields, J. Ramanujan Math. Soc. 16 (2001), 1-86.
- [44] STEIN, A.-TESKE, E.: Explicit bounds and heuristics on class numbers in hyperelliptic function fields, Math. Comp. 71 (2002), 837-861.
- [45] , The parallelized Pollard kangaroo method in real quadratic function fields, Math. Comp. 71 (2002), 793-814.
- [46] , Optimized baby-step giant-step methods in hyperelliptic function fields, J. Ramanujan Math. Soc. 20 (2005), 1-32.
- [47] STICHTENOTH, H.:(2nd ed.), Springer, Berlin, 2009.
- [48] VELICHKA, M. D.:. Master’s thesis, University of Calgary, Calgary, Canada, 2008.
DOI: https://doi.org/10.2478/v10127-010-0030-9 | Journal eISSN: 1338-9750 (formerly 1210-3195) | Journal ISSN: 1210-3195
Language: English
Page range: 31 - 65
Published on: Nov 13, 2012
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
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© 2012 Michael J. Jacobson, Renate Scheidler, Andreas Stein, published by Slovak Academy of Sciences, Mathematical Institute
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