Comparison of scalar multiplication on real hyperelliptic curves

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November 2014, 8(4): 389-406. doi: 10.3934/amc.2014.8.389

Comparison of scalar multiplication on real hyperelliptic curves

Michael J. Jacobson, Jr. ^1, , Monireh Rezai Rad ^2, and Renate Scheidler ^2,

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

Received March 2014 Revised September 2014 Published November 2014

Abstract
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Real hyperelliptic curves admit two structures suitable for cryptography --- the Jacobian (a finite abelian group) and the infrastructure. Mireles Morales described precisely the relationship between these two structures, and made the assertion that when implemented with balanced divisor arithmetic, the Jacobian generically yields more efficient arithmetic than the infrastructure for cryptographic applications. We confirm that this assertion holds for genus two curves, through rigorous analysis and the first detailed numerical performance comparisons, showing that cryptographic key agreement can be performed in the Jacobian without any extra operations beyond those required for basic scalar multiplication. We also present a modified version of Mireles Morales' map that more clearly reveals the algorithmic relationship between the two structures.

Keywords: scalar multiplication, Jacobian, infrastructure, Real hyperelliptic curve, cryptographic key agreement., balanced divisor.

Mathematics Subject Classification: Primary: 11G20, 11R11, 11R29, 11Y40; Secondary: 11Y16, 14G5.

Citation: Michael J. Jacobson, Jr., Monireh Rezai Rad, Renate Scheidler. Comparison of scalar multiplication on real hyperelliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 389-406. doi: 10.3934/amc.2014.8.389

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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