Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 4547 LNCS, Issue , 2007, Pages 202-218

  Erickson, Stefan ^a   Jacobson Jr , Michael J ^b   Shang, Ning ^c   Shen, Shuo ^c   Stein, Andreas ^d  

^a Department of Psychology   (United States)

^b UNIVERSITY OF CALGARY   (Canada)

^c PURDUE UNIVERSITY   (United States)

^d UNIVERSITY OF WYOMING   (United States)

Author keywords

Cantor's algorithm; Cryptographic key exchange; Efficient implementation; Explicit formulas; Hyperelliptic curve; Infrastructure and distance; Reduced divisor

Indexed keywords

CURVE FITTING; FINITE ELEMENT METHOD; GENETIC ALGORITHMS; MATHEMATICAL MODELS; OPTIMIZATION; PUBLIC KEY CRYPTOGRAPHY;

CANTOR'S ALGORITHMS; CRYPTOGRAPHIC KEY EXCHANGE; EFFICIENT IMPLEMENTATION; EXPLICIT FORMULAS; HYPERELLIPTIC CURVES; REDUCED DIVISOR;

LINEAR SYSTEMS;

EID: 38349058155     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-73074-3_16     Document Type: Conference Paper

References (21)

1. Avanzi, R.M.: Aspects of hyperelliptic curves over large prime fields in software implementations. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 148-162. Springer, Heidelberg (2004)
2. Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications, vol. 34. Chapman & Hall/CRC, Sydney, Australia (2005)
3. Enge, A.: How to distinguish hyperelliptic curves in even characteristic. In: Alster, K., Urbanowicz, J., Williams, H.C., (eds.) Public-Key Cryptography and Computational Number Theory, pp. 49-58, De Gruyter, Berlin (2001)
4. Gaudry, P.: On breaking the discrete log on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19-34. Springer, Heidelberg (2000)
5. Gaudry, P., Thomé, E., Thériault, N., Diem, C.: A double large prime variation for small genus hyperelliptic index calculus. Mathematics of Computation 76, 475-492 (2007)
6. Jacobson Jr., M.J., Menezes, A.J., Stein, A.: Hyperelliptic curves and cryptography. In: High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications Series, vol. 41, pp. 255-282. American Mathematical Society (2004)
7. Jacobson Jr., M.J., Scheidler, R., Stein, A.: Cryptographic protocols on real and imaginary hyperelliptic curves. Accepted to Advances in Mathematics of Communications pending revisions (2007)
8. Jacobson Jr., M. J., Scheidler, R., Stein, A.: Fast Arithmetic on Hyperelliptic Curves Via Continued Fraction Expansions. To appear in Advances in Coding Theory and Cryptology. In: Shaaska, T., Huffman, W.C., Joyner, D., Ustimenko, V. (eds.) Series on Coding, Theory and Cryptology vol. 2, World Scientific Publishing (2007)
9. Koblitz, N.: Hyperelliptic cryptosystems. Journal of Cryptology 1, 139-150 (1988)
10. Lange, T.: Formulae for arithmetic on genus 2 hyperelliptic curves. Applicable Algebra in Engineering, Communication, and Computing 15, 295-328 (2005)
11. Menezes, A.J., Wu, Y., Zuccherato, R.J.: An elementary introduction to hyperelliptic curves. Technical Report CORR 96-19, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, 1996. In: Koblitz, N. (ed.) Algebraic Aspects of Cryptography, Springer-Verlag, Berlin Heidelberg New York (1998)
12. Müller, V., Stein, A., Thiel, C.: Computing discrete logarithms in real quadratic congruence function fields of large genus. Mathematics of Computation 68, 807-822 (1999)
13. Mumford, D.: Tata Lectures on Theta I, II. Birkhäuser, Boston (1983/84)
14. National Institute of Standards and Technology (NIST). Recommendation on key establishment schemes. NIST Special Publication 800-56 (January 2003)
15. Paulus, S., Rück, H.-G.: Real and imaginary quadratic representations of hyperelliptic function fields. Mathematics of Computation 68, 1233-1241 (1999)
16. Pelzl, J., Wollinger, T., Paar, C.: Low cost security: explicit formulae for genus-4 hyperelliptic curves. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 1-16. Springer, Heidelberg (2003)
17. Scheidler, R.: Cryptography in quadratic function fields. Designs, Codes and Cryptography 22, 239-264 (2001)
18. Scheidler, R., Stein, A., Williams, H.C.: Key-exchange in real quadratic congruence function fields. Designs, Codes and Cryptography 7, 153-174 (1996)
19. V. Shoup. NTL: A library for doing number theory. Software (2001) See http://www.shoup.net/ntl.
20. Stein, A.: Sharp upper bounds for arithmetics in hyperelliptic function fields. Journal of the Ramanujan Mathematical Society 9-16(2), 1-86 (2001)
21. Wollinger, T., Pelzl, J., Paar, C.: Cantor versus Harley: optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems. IEEE Transactions on Computers 54, 861-872 (2005)