Reducing logarithms in totally non-maximal imaginary quadratic orders to logarithms in finite fields

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

Volumn 1716, Issue , 1999, Pages 220-231

  Huhnlein, Detlef ^a   Takagi, Tsuyoshi ^b  

^a Westfälische Hochschule und secunet Security Networks AG   (Germany)

^b NTT Information Sharing Platform Laboratories   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CODES (SYMBOLS); COMPUTATION THEORY; COMPUTATIONAL EFFICIENCY; PUBLIC KEY CRYPTOGRAPHY; SECURITY OF DATA;

BIT-OPERATIONS; CLASS NUMBERS; DISCRETE LOGARITHM PROBLEMS; DISCRETE LOGARITHMS; FINITE FIELDS; NUMBER FIELD SIEVE; PUBLIC KEY CRYPTOSYSTEMS; SUB-EXPONENTIAL ALGORITHMS;

C (PROGRAMMING LANGUAGE);

EID: 46749154744     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-48000-6_18     Document Type: Conference Paper

References (31)

1. E. Bach, J. Shallit: Algorithmic number theory, vol. 1 - Efficient Algorithms, Foundations of computing, MIT press, ISBN 0-262-02405-5, (1996)
2. I. Biehl and J. Buchmann: “An analysis of the reduction algorithms for binary quadratic forms,” Voronoi’s Impact on Modern Science, vol. 1, Institute of Mathematics of National Academy of Sciences, Kyiv, Ukraine, (1998)
3. I. Biehl, S. Paulus and T. Takagi: “An efficient undeniable signature scheme based on non-maximal imaginary quadratic orders,” Proceedings of Mathematics of Public Key Cryptography, Toronto, (1999)
4. R. Brent: ECM champs, ftp.comlab.ox.ac.uk/pub/Documents/techpapers/ Richard.Brent/champs.ecm
5. J. Buchmann and S. D¨ullmann: “On the computation of discrete logarithms in class groups,” Advances in Cryptology - CRYPTO’90, Springer, LNCS 773, (1991), pp.134-139
6. J. Buchmann, S. D¨ullmann, and H.C. Williams: “On the complexity and efficiency of a new key exchange system,” Advances in Cryptology - EUROCRYPT’89, Springer, LNCS 434, (1990), pp.597-616
7. Z.I. Borevich and I.R. Shafarevich: Number Theory, Academic Press, New York, (1966)
8. J. Buchmann and H.C. Williams: “A key-exchange system based on imaginary quadratic fields,” Journal of Cryptology, 1, (1988), pp.107-118
9. J. Cowie, B. Dodson, M. Elkenbracht-Huizing, A.K. Lenstra, P.L. Montgomery and J. Zayer: “A worldwide number field sieve factoring record: on to 512 bits,” Advances in Cryptology - ASIACRYPT’96, Springer, LNCS 1163, (1996), pp.382-394
10. H. Cohen: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer: Berlin, (1993)
11. D. Coppersmith, A.M. Odlyzko and R. Schroeppel: “Discrete logarithms in GF(p),” Algorithmica, 1, (1986), pp.1-15
12. 2, John Wiley & Sons, New York, (1989)
13. W. Diffie and M. Hellman: “New directions in cryptography,” IEEE Transactions on Information Theory 22, (1976), pp.472-492
14. S. D¨ullmann: Ein Algorithmus zur Bestimmung der Klassenzahl positiv definiter binärer quadratischer Formen, PHD-thesis, University of Saarbr¨ucken, (1991)
15. D.M. Gordon: “Discrete logarithms in GF(p) using the number field sieve,” SIAM Journal on Discrete Mathematics, 6, (1993), pp.124-138 222
16. J.L. Hafner and K.S. McCurley: “A rigorous subexponential algorithm for computation of class groups,” Journal of the American Mathematical Society, 2, (1989), pp.837-850
17. L.K. Hua: Introduction to Number Theory. Springer-Verlag, New York, (1982)
18. M. Hartmann, S. Paulus and T. Takagi: “NICE - New Ideal Coset Encryption -,” to appear in Workshop on Cryptographic Hardware and Embedded Systems.
19. D. Hühnlein, M.J. Jacobson, S. Paulus and T. Takagi: “A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption,” Advances in Cryptology - EUROCRYPT’98, LNCS 1403, Springer, (1998), pp.294-307
20. D. Hühnlein, A. Meyer and T. Takagi: “Rabin and RSA analogues based on non-maximal imaginary quadratic orders,” Proceedings of ICICS’98, ISBN 89-8530514-X, (1998), pp.221-240 223
21. D. Hühnlein: “Efficient implementation of cryptosystems based on non-maximal imaginary quadratic orders,” T.R. TI-6, Technische Universtät Darmstadt, (1999), available at http://www.informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/ Welcome.html#1999
22. M.J. Jacobson Jr.: Subexponential Class Group Computation in Quadratic Orders, PhD thesis, Technische Universtät Darmstadt, to appear, (1999)
23. H.W. Lenstra: “On the computation of regulators and class numbers of quadratic fields,” London Math. Soc. Lecture Notes, 56, (1982), pp.123-150
24. A.K. Lenstra and H.W. Lenstra Jr. (eds.): The development of the number field sieve, Lecture Notes in Mathematics, Springer, (1993)
25. National Institute of Standards and Technology (NIST): Digital Signature Standard (DSS). Federal Information Processing Standards Publication 186, FIPS-186, 19th May, (1994)
26. S. Paulus and T. Takagi: “A new public-key cryptosystem over the quadratic order with quadratic decryption time,” to appear in Journal of Cryptology
27. H.J.J. te Riele: Factorization of RSA-140 with the Number Field Sieve, posting in sci.crypt.research, February (1999)
28. R. Rivest, A. Shamir and L. Adleman: “A method for obtaining digital signatures and public key-cryptosystems,” Communications of the ACM,21, (1978), pp.120-126
29. R.D. Silverman: “The multiple polynomial quadratic sieve,” Math. Comp. 48, (1987), pp.329-229
30. R.J. Schoof: “Quadratic Fields and Factorization,” Computational Methods in Number Theory. Math. Centrum Tracts 155. Part II. Amsterdam, (1983), pp.235-286.
31. D. Weber: “Computing discrete logarithms with quadratic number rings,” Advances in Cryptology - EUROCRYPT’98, LNCS 1403, Springer, 1998, pp. 171-183