Algorithms for Exponentiation in Finite Fields

Journal of Symbolic Computation

Volumn 29, Issue 6, 2000, Pages 879-889

  Gao, Shuhong ^a   Von Zur Gathen, Joachim ^b   Panario, Daniel ^c   Shoup, Victor ^d  

^a CLEMSON UNIVERSITY   (United States)

^b UNIVERSITY OF PADERBORN   (Germany)

^c UNIVERSITY OF TORONTO   (Canada)

^d IBM RESEARCH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000185837     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.1999.0309     Document Type: Article

References (47)

1. Adleman L., Lenstra H. Finding irreducible polynomials over finite fields. Proceedings of the 18th ACM Symposium on the Theory of Computing. 1986;. p. 350-355.
2. Agnew G., Mullin R., Onyszchuk I., Vanstone S. An implementation for a fast public key cryptosystem. J. Cryptol. 3:1991;63-79.
3. G. Agnew, R. Mullin, S. Vanstone, Advances in Cryptology - EUROCRYPT'88, Günther, C. G. 1988, Springer, Berlin, 251, 255.
4. Agnew G., Mullin R., Vanstone S. An implementation of elliptic curve cryptosystem over F 2155. IEEE J. Selected Areas Commun. 11:1993;804-813.
5. Aho A., Hopcroft J., Ullman J. The Design and Analysis of Computer Algorithms. 1974;Addison-Wesley, Reading.
6. Ash D., Blake I., Vanstone S. Low complexity normal bases. Discrete Applied Math. 25:1989;191-210.
7. Bach E., Shallit J. Factoring with cyclotomic polynomials. Math. Comput. 52:1989;201-219.
8. Berlekamp E. Bit-serial Reed-Solomon encoders. IEEE Trans. Inf. Theory. 28:1982;869-874.
9. Brent R., Kung H. Fast algorithms for manipulating formal power series. J. Assoc. Comput. Mach. 25:1978;581-595.
10. E. Brickell, D. Gordon, K. McCurley, D. Wilson, Proceedings of Eurocrypt'92, 1992, Balatonfüred, Hungary.
11. Calmos.
12. Cantor D. On arithmetical algorithms over finite fields. J. Comb. Theory, Ser. A. 50:1989;285-300.
13. Cantor D., Kaltofen E. On fast multiplication of polynomials over arbitrary algebras. Acta Inform. 28:1991;693-701.
14. Coppersmith D., Winograd S. Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9:1990;251-280.
15. Gao S., Von zur Gathen J., Panario D. Gauss periods: orders and cryptographical applications. Math. Comput. 67:1998;343-352.
16. Gao S., Howell J., Panario D. Irreducible polynomials of given forms. Mullin R., Mullen G. Finite Fields: Theory, Applications and Algorithms. 1999;43-54 American Mathematical Society, New York.
17. Gao S., Lenstra H. Jr. Optimal normal bases. Des. Codes Cryptogr. 2:1992;315-323.
18. Gao S., Vanstone S. On orders of optimal normal basis generators. Math. Comput. 64:1994;1227-1233.
19. Von zur Gathen J. Efficient and optimal exponentiation in finite fields. Comput. Complexity. 1:1991;360-394.
20. J. Von zur Gathen, J. Gerhard, Proceedings of ISSAC'96, Zürich, Switzerland, Y. N. L. 1996, ACM press, New York, 1, 9.
21. Von zur Gathen J., Gerhard J. Modern Computer Algebra. 1999;Cambridge University Press, Cambridge.
22. Von zur Gathen J., Nöcker M. Exponentiation in finite fields: theory and practice. Proceedings of AAECC 12, LNCS 1255. 1997;Springer-Verlag, Berlin/Heidelberg. p. 88-113.
23. J. Von zur Gathen, F. Pappalardi.
24. Von zur Gathen, Shparlinski I. E. Order of Gauss periods in finite fields. Appl. Algebra Eng. Commun. Comput. 9:1998;15-24.
25. Gauss C. Disquisitiones Arithmeticae. 1801;Braunschweig.
26. Geiselmann W., Gollmann D. Symmetry and duality in normal basis multiplication. AAECC-6. 1989;Springer-Verlag, Berlin/Heidelberg. p. 230-238.
27. Jungnickel D. Finite Fields: Structure and Arithmetics. 1993;Wissenschaftsverlag, Mannhein-Leipzig-Wien-Zurich.
28. Kaltofen E., Shoup V. Subquadratic-time factoring of polynomials over finite fields. Math. Comput. 67:1998;1179-1197.
29. Karatsuba A., Ofman Y. Umnozhenie mnogoznachnykh chisel na abto matakh. Dokl. Akad. Nauk USSR. 145:1962;293-294.
30. Sov. Phys. Dokl. 7:1963;595-596.
31. Lempel A., Weinberger M. Self-complementary normal bases in finite fields. SIAM J. Discrete Math. 1:1988;193-198.
32. J. Massey, J. Omura.
33. Menezes A., Blake I., Gao X., Mullin R., Vanstone S., Yaghoobian T. Applications of Finite Fields. 1993;Kluwer Academic Publishers, Boston, Dordrecht, Lancaster.
34. 216091, X.
35. Mullin R., Onyszchuk I., Vanstone S., Wilson R. Optimal normal bases in Fpn. Discrete Appl. Math. 22:1989;149-161.
36. I. Onyszchuk, R. Mullin, S. Vanstone.
37. Pohst M., Zassenhaus H. Algorithmic Algebraic Number Theory. 1989;Cambridge University Press, Cambridge.
38. Rosati T. A high speed data encryption processor for public key cryptography. Proceedings of IEEE Custom Integrated Circuits Conference. 1989;. p. 12.3.1-12.3.5.
39. Schönhage A. Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Inform. 7:1977;395-398.
40. Schönhage A., Strassen V. Schnelle Multiplikation großer Zahlen. Computing. 7:1971;281-292.
41. V. Shoup, Proceedings of the International IMACS Symposium on Symbolic Computation, New Trends and Developments, 1993, Lille, France, 124, 129.
42. Stinson D. Some observations on parallel algorithms for fast exponentiation in F 2 n. SIAM J. Comput. 19:1990;711-717.
43. Storer T. Cyclotomy and Difference Sets. 1967;Markham, Chicago.
44. Strassen V. Gaussian elimination is not optimal. Numer. Math. 13:1969;354-356.
45. Wang C. An algorithm to design finite field multipliers using a self-dual normal basis. IEEE Trans. Comput. 38:1989;1457-1460.
46. Wassermann A. Konstruktion von Normalbasen. Bayreuther Mathematische Schriften. 31:1990;155-164.
47. Wassermann A. Zur Arithmetik in endlichen Körpern. Bayreuther Mathematische Schriften. 44:1993;147-251.