On the Computation of Square Roots in Finite Fields

Designs, Codes, and Cryptography

Volumn 31, Issue 3, 2004, Pages 301-312

  Müller, Siguna ^a  

^a UNIVERSITY OF CALGARY   (Canada)

Author keywords

Complexity; Efficient computation; Finite fields; Square roots

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; MATHEMATICAL MODELS; PROBLEM SOLVING;

COMPLEXITY; EFFICIENT COMPUTATION; FINITE FIELDS; SQUARE ROOTS;

CRYPTOGRAPHY;

EID: 11144356697     PISSN: 09251022     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:DESI.0000015890.44831.e2     Document Type: Article

References (26)

1. A. O. L. Atkin, Probabilistic primality testing, Summary by F. Morain, INRIA Res. Rep. 1779, (1992) pp. 159-163.
2. A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp., Vol. 61 (1993) pp. 29-68.
3. E. Bach and K. Huber, Note on taking square-roots modulo N, IEEE Trans. Inf. Theory, Vol. 45, No. 2 (1999) pp. 807-809.
4. E. Bach and J. Shallit, Algorithmic Number Theory, Efficient Algorithms, Vol. 1, MIT Press Cambridge, MA (1996).
5. E. R. Berlekamp, Factoring polynomials over large finite fields, Math. Comp., Vol. 24 (1970) pp. 713-735.
6. D. Bressoud and S. Wagon, Computational Number Theory, Springer (2000).
7. M. Cipolla, Un metodo per la risolutione della congruenza di secondo grado, Rendiconto dell'Accademia Scienze Fisiche e Matematiche, Napoli, Ser. 3, Vol. IX (1903) pp. 154-163.
8. R. Crandall and C. Pomerance, Prime Numbers. A Computational Perspective, Springer (2001).
9. H. Dubner and W. Keller, Factors of generalized fermat numbers, Math. Comp., Vol. 64, No. 209 (1995) pp. 397-405.
10. D. Gordon, A survey of fast exponentiation methods, Journal of Algorithms, Vol. 27 (1998) pp. 129-146.
11. N. Koblitz, A Course in Number Theory and Cryptography, Springer, New York (1994).
12. D. H. Lehmer, Computer technology applied to the theory of numbers, In Studies in Number Theory, Prentice-Hall, Englewood Cliffs, NJ (1969) pp. 117-151.
13. S. Lindhurst, An analysis of Shanks's algorithm for computing square roots in finite fields, CRM Proceedings and Lecture Notes, Vol. 19 (1999) pp. 231-242.
14. A. Menezes, P. C. van Oorschot and S. Vanstone, Handbook of Applied Cryptography, CRC (1997).
15. S. M. Meyer and J. Sorenson, Efficient algorithms for computing the Jacobi symbol, Algorithmic Number Theory, ANTS II, Proceedings (ed. Henri Cohen), LNCS 1122 (1996) pp. 225-239.
16. S. Müller, On probable prime testing and the computation of square roots mod n, Algorithmic Number Theory, ANTS IV, Proceedings (ed. Wieb Bosma), LNCS 1838 (2000) pp. 423-437.
17. q, Finite Fields and Applications (eds. H. Niederreiter and A. Enge), Springer (2001) pp. 390-408.
18. W. B. Müller and R. Nöbauer, Cryptanalysis of the Dickson-scheme, Advances in Cryptology, EUROCRYPT'85, 50-61, Springer, Berlin (1986).
19. C. Pomerance, Analysis and comparison of some integer factoring algorithms. In Computational Methods in Number Theory, Part I (eds. H. Lenstra Jr and R. Tijdeman), Vol. 154 of Math. Centre Tracts, Math. Centrum (1982) pp. 89-139.
20. M. O. Rabin, Probabilistic algorithms in finite fields, SIAM J. Comput., Vol. 9 (1980) pp. 273-280.
21. H. Postl, Fast evaluation of Dickson Polynomials, Contrib. to General Algebra, Vol. 6 (1988) pp. 223-225.
22. H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser (1994).
23. R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp., Vol. 44, No. 170 (1985) pp. 483-494.
24. D. Shanks, Five number-theoretic algorithms, In Proc. 2nd Manitoba Conf. Numer. Math., Manitoba, Canada (1972) pp. 51-70.
25. A. Tonelli, Bemerkung über die Auflösung quadratischer Congruenzen, Göttinger Nachrichten (1891) pp. 344-346.
26. H. C. Williams, Édouard Lucas and Primality Testing, John Wiley & Sons (1998).