Note on taking square-roots modulo N

IEEE Transactions on Information Theory

Volumn 45, Issue 2, 1999, Pages 807-809

  Bach, Eric ^a   Huber, Klaus ^a  

^a University of Wisconsin   (United States)

Author keywords

Cryptography; Factoring; Gauss sums; Square roots modulo n

Indexed keywords

ALGORITHMS; CRYPTOGRAPHY; PROBLEM SOLVING;

FACTORING; GAUSS' CYCLOTOMIC SUM FORMULA;

NUMBER THEORY;

EID: 0033097625     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.749034     Document Type: Article

References (25)

1. A.O. L. Atkin, "Probabilistic primality testing," summary François Morain, INRIA Res. Rep. 1779, pp. 159-163, Oct. 1992.
2. E. Bach, "A note on square roots in finite fields," IEEE Trans. Inform. Theory, vol. 36, pp. 1494-1498, Nov. 1990.
3. E. Bach and J. Shallit, "Algorithmic number theory," Efficient Algorithms. Cambridge, MA: MIT Press, 1996, vol. I.
4. M. Ben-Or, "Probabilistic algorithms in finite fields," in Proc. IEEE Symp. Found. Comp. Sei, 1981, pp. 394-398.
5. E.R. Berlekamp, "Factoring polynomials over large finite fields," Math. Comp., vol. 24, pp. 713-735, 1970.
6. M. Cipolla, "Un metodo per la risolutione della congruenza di secondo grado," Rendiconto dell'Accademia Scienze Fisiche e Matematiche, Napoli, Ser. 3, vol IX, pp. 154-163, 1903.
7. A. Fiat and A. Shamir, "How to prove yourself: Practical solutions to identification and signature problems," in Proc. Advances in CryptologyCrypto. New York: Springer-Verlag, 1987, pp. 186-194.
8. C.F. Gauss, "Untersuchungen über Höhere Arithmetik," in Disquisitiones Arithmeticae, H. Maser, Ed. New York: Chelsea, 1889; 2nd reprint, 1981 (German transi.).
9. G.H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford, U.K.: Oxford, 1979.
10. D.E. Knuth, "The Art of Computer Programming," Seminumerical Algorithms, 2nd ed. Reading, MA: Addison Wesley, 1981, vol. 2.
11. J.L. Lagrange, "Sur la solution des problèmes indéterminés du second degré," Histoire Acad. Roy. Sei. Belle-Lett., Berlin, Germany, 1769, pp. 165-310 (reprint Oeuvres, vol. 2, pp. 377-535).
12. E. Landau, "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie," Hirzel 1927, Vierter Teil, Kapitel 6 Gausche Summen, reprint, Chelsea, New York, 1950.
13. D.H. Lehmer, "Computer technology applied to the theory of numbers," in Studies in Number Theory. Englewood Cliffs, NJ: Prentice-Hall, 1969, pp. 117-151.
14. S. Lindhurst, "An analysis of Shanks's algorithm for computing square roots in finite fields," in Proc. 5th Conf. Canadian Number Theory Assoc.. 1995.
15. R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications. Cambridge, U.K.: Cambridge Univ. Press, 1984, vol. 20.
16. J.L. Massey, "Cryptography: Fundamentals and applications," copies of transparencies, in Adv. Technol. Seminars, Zürich, Switzerland, pp. 6.42-6.44.
17. R. Peralta, "A simple and fast probabilistic algorithm for computing square roots modulo a prime number," IEEE Trans. Inform. Theory, vol. IT-32, pp. 846-847, Nov. 1986.
18. H.C. Pocklington, "The direct solution of the quadratic and cubic binomial congruences with prime moduli," in Proc. Cambridge Phil. Soc., Cambridge, U.K., 1917, vol. 19, pp. 57-59.
19. M.O. Rabin, "Probabilistic algorithms in finite fields," SIAM J. Comput., vol. 9, pp. 273-280, 1980.
20. H. Riesel, Prime Numbers and Computer Methods for Factorization. Boston, MA: Birkhäuser, 1985.
21. 2 = a (mod P) în cazuri pĩnâ acum necunoscute si aplicarea lor pentru a détermina direct râdâcinile primitive ale unor numere prime," Acad. R. P. Rornine Fit. Ctuj. Stud. Cere. Mat. Fiz., vol. 7, 1956, pp. 51-58(, in Romainian, with Russian, French summaries).
22. R. Schoof, "Elliptic curves over finite fields and the computation of square roots mod p" Math. Computat., vol. 44, no. 170, pp. 483-194, Apr. 1985.
23. D. Shanks, "Five number-theoretic algorithms," in Proc. 2nd Manitoba Conf. Numer. Math., Manitoba, Canada, 1972, pp. 51-70.
24. A.Tonelli, "Bemerkung über die Auflösung quadratischer Congruenzen," Göttinger Nachrichten, pp. 344-346, 1891.
25. H.C. Williams, "A modification of the RSA public-key encryption procedure," IEEE Trans. Inform. Theory, vol. IT-26, pp. 726-729, 1980.