ABSTRACT
Deterministic algorithms are presented for the efficient solution of diagonal homogeneous equations in many variables over finite fields. As auxiliary algorithms, it is shown how to compute a field generator that is an nth power, and how to write elements as sums of nth powers, for a given integer n. All these algorithms take polynomial time in n and in the logarithm of the field size, and are practical as stated.
- Leonard Adleman, Kenneth Manders, and Gary Miller. On taking roots in finite fields. In 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), pages 175--178. IEEE Comput. Sci., Long Beach, Calif., 1977.Google ScholarDigital Library
- Eric Bach. A note on square roots in finite fields. IEEE Trans. Inform. Theory, 36(6):1494--1498, 1990.Google ScholarDigital Library
- Eric Bach, Joachim von zur Gathen, and Hendrik~W. Lenstra, Jr. Factoring polynomials over special finite fields. Finite Fields Appl., 7(1):5--28, 2001. Dedicated to Professor Chao Ko on the occasion of his 90th birthday. Google ScholarDigital Library
- Henri Cohen. A course in computational algebraic number theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993. Google ScholarDigital Library
- V. B. Dem′yanov. On representation of a zero of forms of the form ∑ m i=1a ix i n. Dokl. Akad. Nauk SSSR (N.S.), 105:203--205, 1955.Google Scholar
- G.H. Hardy and E.M. Wright. An introduction to the theory of numbers. Oxford, at the Clarendon Press, 1965. Fourth edition, 3rd corrected printing.Google Scholar
- Jean-René Joly. Équations et variéé algébriques sur un corps fini. Enseignement Math. (2), 19:1--117, 1973.Google Scholar
- T. Y. Lam. The algebraic theory of quadratic forms. W. A. Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series.Google Scholar
- H. W. Lenstra, Jr. Finding isomorphisms between finite fields. Math. Comp., 56(193):329--347, 1991.Google ScholarCross Ref
- René Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483--494, 1985.Google Scholar
- Štefan Schwarz. On universal forms in finite fields. Časopis Pěst. Mat. Fys., 75:45--50, 1950.Google Scholar
- Daniel Shanks. Five number-theoretic algorithms. In Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972), pages 51--70. Congressus Numerantium, No. VII, Winnipeg, Man., 1973. Utilitas Math.Google Scholar
- Charles Small. Diagonal equations over large finite fields. Canad. J. Math., 36(2):249--262, 1984.Google ScholarCross Ref
- Alberto Tonelli. Bemerkung Über die Auflösung quadratischer Congruenzen. Nachr. Göttingen, (10):344--346, 1891. Reported in Dickson's History, Vol. 1, Ch. VII, item 193, p. 215.Google Scholar
- Leonard Tornheim. Sums of n-th powers in fields of prime characteristic. Duke Math. J., 4:359--362, 1938.Google ScholarCross Ref
- Joachim von zur Gathen and Jürgen Gerhard. Modern computer algebra. Cambridge University Press, Cambridge, second edition, 2003. Google ScholarDigital Library
- André Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497--508, 1949.Google ScholarCross Ref
- Arne Winterhof. On Waring's problem in finite fields. Acta Arith., 87(2):171--177, 1998.Google ScholarCross Ref
Index Terms
- Deterministic equation solving over finite fields
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