POLYNOMIAL REPRESENTATIONS FOR n-TH ROOTS IN FINITE FIELDS

Computing square, cube and n-th roots in general, in finite fields, are important computational problems with significant applications to cryptography. One interesting approach to computational problems is by using polynomial representations. Agou, Deleglise and Nicolas proved results concerning the lower bounds for the length of polynomials representing square roots modulo a prime p. We generalize the results by considering n-th roots over finite fields for arbitrary n > 2.

[1]  Igor E. Shparlinski,et al.  On Polynomial Approximation of the Discrete Logarithm and the Diffie—Hellman Mapping , 2015, Journal of Cryptology.

[2]  Zhe-xian Wan.  A shorter proof for an explicit formula for discrete logarithms in finite fields , 2008, Discret. Math..

[3]  Takakazu Satoh,et al.  Closed formulae for the Weil pairing inversion , 2008, Finite Fields Their Appl..

[4]  Paulo S. L. M. Barreto,et al.  Efficient Computation of Roots in Finite Fields , 2006, Des. Codes Cryptogr..

[5]  Takakazu Satoh,et al.  On Polynomial Interpolations related to Verheul Homomorphisms , 2006 .

[6]  Harald Niederreiter,et al.  A short proof for explicit formulas for discrete logarithms in finite fields , 1990, Applicable Algebra in Engineering, Communication and Computing.

[7]  Eike Kiltz,et al.  On the interpolation of bivariate polynomials related to the Diffie-Hellman mapping , 2004, Bulletin of the Australian Mathematical Society.

[8]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[9]  Tanja Lange,et al.  Interpolation of the Discrete Logarithm in Fq by Boolean Functions and by Polynomials in Several Variables Modulo a Divisor of Q-1 , 2003, Discret. Appl. Math..

[10]  Marc Deléglise,et al.  Short Polynomial Representations for Square Roots Modulo p , 2003, Des. Codes Cryptogr..

[11]  Tanja Lange,et al.  Polynomial Interpolation of the Elliptic Curve and XTR Discrete Logarithm , 2002, COCOON.

[12]  Arne Winterhof,et al.  Polynomial Interpolation of the Discrete Logarithm , 2002, Des. Codes Cryptogr..

[13]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[14]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[15]  Gary L. Mullen,et al.  A polynomial representation for logarithms in GF(q) , 1986 .

[16]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[17]  Rudolf Lide,et al.  Finite fields , 1983 .

[18]  Gary L. Miller,et al.  On taking roots in finite fields , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).