We present a method for enumerating the permutation trinomials and quadrinomials using various symmetries and algebraic properties to reduce the search space. Applying this method, we enumerated all permutation trinomials and quadrinomials for the prime finite fields with orders up to 3000 and 500, respectively. Using the enumeration results, we stated a conjecture about the classification of permutation polynomials over prime finite fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ch. Hermite, Sur les fonctions de sept lettres (1905).
L. E. Diskson, “The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group,” Ann. Math., 11, 161–183 (1896).
R. Lidl and G. L. Mullen, “When does a polynomial over a finite field permute the elements of the field?” Amer. Math. Monthly, 95, No. 3, 243 (1988).
R. Lidl and G. L. Mullen, “When does a polynomial over a finite field permute the elements of the field? II,” Amer. Math. Monthly, 100, No. 1, 71 (1993).
N. Vasiliev and M. Rybalkin, “Permutation binomials and their groups,” J. Math. Sci., 179, 679–689 (2011).
D. Wan and R. Lidl, “Permutation polynomials of the form and their group structure,” Monatsh. Math., 112, No. 2, 149–163 (1991).
M. E. Zieve, “On some permutation polynomials over GF(q) of the form xrh(x(q−1)/d),” Proc. Amer. Math. Soc., 137, 2209–2216 (2009).
Q. Wang, “On inverse permutation polynomials,” Finite Fields Appl., 15, No. 2, 207–213 (2009).
A. M. Masuda and M. E. Zieve, “Permutation binomials over finite fields,” Trans. Amer. Math. Soc., 361, 4169–4180 (2009).
R. P. Singh, B. K. Sarma, and A. Saikia, “Public key cryptography using permutation p-polynomials over finite fields,” IACR Cryptology Print Archive, 2009, 208 (2009).
G. Castagnos and D. Vergnaud, “Trapdoor permutation polynomials of Z/nZ and public key cryptosystems,” Lect. Notes Comput. Sci., 4779, 333–350 (2007).
R. Lidl and W. B. Müller, “Permutation polynomials in RSA-cryptosystems,” in: Advances in Cryptology (Santa Barbara, California, 1983), Plenum, New York (1984), pp. 293–301.
S. R. Finch, Mathematical Constants, Cambridge Univ. Press, Cambridge (2003). 741
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 421, 2014, pp. 152–165.
Rights and permissions
About this article
Cite this article
Rybalkin, M.A. Classification of Permutation Trinomials and Quadrinomials Over Prime Fields. J Math Sci 200, 734–741 (2014). https://doi.org/10.1007/s10958-014-1966-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1966-0