Extended Finite Fields in Cryptographic Information Protection

2016;
: pp. 125 - 129
https://doi.org/10.23939/acps2016.02.125
Accepted: November 21, 2016
Authors:
1
Lviv Polytechnic National University

The use of extended finite fields for cryptographic information protection is focused on. In particular, explicit construction in finite fields elements of high multiplicative order is described. The obtained correspondent lower bounds on the order are provided.

[1] M. Agrawal, N. Kayal, N. Saxena, PRIMES is in P, Annals of Mathematics, vol. 160, no. 2, pp. 781-793, 2004.
https://doi.org/10.4007/annals.2004.160.781

[2] O. Ahmadi, I. E. Shparlinski, J. F. Voloch, Multiplicative order of Gauss periods, Int. J. Number Theory, vol. 6, no. 4, pp. 877-882, 2010.
https://doi.org/10.1142/S1793042110003290

[3] Q. Cheng, On the construction of finite field elements of large order, Finite Fields Appl., vol. 11, no. 3, pp. 358-366, 2005.
https://doi.org/10.1016/j.ffa.2005.06.001

[4] S. Gao, Elements of provable high orders in finite fields, Proc. Amer. Math. Soc., vol. 127, no. 6, pp. 1615-1623, 1999.
https://doi.org/10.1090/S0002-9939-99-04795-4

[5] R. Lidl, H. Niederreiter, Finite Fields. Cambridge University Press, 1997, 755 p.
https://doi.org/10.1017/CBO9780511525926

[6] A. Menezes, P. Van Oorschot, S. Vanstone. Handbook of Applied Cryptography, London, CRC Press, 1996, 794 p.

[7] G.L. Mullen, D. Panario, Handbook of finite Fields. Boca Raton: CRC Press, 2013, 1068 p.
https://doi.org/10.1201/b15006

[8] R. Popovych, Elements of high order in finite fields of the form Fq[x]/ r (x) F , Finite Fields Appl.,. vol. 18, no. 4, pp. 700-710, 2012.

[9] R. Popovych, Elements of high order in finite fields of the form F [x] /(xm a) q - , Finite Fields Appl., vol. 19, no. 1, pp. 86-92, 2013.

[10] R. Popovych, Sharpening of explicit lower bounds on elements order for finite field extensions based on cyclotomic polynomials, Ukr. Math. Journ., vol. 66, no. 6, pp. 815-825, 2014.
https://doi.org/10.1007/s11253-014-0981-0

[11] R. Popovych On elements of high order in general finite fields, Algebra and Discrete Mathematics, vol. 18, no.2, pp. 295-300, 2014.

[12] R. B. Popovych, Some primitive elements for the Artin-Schreier extensions of finite fields, Journal of Mathematical Sciences, vol. 210, no. 1, pp. 67-75, 2015.
https://doi.org/10.1007/s10958-015-2548-5