Designs, Codes and Cryptography

, Volume 47, Issue 1–3, pp 225–235

# Two results on maximum nonlinear functions

• Doreen Hertel
• Alexander Pott
Article

## Abstract

Maximum nonlinear functions $$F: \mathbb F_{2^m}\to \mathbb F_{2^m}$$ are widely used in cryptography because the coordinate functions F β (x) := tr(β F(x)), $$\beta \in \mathbb F^{*}_{2^m}$$ , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, $$a,b \in \mathbb F_{2^m}, b\neq 0$$ , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions $$x^{2^{k}+1}$$ , gcd(k, m) = 1, and the Kasami power functions $$x^{2^{2k}-2^{k}+1}$$ , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in $$\mathbb F_{2^m}$$ . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.

## Keywords

Maximum nonlinear Gold power function Walsh transform Difference set Finite field Kasami power function Almost perfect nonlinear

05B10 05B25

## References

1. 1.
Beth T., Jungnickel D. and Lenz H. (1999). Design, Theory, 2nd ed. Cambridge University Press, Cambridge Google Scholar
2. 2.
Bose R.C. (1942). An affine analogue of Singer’s theorem. J. Indian Math. Soc. (N.S.) 6: 1–15
3. 3.
Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis, Santis A.D (ed.) In Advances in cryptology – EUROCRYPT 94, vol. 950 of  Lecture Notes in Computer Science, pp. 356–365 Springer-Verlag, New York, (1995).Google Scholar
4. 4.
De Clerck F., Hamilton N., O’Keefe C.M. and Penttila T. (2000). Quasi-quadrics and related structures. Australas. J. Comb. 22: 151–166
5. 5.
Dillon J. and Dobbertin H. (2004). New cyclic difference sets with Singer parameters. Finite Fields Appl. 10: 342–389
6. 6.
Dillon J.F. (1999). Multiplicative difference sets via additive characters. Des. Codes Cryptogr. 17: 225–235
7. 7.
Games R.A. (1986). The geometry of quadrics and correlations of sequences. IEEE Trans. Inform Thoery 32: 423–426
8. 8.
Gold R. (1968). Maximal recursive sequences with 3-valued recursive cross-correlation function. IEEE Trans. Inform Thoery 14: 154–156
9. 9.
Golomb S.W. and Gong G. (2005). Signal Design for Good Correlation. Cambridge University Press, Cambridge
10. 10.
Helleseth T., Kumar P.V.: Sequences with low correlation. In Handbook of Coding Theory, vol. I, II, pp. 1065–1138 North-Holland, Amsterdam (1998).Google Scholar
11. 11.
Hirschfeld  J.: Projective Geometries Over Finite Fields, 2nd ed. Oxford Mathematical Monographs. Clarendon Press, Oxford (1998).Google Scholar
12. 12.
Kim S.-H., No J.-S., Chung H. and Helleseth T. (2005). New cyclic difference sets constructed from d- homogeneous functions with difference balanced property. IEEE Trans. Inf. Th. 15: 1155–1163
13. 13.
Langevin P. and Véron P. (2006). On the nonlinearity of power functions. Des. Codes Cryptogr. 37: 31–43
14. 14.
Lidl R., Niederreiter H.: Finite fields, vol. 20 of Encyclopedia of Mathematics and its Applications, 2nd ed. Cambridge University Press (1997).Google Scholar
15. 15.
No J.-S., Chung H. and Yun M.-S. (1998). Binary pseudorandom sequences of period 2m − 1 with ideal autocorrelation generated by the polynomial z d + (z + 1)d. IEEE Trans. Inform Theory 44: 1278–1282