No Access

Nonlinearity of $k$ $k$ -cycle permutations on $ℤ_{n}$

Yogesh Kumar

Department of Mathematics, IIT Delhi, New Delhi 110016, India

E-mail Address: yogesh_lather@yahoo.co.in

Corresponding author.

Search for more papers by this author

P. R. Mishra

SAG, DRDO, Metcalfe House, Delhi 110 054, India

E-mail Address: prasanna.r.mishra@gmail.com

Search for more papers by this author

, and

R. K. Sharma

Department of Mathematics, IIT Delhi, New Delhi 110016, India

E-mail Address: rksharma@maths.iitd.ac.in

Search for more papers by this author

https://doi.org/10.1142/S1793557118500201Cited by:5 (Source: Crossref)

Abstract

In this paper, we explore the existence and nonlinearity of affine $k$ -cycle permutations on $ℤ_{n}$ for different values of $k$ and $n$ . It is proved that a transposition $f$ on $ℤ_{n}$ has nonlinearity $2$ for $n > 4$ , a $3$ -cycle permutation on $ℤ_{p}$ has nonlinearity $3$ for $p > 5$ , $p$ a prime. The number of affine $k$ -cycle permutations on $ℤ_{n}$ and a bound for their nonlinearity have been obtained. Some results on the cyclic decomposition of a permutation on $ℤ_{p}$ , $p$ a prime, have also been proved.

Communicated by B. K. Dass

Keywords:

AMSC: 05A05, 06E30, 94A60

References

1. T. Beth and C. Ding, On almost perfect nonlinear permutations, in Advances in Cryptology — Eurocrypt’93, Lecture Notes in Computer Science, Vol. 765 (Springer, 1994), pp. 65–76. Google Scholar
2. L. Budaghyan, Construction and Analysis of Cryptographic Functions (Springer, 2014). Crossref, Google Scholar
3. C. Carlet, Boolean functions for cryptography and error correcting codes, Boolean Models and Methods in Mathematics, Computer Science, and Engineering (Cambridge University Press, 2010). Crossref, Google Scholar
4. C. Carlet and C. Ding, Highly nonlinear mappings, J. Complex. 20(2–3) (2004) 205–244. Crossref, Web of Science, Google Scholar
5. T. W. Cusik and P. Stanica, Cryptographic Boolean Functions and Applications, 1st edn. (Academic Press, 2009). Google Scholar
6. S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University Press, 2005). Crossref, Google Scholar
7. S. Kitaev, Patterns in Permutations and Words, Monographs in Theoretical Computer Science. An EATCS Series (Springer-Verlag, Berlin, 2011). Crossref, Google Scholar
8. Y. Kumar, P. R. Mishra, N. R. Pillai and R. K. Sharma, Affine equivalence and non-linearity of permutation over $Z_{n}$ , Appl. Algebra Engrg. Comm. Comput., doi:10.1007/500200-016-0307-6. Google Scholar
9. O. A. Logachev, A. A. Salnikov and V. V. Yashchenko, Boolean Functions in Coding Theory and Cryptography (American Mathematical Society, Providence, RI, 2012). Crossref, Google Scholar
10. F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1997). Google Scholar
11. A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryptography (CRC Press, 1997). Google Scholar
12. P. R. Mishra, I. Gupta and N. R. Pillai, Non-linearity and affine equivalence of permutations (2014), http://eprint.iacr.org/2014/974.pdf. Google Scholar