# American Institute of Mathematical Sciences

May  2017, 11(2): 301-306. doi: 10.3934/amc.2017022

## Explicit formulas for monomial involutions over finite fields

 1 Department of Mathematics, University of Puerto Rico, Río Piedras, Box 70377, S.J., PR 00936-8377 2 Department of Computer Science, University of Puerto Rico, Río Piedras, Box 70377, S.J., PR 00936-8377

* Corresponding author

Received  February 2016 Revised  March 2016 Published  May 2017

Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of ${\mathbb{ F\!}}_q$ that are given by monomials and for their fixed points.

Citation: Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022
##### References:
 [1] C. Corrada and I. Rubio, Deterministic interleavers for Turbo codes with random-like performance and simple implementation, in Proc. 3rd Int. Symp. Turbo Codes Related Topics, (2003), 555-558.   Google Scholar [2] C. Corrada and I. Rubio, Cyclic decomposition of permutations of finite fields obtained using monomials, in Finite Fields and Applications, (2004), 254-261.  doi: 10.1007/978-3-540-24633-6_19.  Google Scholar [3] P. Charpin, S. Mesnager and S. Sarkar, On involutions of finite fields, in Int. Symp. Inf. Theory-ISIT, 80 (2016), 379-393.   Google Scholar [4] P. Charpin, S. Mesnager and S. Sarkar, Involutions over the Galois field $\mathbb F_{2^n}$, IEEE Trans. Inf. Theory, 62 (2016), 2266-2276.  doi: 10.1109/TIT.2016.2526022.  Google Scholar [5] A. Sakzad, D. Panario, M. Sadeghi and N. Eshghi, Self-inverse interleavers based on permutation functions for Turbo codes, in 2010 48th Ann. Allerton Conf. Commun. Control Comp., IEEE, 2010, 22–28. Google Scholar [6] O. Takeshita, On maximum contention-free interleavers and permutation polynomials over integer rings, IEEE Trans. Inf. Theory, 52 (2006), 1249-1253.  doi: 10.1109/TIT.2005.864450.  Google Scholar [7] Q. Wang, A note on inverses of cyclotomic mapping permutation polynomials over finite fields, Finite Fields Appl., 45 (2017), 422-427.  doi: 10.1016/j.ffa.2017.01.006.  Google Scholar

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##### References:
 [1] C. Corrada and I. Rubio, Deterministic interleavers for Turbo codes with random-like performance and simple implementation, in Proc. 3rd Int. Symp. Turbo Codes Related Topics, (2003), 555-558.   Google Scholar [2] C. Corrada and I. Rubio, Cyclic decomposition of permutations of finite fields obtained using monomials, in Finite Fields and Applications, (2004), 254-261.  doi: 10.1007/978-3-540-24633-6_19.  Google Scholar [3] P. Charpin, S. Mesnager and S. Sarkar, On involutions of finite fields, in Int. Symp. Inf. Theory-ISIT, 80 (2016), 379-393.   Google Scholar [4] P. Charpin, S. Mesnager and S. Sarkar, Involutions over the Galois field $\mathbb F_{2^n}$, IEEE Trans. Inf. Theory, 62 (2016), 2266-2276.  doi: 10.1109/TIT.2016.2526022.  Google Scholar [5] A. Sakzad, D. Panario, M. Sadeghi and N. Eshghi, Self-inverse interleavers based on permutation functions for Turbo codes, in 2010 48th Ann. Allerton Conf. Commun. Control Comp., IEEE, 2010, 22–28. Google Scholar [6] O. Takeshita, On maximum contention-free interleavers and permutation polynomials over integer rings, IEEE Trans. Inf. Theory, 52 (2006), 1249-1253.  doi: 10.1109/TIT.2005.864450.  Google Scholar [7] Q. Wang, A note on inverses of cyclotomic mapping permutation polynomials over finite fields, Finite Fields Appl., 45 (2017), 422-427.  doi: 10.1016/j.ffa.2017.01.006.  Google Scholar
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