# American Institute of Mathematical Sciences

May  2017, 11(2): 389-396. doi: 10.3934/amc.2017033

## Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case

 1 Department of Mathematics, UPR-Cayey, Puerto Rico (PR), 00736 USA 2 Department of Mathematics, UPR-Rio Piedras, San Juan, PR 00931 USA

Received  February 2016 Revised  March 2016 Published  May 2017

An almost perfect nonlinear (APN) function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami-Welch number $(2^{2k}-2^k+1)$. When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when $\deg{(h(x))}$ is odd. We also show exactly when the corresponding multivariate polynomial $φ(x, y, z)$ is absolutely irreducible. Also, there is only one result known when $f(x)=x^{2^{k}+1} + h(x)$, and $\deg(h(x))$ is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.

Citation: Moises Delgado, Heeralal Janwa. Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case. Advances in Mathematics of Communications, 2017, 11 (2) : 389-396. doi: 10.3934/amc.2017033
##### References:
 [1] Y. Aubry, G. McGuire and F. Rodier, A few more functions that are not APN infinitely often, in Finite Fields: Theory and Applications, 2010, 23-31. doi: 10.1090/conm/518/10193. [2] R. D. Baker, J. H. Van Lint and R. M. Wilson, On the preparata and goethals codes, IEEE Trans. Inf. Theory, 29 (1983), 342-345.  doi: 10.1109/TIT.1983.1056675. [3] T. P. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear functions over $F_{2n}$, IEEE Trans. Inf. Theory, 52 (2006), 4160-4170.  doi: 10.1109/TIT.2006.880036. [4] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968. [5] C. Blondeau and K. Nyberg, Perfect nonlinear functions and cryptography, Finite Fields Appl., 32 (2015), 120-147.  doi: 10.1016/j.ffa.2014.10.007. [6] L. Budaghyan, C. Carlet and G. Leander, Constructing new APN functions from known ones, Finite Fields Appl., 15 (2009), 150-159.  doi: 10.1016/j.ffa.2008.10.001. [7] E. Byrne and G. McGuire, Quadratic binomial APN functions and absolutely irreducible polynomials, preprint, arXiv: 0810.4523 [8] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for des-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [9] F. Caullery, Polynomial functions of degree 20 which are APN infinitely often, preprint, arXiv: 1212.4638 [10] F. Caullery, A new large class of functions not APN infinitely often, Des. Codes Crypt., 73 (2014), 601-614.  doi: 10.1007/s10623-014-9956-2. [11] M. Delgado and H. Janwa, On the conjecture on APN functions, preprint, arXiv: 1207.5528 [12] M. Delgado and H. Janwa, Further results on exceptional APN functions, 2013. [13] M. Delgado and H. Janwa, On the conjecture on APN functions and absolute irreducibility of polynomials, Des. Codes Crypt., (2016), 1-11.  doi: 10.1007/s10623-015-0168-1. [14] M. Delgado and H. Janwa, Progress towards the conjecture on APN functions and absolutely irreducible polynomials, preprint, arXiv: 1602.02576 [15] Y. Edel, G. Kyureghyan and A. Pott, A new APN function which is not equivalent to a power mapping, IEEE Trans. Inf. Theory, 52 (2006), 744-747.  doi: 10.1109/TIT.2005.862128. [16] E. Férard, R. Oyono and F. Rodier, Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents, in Arithmetic, Geometry, Cryptography and Coding Theory, 2012, 27-36. doi: 10.1090/conm/574/11423. [17] W. Fulton, Algebraic Curves: An INTRODUCTION to Algebraic Geometry , 2008. [18] S. R. Ghorpade and G. Lachaud, Étale cohomology, lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2 (2002), 589-631. [19] F. Hernando and G. McGuire, Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions, J. Algebra, 343 (2011), 78-92.  doi: 10.1016/j.jalgebra.2011.06.019. [20] H. Janwa, G. Mcguire and R. M. Wilson, Double-error-correcting cyclic codes and absolutely irreducible polynomials over gf (2), J. Algebra, 178 (1995), 665-676.  doi: 10.1006/jabr.1995.1372. [21] H. Janwa and R. M. Wilson, Hyperplane sections of fermat varieties in p 3 in char. 2 and some applications to cyclic codes, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Corr. Codes, Springer, 1993,180-194. doi: 10.1007/3-540-56686-4_43. [22] Y. Niho, Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph. D thesis, Univ. Southern California, 1972. [23] K. Nyberg, Differentially uniform mappings for cryptography, in Worksh. Theory Appl. Crypt. Techn., Springer, 1993, 55-64. doi: 10.1007/3-540-48285-7_6. [24] A. Pott, Almost perfect and planar functions, Des. Codes Crypt., 78 (2016), 141-195.  doi: 10.1007/s10623-015-0151-x. [25] F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires, Contemp. Math., 14 (2009), 169.  doi: 10.1090/conm/487/09531. [26] F. Rodier, Some more functions that are not APN infinitely often. the case of Kasami exponents, preprint, arXiv: 1101.6033

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##### References:
 [1] Y. Aubry, G. McGuire and F. Rodier, A few more functions that are not APN infinitely often, in Finite Fields: Theory and Applications, 2010, 23-31. doi: 10.1090/conm/518/10193. [2] R. D. Baker, J. H. Van Lint and R. M. Wilson, On the preparata and goethals codes, IEEE Trans. Inf. Theory, 29 (1983), 342-345.  doi: 10.1109/TIT.1983.1056675. [3] T. P. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear functions over $F_{2n}$, IEEE Trans. Inf. Theory, 52 (2006), 4160-4170.  doi: 10.1109/TIT.2006.880036. [4] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968. [5] C. Blondeau and K. Nyberg, Perfect nonlinear functions and cryptography, Finite Fields Appl., 32 (2015), 120-147.  doi: 10.1016/j.ffa.2014.10.007. [6] L. Budaghyan, C. Carlet and G. Leander, Constructing new APN functions from known ones, Finite Fields Appl., 15 (2009), 150-159.  doi: 10.1016/j.ffa.2008.10.001. [7] E. Byrne and G. McGuire, Quadratic binomial APN functions and absolutely irreducible polynomials, preprint, arXiv: 0810.4523 [8] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for des-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [9] F. Caullery, Polynomial functions of degree 20 which are APN infinitely often, preprint, arXiv: 1212.4638 [10] F. Caullery, A new large class of functions not APN infinitely often, Des. Codes Crypt., 73 (2014), 601-614.  doi: 10.1007/s10623-014-9956-2. [11] M. Delgado and H. Janwa, On the conjecture on APN functions, preprint, arXiv: 1207.5528 [12] M. Delgado and H. Janwa, Further results on exceptional APN functions, 2013. [13] M. Delgado and H. Janwa, On the conjecture on APN functions and absolute irreducibility of polynomials, Des. Codes Crypt., (2016), 1-11.  doi: 10.1007/s10623-015-0168-1. [14] M. Delgado and H. Janwa, Progress towards the conjecture on APN functions and absolutely irreducible polynomials, preprint, arXiv: 1602.02576 [15] Y. Edel, G. Kyureghyan and A. Pott, A new APN function which is not equivalent to a power mapping, IEEE Trans. Inf. Theory, 52 (2006), 744-747.  doi: 10.1109/TIT.2005.862128. [16] E. Férard, R. Oyono and F. Rodier, Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents, in Arithmetic, Geometry, Cryptography and Coding Theory, 2012, 27-36. doi: 10.1090/conm/574/11423. [17] W. Fulton, Algebraic Curves: An INTRODUCTION to Algebraic Geometry , 2008. [18] S. R. Ghorpade and G. Lachaud, Étale cohomology, lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2 (2002), 589-631. [19] F. Hernando and G. McGuire, Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions, J. Algebra, 343 (2011), 78-92.  doi: 10.1016/j.jalgebra.2011.06.019. [20] H. Janwa, G. Mcguire and R. M. Wilson, Double-error-correcting cyclic codes and absolutely irreducible polynomials over gf (2), J. Algebra, 178 (1995), 665-676.  doi: 10.1006/jabr.1995.1372. [21] H. Janwa and R. M. Wilson, Hyperplane sections of fermat varieties in p 3 in char. 2 and some applications to cyclic codes, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Corr. Codes, Springer, 1993,180-194. doi: 10.1007/3-540-56686-4_43. [22] Y. Niho, Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph. D thesis, Univ. Southern California, 1972. [23] K. Nyberg, Differentially uniform mappings for cryptography, in Worksh. Theory Appl. Crypt. Techn., Springer, 1993, 55-64. doi: 10.1007/3-540-48285-7_6. [24] A. Pott, Almost perfect and planar functions, Des. Codes Crypt., 78 (2016), 141-195.  doi: 10.1007/s10623-015-0151-x. [25] F. Rodier, Borne sur le degré des polynômes presque parfaitement non-linéaires, Contemp. Math., 14 (2009), 169.  doi: 10.1090/conm/487/09531. [26] F. Rodier, Some more functions that are not APN infinitely often. the case of Kasami exponents, preprint, arXiv: 1101.6033
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