# American Institute of Mathematical Sciences

February  2012, 6(1): 95-106. doi: 10.3934/amc.2012.6.95

## Partitioning CCZ classes into EA classes

 1 RMIT University, G.P.O. Box 2476, Melbourne, VIC 3001, Australia, Australia

Received  January 2011 Revised  September 2011 Published  January 2012

EA equivalence classes and the coarser CCZ equivalence classes of functions over $GF(p^n)$ each preserve measures of nonlinearity desirable in cryptographic functions. We identify very precisely the condition on a linear permutation defining a CCZ isomorphism between functions which ensures that the CCZ isomorphism can be rewritten as EA isomorphism. We introduce new algebraic invariants $n(f)$ of the EA isomorphism class of $f$ and $s(f)$ of the CCZ isomorphism class of $f$, with $n(f) < s(f)$, and relate them to the differential uniformity of $f$. We formulate three questions about partitioning CCZ classes into EA classes and relate these to a conjecture of Edel's about quadratic APN functions.
Citation: Kathy Horadam, Russell East. Partitioning CCZ classes into EA classes. Advances in Mathematics of Communications, 2012, 6 (1) : 95-106. doi: 10.3934/amc.2012.6.95
##### References:
 [1] B. Aslan, M. T. Sakalli and E. Bulus, Classifying 8-bit to 8-bit S-boxes based on power mappings from the point of DDT and LAT distributions, in "Proc. WAIFI 2008'' (eds. J. von zur Gathen), Springer, Berlin, (2008), 123-133.  Google Scholar [2] C. Bracken, E. Byrne, G. McGuire and G. Nebe, On the equivalence of quadratic APN functions, Des. Codes Cryptogr., 61 (2011), 261-272. doi: 10.1007/s10623-010-9475-8.  Google Scholar [3] M. Brinkmann and G. Leander, On the classification of APN functions up to dimension 5, Des. Codes Cryptogr., 49 (2008), 273-288. doi: 10.1007/s10623-008-9194-6.  Google Scholar [4] K. A. Browning, J. F. Dillon, R. E. Kibler and M. T. McQuistan, APN polynomials and related codes, J. Comb. Inf. Syst. Sci., 34 (2009), 135-159. Google Scholar [5] L. Budaghyan, C. Carlet and A. Pott, New classes of almost bent and almost perfect nonlinear polynomials, IEEE Trans. Inform. Theory, 52 (2006), 1141-1152. doi: 10.1109/TIT.2005.864481.  Google Scholar [6] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.  Google Scholar [7] R. East, "Nonlinear Functions over Finite Fields,'' Honours thesis, RMIT University, 2008, (unpublished). Google Scholar [8] Y. Edel, APN functions and dual hyperovals, in "NATO Advanced Research Workshop,'' Veliko Tarnovo, Bulgaria, (2008). Google Scholar [9] Y. Edel, Personal correspondence, March 2010. Google Scholar [10] Y. Edel, G. Kyureghyan and A. Pott, A new APN function which is not equivalent to a power mapping, IEEE Trans. Inform. Theory, 52 (2006), 744-747. doi: 10.1109/TIT.2005.862128.  Google Scholar [11] Y. Edel and A. Pott, A new almost perfect nonlinear function which is not quadratic, Adv. Math. Commun., 3 (2009), 59-81. doi: 10.3934/amc.2009.3.59.  Google Scholar [12] K. J. Horadam, "Hadamard Matrices and their Applications,'' Princeton University Press, Princeton, 2007.  Google Scholar [13] K. J. Horadam, Relative difference sets, graphs and inequivalence of functions between groups, J. Combin. Des., 18 (2010), 260-273.  Google Scholar [14] K. J. Horadam, Equivalence classes of functions between finite groups, J. Algebr. Comb., (2012), to appear, DOI 10.1007/s10801-011-0310-8. doi: 10.1007/s10801-011-0310-8.  Google Scholar [15] K. J. Horadam and D. G. Farmer, Bundles, presemifields and nonlinear functions, Des. Codes Cryptogr., 49 (2008), 79-94. doi: 10.1007/s10623-008-9172-z.  Google Scholar [16] G. M. Kyureghyan and A. Pott, Some theorems on planar mappings, in "Proc. WAIFI 2008'' (eds. J. von zur Gathen et al), Springer, Berlin, (2008), 117-122.  Google Scholar [17] K. Nyberg, Differentially uniform mappings for cryptography, in "EUROCRYPT-93,'' Springer, New York, (1994), 55-64.  Google Scholar

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##### References:
 [1] B. Aslan, M. T. Sakalli and E. Bulus, Classifying 8-bit to 8-bit S-boxes based on power mappings from the point of DDT and LAT distributions, in "Proc. WAIFI 2008'' (eds. J. von zur Gathen), Springer, Berlin, (2008), 123-133.  Google Scholar [2] C. Bracken, E. Byrne, G. McGuire and G. Nebe, On the equivalence of quadratic APN functions, Des. Codes Cryptogr., 61 (2011), 261-272. doi: 10.1007/s10623-010-9475-8.  Google Scholar [3] M. Brinkmann and G. Leander, On the classification of APN functions up to dimension 5, Des. Codes Cryptogr., 49 (2008), 273-288. doi: 10.1007/s10623-008-9194-6.  Google Scholar [4] K. A. Browning, J. F. Dillon, R. E. Kibler and M. T. McQuistan, APN polynomials and related codes, J. Comb. Inf. Syst. Sci., 34 (2009), 135-159. Google Scholar [5] L. Budaghyan, C. Carlet and A. Pott, New classes of almost bent and almost perfect nonlinear polynomials, IEEE Trans. Inform. Theory, 52 (2006), 1141-1152. doi: 10.1109/TIT.2005.864481.  Google Scholar [6] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.  Google Scholar [7] R. East, "Nonlinear Functions over Finite Fields,'' Honours thesis, RMIT University, 2008, (unpublished). Google Scholar [8] Y. Edel, APN functions and dual hyperovals, in "NATO Advanced Research Workshop,'' Veliko Tarnovo, Bulgaria, (2008). Google Scholar [9] Y. Edel, Personal correspondence, March 2010. Google Scholar [10] Y. Edel, G. Kyureghyan and A. Pott, A new APN function which is not equivalent to a power mapping, IEEE Trans. Inform. Theory, 52 (2006), 744-747. doi: 10.1109/TIT.2005.862128.  Google Scholar [11] Y. Edel and A. Pott, A new almost perfect nonlinear function which is not quadratic, Adv. Math. Commun., 3 (2009), 59-81. doi: 10.3934/amc.2009.3.59.  Google Scholar [12] K. J. Horadam, "Hadamard Matrices and their Applications,'' Princeton University Press, Princeton, 2007.  Google Scholar [13] K. J. Horadam, Relative difference sets, graphs and inequivalence of functions between groups, J. Combin. Des., 18 (2010), 260-273.  Google Scholar [14] K. J. Horadam, Equivalence classes of functions between finite groups, J. Algebr. Comb., (2012), to appear, DOI 10.1007/s10801-011-0310-8. doi: 10.1007/s10801-011-0310-8.  Google Scholar [15] K. J. Horadam and D. G. Farmer, Bundles, presemifields and nonlinear functions, Des. Codes Cryptogr., 49 (2008), 79-94. doi: 10.1007/s10623-008-9172-z.  Google Scholar [16] G. M. Kyureghyan and A. Pott, Some theorems on planar mappings, in "Proc. WAIFI 2008'' (eds. J. von zur Gathen et al), Springer, Berlin, (2008), 117-122.  Google Scholar [17] K. Nyberg, Differentially uniform mappings for cryptography, in "EUROCRYPT-93,'' Springer, New York, (1994), 55-64.  Google Scholar
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