Partitioning CCZ classes into EA classes

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

Partitioning CCZ classes into EA classes

Kathy Horadam ^1, and Russell East ^1,

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

Received January 2011 Revised September 2011 Published January 2012

Abstract
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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.

Keywords: EA equivalence, APN function, CCZ equivalence, dierential uniformity..

Mathematics Subject Classification: Primary: 94A60; Secondary: 05B10, 20K3.

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.
[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.
[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.
[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.
[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.
[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.
[7]	R. East, "Nonlinear Functions over Finite Fields,'' Honours thesis, RMIT University, 2008, (unpublished).
[8]	Y. Edel, APN functions and dual hyperovals, in "NATO Advanced Research Workshop,'' Veliko Tarnovo, Bulgaria, (2008).
[9]	Y. Edel, Personal correspondence, March 2010.
[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.
[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.
[12]	K. J. Horadam, "Hadamard Matrices and their Applications,'' Princeton University Press, Princeton, 2007.
[13]	K. J. Horadam, Relative difference sets, graphs and inequivalence of functions between groups, J. Combin. Des., 18 (2010), 260-273.
[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.
[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.
[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.
[17]	K. Nyberg, Differentially uniform mappings for cryptography, in "EUROCRYPT-93,'' Springer, New York, (1994), 55-64.

show all references

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.
[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.
[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.
[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.
[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.
[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.
[7]	R. East, "Nonlinear Functions over Finite Fields,'' Honours thesis, RMIT University, 2008, (unpublished).
[8]	Y. Edel, APN functions and dual hyperovals, in "NATO Advanced Research Workshop,'' Veliko Tarnovo, Bulgaria, (2008).
[9]	Y. Edel, Personal correspondence, March 2010.
[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.
[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.
[12]	K. J. Horadam, "Hadamard Matrices and their Applications,'' Princeton University Press, Princeton, 2007.
[13]	K. J. Horadam, Relative difference sets, graphs and inequivalence of functions between groups, J. Combin. Des., 18 (2010), 260-273.
[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.
[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.
[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.
[17]	K. Nyberg, Differentially uniform mappings for cryptography, in "EUROCRYPT-93,'' Springer, New York, (1994), 55-64.

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