Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 9627840, 3 pages https://doi.org/10.1155/2017/9627840
Research Article Class of Quadratic Almost Bent Functions That Is EA-Inequivalent to Permutations Xinyang Zhang and Meng Zhou School of Mathematics and Systems Science, Beihang University, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China Correspondence should be addressed to Xinyang Zhang;
[email protected] Received 15 February 2017; Revised 9 April 2017; Accepted 11 April 2017; Published 13 August 2017 Academic Editor: Allan C. Peterson Copyright ยฉ 2017 Xinyang Zhang and Meng Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The permutation relationship for the almost bent (AB) functions in the finite field is a significant issue. Li and Wang proved that a class of AB functions with algebraic degree 3 is extended affine- (EA-) inequivalent to any permutation. This study proves that another class of AB functions, which was developed in 2009, is EA-inequivalent to any permutation. This particular AB function is the first known quadratic class EA-inequivalent to permutation.
1. Introduction Almost perfect nonlinear (APN) and almost bent (AB) functions and significant theoretical meanings have been extensively applied in finite field theory. The search for new APN (see Definition 1) and AB (which also implies APN property) functions has become an interesting topic. Power functions have six known classes of APN functions, namely, Gold [1], Kasami [2], Welch [3, 4], Niho [4], Inverse, and Dobbertin [5]. Apart from power functions, APN function also has several known classes. Accordingly, [6โ11] show that all results are quadratic functions (the meaning of degree is a little different, see Definitions 2 and 3). In the design of a block cipher, permutations over ๐น2๐ with an even ๐ are preferred due to hardware and software requirements. No APN permutation over ๐น2๐ with an even ๐ was determined until Dillon [12] in 2009. Thus, the Big APN problem emerged: Does such function exist? This problem is still open for ๐ โฅ 8. Berger et al. in [13] provided a significant solution for the Big APN problem: if the components of an APN function over ๐น2๐ with an even ๐ are plateaued, then a bent component exists, which is not permuted. This result is negative for quadratic functions because quadratic implies reaching a plateau [14]. If ๐น2๐ is changed by an odd ๐, then the plateaued APN functions are equal to the AB functions based on the result of [3]. The AB functions are conjectured to be EA-equivalent (see Definition 4) to the permutations.
In 2013, Li and Wang [15] proved that the infinite class in [7] is EA-inequivalent to any permutation. Definition 1. ๐น(๐ฅ) is called almost perfect nonlinear (APN) function on ๐น2๐ if ๐ท๐ ๐น(๐ฅ) = ๐น(๐ฅ + ๐) โ ๐น(๐ฅ) are 2โ1 on ๐น2๐ (i.e., ๐ท๐ ๐น(๐ฅ) = ๐ท๐ ๐น(๐ฆ) if and only if ๐ฅ = ๐ฆ or ๐ฅ + ๐ = ๐ฆ) for all ๐ โ ๐น2๐ \ {0}. Almost bent (AB) function is a kind of APN function. Definition 2. Every mapping ๐น : ๐น2๐ โ ๐น2๐ can be unique ๐ represented in the form ๐น(๐ฅ) = โ2๐=0โ1 ๐๐ ๐ฅ๐ , called the algebraic normal form (ANF) of mapping ๐น. ANF is zero (i.e., ๐๐ = 0 for any 0 โค ๐ โค 2๐ โ 1) if and only if mapping is zero (i.e., ๐น(๐ฅ) = 0 for any ๐ฅ). Definition 3. Every 0 โค ๐ โค 2๐ โ 1 is equal to an ๐ tuple ๐1 โ
โ
โ
๐๐ โ ๐2๐ as ๐ = โ๐๐=1 ๐๐ 2๐โ๐ , called the ๐ bits binary representation of ๐. Integer ๐ and ๐ tuple ๐1 โ
โ
โ
๐๐ will be regarded as the same from here. The degree of monomial ๐ฅ๐ on ๐น2๐ is not ๐ itself but the number of support supp(๐) = {๐ : ๐๐ =ฬธ 0}, called the weight of ๐ and denoted as ๐ค๐ก(๐). The degree of mapping ๐น : ๐น2๐ โ ๐น2๐ is deg ๐น = max{๐ค๐ก(๐) : ๐๐ =ฬธ 0}, the highest degree of all nonzero monomials in its ANF. For example, linear mappings on ๐น2๐ are in the form ๐โ1 2๐ 2๐ ๐ฟ(๐ฅ) = โ๐โ1 ๐=0 ๐๐ ๐ฅ . Trace mapping tr(๐ฅ) = โ๐=0 ๐ฅ is a linear mapping only of values 0 or 1.
2
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Definition 4. Two mappings ๐น๓ธ and ๐น over ๐น2๐ are called extended affine- (EA-) equivalent if there are affine permutations ๐ด 1 , ๐ด 2 , and ๐ด over ๐น2๐ such that ๐น๓ธ = ๐ด 1 โ ๐น โ ๐ด 2 + ๐ด and affine-equivalent if ๐ด = 0.
[tr(๐3 ) + 1] tr(๐3 ๐ฟ(๐โ1 )) = 0 for any ๐. Obviously, ๐ฟ(๐ฅ) = ๐2 ๐ฅ + ๐๐ฅ2 and ๐ฟ(๐ฅ) = tr(๐๐ฅ) satisfy the identity. Theorem 8 will show that all kinds of ๐ฟ(๐ฅ) satisfying the identity are the adding of the two kinds above.
2. Methods and Tools
2 Theorem 8. One assumes that ๐ โฅ 9 and ๐ฟ(๐ฅ) = โ๐โ1 ๐=0 ๐๐ ๐ฅ 3 3 โ1 over ๐น2๐ . If (tr(๐ ) + 1) tr(๐ ๐ฟ(๐ )) โก 0, then
๐
Lemma 5. ๐น(๐ฅ) over ๐น2๐ is EA-equivalent to permutation if and only if there is linear mapping ๐ฟ(๐ฅ) such that ๐น(๐ฅ) + ๐ฟ(๐ฅ) is permuted on ๐น2๐ .
2
๐ฟ (๐ฅ) = (๐1 + ๐) ๐ฅ + (๐1 + ๐) ๐ฅ2 + tr (๐๐ฅ) . Proof. Initially,
Proof. If ๐ด 1 โ ๐น โ ๐ด 2 + ๐ด is permuted, then ๐ดโ1 1 โ (๐ด 1 โ ๐น โ โ1 โ1 ๐ด 2 + ๐ด) โ ๐ดโ1 2 = ๐น + ๐ด 1 โ ๐ด โ ๐ด 2 is also permuted over ๐น2๐ . ๐โ1 ๐โ1 โ1 2๐ 2๐ ๐ดโ1 1 โ ๐ด โ ๐ด 2 = โ๐=0 ๐๐ ๐ฅ + ๐ denotes that ๐น(๐ฅ) + โ๐=0 ๐๐ ๐ฅ , thereby permuted over ๐น2๐ .
(tr (๐3 ) + 1) tr (๐3 ๐ฟ (๐โ1 )) = tr (๐3 (tr (๐3 ) + 1) ๐ฟ (๐โ1 )) ๐โ1 ๐=0
The exponents 3(2๐ +1)+(โ2๐ ) and 3+(โ2๐ ) (plus negative will be denoted as minus for convenience) can be divided into the following orbits:
Definition 6. Consider the circulation mapping ๐ : ๐2๐ โ ๐2๐ defined as ๐(๐1 ๐2 โ
โ
โ
๐๐ ) = ๐2 โ
โ
โ
๐๐ ๐1 , which means 2๐ when ๐ โค 2๐โ1 โ 1 or 2๐ โ 2๐ + 1 when ๐ โฅ 2๐โ1 . ๐(๐) = {๐๐ (๐) : ๐ โ ๐} is called the circulation orbit of ๐ and its order |๐(๐)| = min{๐ โ ๐โ : ๐๐ (๐) = ๐} minimal positive period of ๐.
1: 3โ20 , 3โ21 , 3(20 +1)โ21 , 3(20 +1)โ22 , 3(21 +1)โ20 , 3(21 + 1) โ 23 , 3(2๐โ1 + 1) โ 22 , 3(2๐โ1 + 1) โ 2๐โ1 . 101โ
โ
โ
1: 3 โ 2๐ (2 โค ๐ โค ๐ โ 1), 3(20 + 1) โ 2๐ (๐ = 0 or 3 โค ๐ โค ๐ โ 1), 3(2๐ + 1) โ 2๐ (2 โค ๐ โค ๐ โ 2, ๐ = ๐ + 2) or ๐ = ๐ + 2 and 101 of 3(21 + 1) โ 22 and 3(2๐โ1 + 1) โ 21 and 1011 of 3(23 + 1) โ 24 and 3(2๐โ3 + 1) โ 21 .
The lemma below is obvious since tr(๐๐ ๐ฅ๐ ) and tr(๐๐ ๐ฅ๐ ) have no similar terms when ๐ โ ๐(๐). ๐
2 โ1 Lemma 7. tr(โ๐=0 ๐๐ ๐ฅ๐ ) = 0 if and only if tr(โ๐โ๐(๐) ๐๐ ๐ฅ๐ ) = 0 ๐ for every 0 โค ๐ โค 2 โ 1.
10โ
โ
โ
011: 3(2๐ + 1) โ 2๐ (2 โค ๐ โค ๐ โ 2, ๐ = 0, 1, ๐, ๐ + 1), 100011 of 3(21 + 1) โ 2๐โ2 and 3(2๐โ1 + 1) โ 2๐โ3 , 111 of 3(21 + 1) โ 21 and 3(2๐โ1 + 1) โ 20 .
3. Main Result and Proof
101โ
โ
โ
10โ
โ
โ
011: 3(2๐ +1)โ2๐ , 4 โค ๐ โค ๐โ2, 3 โค ๐ โค ๐โ1 or ๐ + 3 โค ๐ โค ๐ โ 1.
If ๐ท๐ ๐น(๐ฅ) = 0 has no solution for any ๐ โ ๐น2๐ \ {0}, then ๐น(๐ฅ) is permuted. If ๐ exists, such that ๐๐ (๐ด๐ฅ + ๐) = 1, then ๐ด๐ฅ + ๐ = 0 has no solution based on linear algebra. If ๐ท๐ ๐น(๐ฅ) is 2 โ 1, then rank ๐ด = ๐ โ 1 and ๐, which satisfies ๐๐ ๐ด = 0, is unique. Furthermore, ๐ satisfies ๐๐ ๐ = 1. If tr(๐3 ) = 0, then tr(๐3 ๐ท๐โ1 (๐ฅ3 + tr(๐ฅ9 ) + ๐ฟ(๐ฅ))) = tr(๐3 ๐ฟ(๐โ1 ) + 1). If tr(๐3 ๐ฟ(๐โ1 )) = 0 when tr(๐3 ) = 0, then
101 โ
โ
โ
1: 101:
10001โ
โ
โ
1: 3(21 + 1) โ 2๐ and 3(2๐โ1 + 1) โ 2๐ with ๐ unequal to ๐ โ 2 and ๐ โ 3. The condition ๐ โฅ 9 can distinguish this class from 10100011. The following equations were formulated based on Lemma 7:
1/4 ๐01/2 + ๐1 + ๐11/4 + ๐21/2 + ๐01/8 + ๐3 + ๐๐โ1 + ๐22 = 0 ๐
1/2 ๐๐ + ๐๐+1 + ๐๐+2 + ๐22 = 0
(2 โค ๐ โค ๐ โ 3)
๐๐โ1 + ๐01/2 + ๐21/2 + ๐1 = 0
1011:
1/2 + ๐0 + ๐21/4 + ๐41/4 + ๐12 = 0 ๐๐โ2 + ๐๐โ1
100011:
2 + ๐41/4 + ๐04 + ๐61/4 + ๐18 = 0 ๐๐โ2 + ๐๐โ3
111: 10 โ
โ
โ
011:
2 =0 ๐1 + ๐02 + ๐3 + ๐14 + ๐01/2 + ๐๐โ2
๐3 + ๐08 + ๐5 + ๐116 = 0 ๐
๐+1
๐๐ + ๐02 + ๐๐+2 + ๐12 101 โ
โ
โ
10 โ
โ
โ
011:
(2)
๐
= tr ((( โ ๐3(2 +1) ) + ๐3 ) ๐ฟ (๐โ1 )) .
Circulation and cycle are introduced to identify and 2๐ โ1 combine similar terms in tr(โ๐=0 ๐๐ ๐ฅ๐ ).
1:
(1)
๐
2 =0 ๐๐ = ๐๐+๐โ๐
=0
(5 โค ๐ โค ๐ โ 3)
(3 โค ๐ โค ๐ โ 3, 2 โค ๐ โค ๐ โ ๐ โ 1) .
(3)
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The last equation implies that ๐๐ = ๐๐2 2 ; ๐๐โ1
3 for all 3 โค ๐, ๐ โค
2๐
then ๐๐ = ๐ for all 3 โค ๐ โค ๐ โ 1. ๐ โ 1. We let ๐ = 1/2 + We let ๐ = ๐ โ 3 in the second equation; thus, ๐๐โ3 + ๐๐โ2 2๐โ3 2๐โ3 4 ๐๐โ1 + ๐2 = ๐๐โ1 + ๐2 = 0. Therefore, ๐2 = ๐ . Moreover, ๐๐โ1 = ๐1/2 and ๐2 = ๐4 are substituted into the third equation; thus, (๐0 + ๐)1/2 = ๐1 + ๐2 . Therefore, ๐ฟ(๐ฅ) = (๐1 + ๐2 )2 ๐ฅ + (๐1 + ๐2 )๐ฅ2 + tr(๐๐ฅ). ๐ฅ3 + tr(๐ฅ9 ) + ๐ฟ(๐ฅ) cannot be permuted unless ๐ฟ(๐ฅ) is in the form ๐๐ฅ2 + ๐2 ๐ฅ + tr(๐๐ฅ). Thus only ๐ฅ3 + tr(๐ฅ9 ) + ๐๐ฅ2 + ๐2 ๐ฅ + tr(๐๐ฅ) should be considered. If ๐ฆ = ๐ฅ + ๐, then it is equal to ๐ฆ3 + tr(๐ฆ9 ) + tr(๐๐ฆ + ๐8 ๐ฆ + ๐๐ฆ8 ) + ๐3 + tr(๐9 + ๐๐), only different in a constant with ๐ฆ3 +tr(๐ฆ9 )+tr(๐๓ธ ๐ฆ), in which ๐๓ธ = ๐ + ๐8 + ๐1/8 . Theorem 9. ๐ฅ3 + tr(๐ฅ9 ) + tr(๐๐ฅ) with ๐ โฅ 5 odd is not permuted. Proof. ๐ฅ3 is permuted on ๐น2๐ because ๐ is odd; its inverse is ๐ฅ๐ก in which supp(๐ก) = {1, 3, . . . , ๐}. So the theorem is equal to ๐ฅ + tr(๐ฅ3 ) + tr(๐๐ฅ๐ก ) which is not permuted. There is ๐ท1 (๐ฅ + tr(๐ฅ3 ) + tr(๐๐ฅ๐ก )) = ๐ท1 tr(๐๐ฅ๐ก ) = tr(๐ โ๐โฮ ๐ฅ๐ ), in which ฮ = {0 โค ๐ โค 2๐ โ 1 : supp(๐) โ supp(๐ก)}. Every ๐ with ๐1 = ๐๐ = 1 satisfies ๐ โ ๐(๐) if ๐ โ ฮ \ {๐}, which means tr(๐๐ฅ๐ ) has no similar terms with tr(๐๐ฅ๐ ); and |๐(๐)| = ๐, which means the terms in tr(๐๐ฅ๐ ) are not similar to each other. Since ๐1 and ๐๐ will be adjacent after circulation, when ๐1 = ๐๐ = 1 there is ฮ โฉ ๐(๐) = {๐}, which means ๐๐ฅ๐ is not similar to other terms in tr(๐ โ๐โฮ ๐ฅ๐ ). Thus ANF of tr(๐ โ๐โฮ ๐ฅ๐ ) is not 1. According to Definitions 2 and 3, there exists ๐ฅ โ ๐น2๐ such that tr(๐๐ฅ๐ก ) = tr(๐(๐ฅ+1)๐ก ). However, ๐ฅ+tr(๐ฅ3 ) is equal to ๐ฅ+1+tr(๐ฅ+1)3 .
4. Conclusions The AB class in [11] when ๐ โฅ 9 is EA-inequivalent to permutations. However, distinguishing whether the AB class is CCZ-equivalent to permutations is still unknown. Furthermore, the relationship of the permutations of the APN classes in [6, 8โ10] and class with tr๐/๐ in [7] is unknown. The solution to these problems will be a significant topic in algebra and cryptography in the future.
Conflicts of Interest The authors declare that they have no conflicts of interest.
Acknowledgments This work is partly supported by NSFC Project 11271040.
References [1] R. Gold, โMaximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.),โ IEEE Transactions on Information Theory, vol. 14, no. 1, pp. 154โ156, 1968. [2] T. Kasami, โThe weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes,โ Information and Control, vol. 18, no. 4, pp. 369โ394, 1971.
[3] A. Canteaut, P. Charpin, and H. Dobbertin, โBinary m-sequences with three-valued crosscorrelation: a proof of Welchโs conjecture,โ IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 4โ8, 2000. [4] Y. Niho, Multi-valued cross-correlation functions between two maximal linear recursive sequences [Ph.D. thesis], 1972. [5] H. Dobbertin, โAlmost perfect nonlinear power functions on GF(2๐ ): a new case for n Divisible by 5,โ in Finite Fields and Applications, pp. 113โ121, Springer, Berlin, Germany, 2001. [6] C. Bracken, E. Byrne, N. Markin, and G. McGuire, โNew families of quadratic almost perfect nonlinear trinomials and multinomials,โ Finite Fields and Their Applications, vol. 14, no. 3, pp. 703โ714, 2008. [7] L. Budaghyan, C. Carlet, and A. Pott, โNew classes of almost bent and almost perfect nonlinear polynomials,โ IEEE Transactions on Information Theory, vol. 52, no. 3, pp. 1141โ1152, 2006. [8] L. Budaghyan, C. Carlet, P. Felke, and G. Leander, โAn infinite class of quadratic APN functions which are not equivalent to power mappings,โ in Proceedings of the IEEE International Symposium on Information Theory (ISIT โ06), pp. 2637โ2641, Seattle, Wash, USA, July 2006. [9] L. Budaghyan, C. Carlet, P. Felke, and G. Leander, โAnother class of quadratic APN binomials over ๐น2๐ : the case divisible by 4,โ in Proceedings of the International Workshop on Coding and Cryptography (WCC โ07), pp. 49โ58, Versailles, France, 2007. [10] L. Budaghyan and C. Carlet, โClasses of quadratic APN trinomials and hexanomials and related structures,โ IEEE Transactions on Information Theory, vol. 54, no. 5, pp. 2354โ2357, 2008. [11] L. Budaghyan, C. Carlet, and G. Leander, โConstructing new APN functions from known ones,โ Finite Fields and Their Applications, vol. 15, no. 2, pp. 150โ159, 2009. [12] J. Dillon, โAPN polynomials: an update,โ http://maths.ucd.ie/ โผgmg/Fq9Talks/Dillon.pdf. [13] T. P. Berger, A. Canteaut, P. Charpin, and Y. Laigle-Chapuy, โOn almost perfect nonlinear functions over ๐น2๐ ,โ IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 4160โ4170, 2006. [14] F. J. MacWilliams and N. J. Sloane, The Theory of ErrorCorrecting Codes, North Holland Publishing, Amsterdam, Netherlands, 1977. [15] Y. Li and M. Wang, โThe nonexistence of permutations EAequivalent to certain AB functions,โ IEEE Transactions on Information Theory, vol. 59, no. 1, pp. 672โ679, 2013.
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