Class of Quadratic Almost Bent Functions That Is EA-Inequivalent to ...

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Apr 11, 2017 - The permutation relationship for the almost bent (AB) functions in the finite field is ... Almost perfect nonlinear (APN) and almost bent (AB) func-.
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, [6โ€“11] 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 ๐น2๐‘› with an even ๐‘› are preferred due to hardware and software requirements. No APN permutation over ๐น2๐‘› 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‘› โ‰ฅ 8. Berger et al. in [13] provided a significant solution for the Big APN problem: if the components of an APN function over ๐น2๐‘› 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 ๐น2๐‘› 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. ๐น(๐‘ฅ) is called almost perfect nonlinear (APN) function on ๐น2๐‘› if ๐ท๐‘Ž ๐น(๐‘ฅ) = ๐น(๐‘ฅ + ๐‘Ž) โˆ’ ๐น(๐‘ฅ) are 2โˆ’1 on ๐น2๐‘› (i.e., ๐ท๐‘Ž ๐น(๐‘ฅ) = ๐ท๐‘Ž ๐น(๐‘ฆ) if and only if ๐‘ฅ = ๐‘ฆ or ๐‘ฅ + ๐‘Ž = ๐‘ฆ) for all ๐‘Ž โˆˆ ๐น2๐‘› \ {0}. Almost bent (AB) function is a kind of APN function. Definition 2. Every mapping ๐น : ๐น2๐‘› โ†’ ๐น2๐‘› can be unique ๐‘› represented in the form ๐น(๐‘ฅ) = โˆ‘2๐‘–=0โˆ’1 ๐‘Ž๐‘– ๐‘ฅ๐‘– , called the algebraic normal form (ANF) of mapping ๐น. ANF is zero (i.e., ๐‘Ž๐‘– = 0 for any 0 โ‰ค ๐‘– โ‰ค 2๐‘› โˆ’ 1) if and only if mapping is zero (i.e., ๐น(๐‘ฅ) = 0 for any ๐‘ฅ). Definition 3. Every 0 โ‰ค ๐‘— โ‰ค 2๐‘› โˆ’ 1 is equal to an ๐‘› tuple ๐‘—1 โ‹… โ‹… โ‹… ๐‘—๐‘› โˆˆ ๐‘2๐‘› as ๐‘— = โˆ‘๐‘›๐‘˜=1 ๐‘—๐‘˜ 2๐‘›โˆ’๐‘˜ , called the ๐‘› bits binary representation of ๐‘—. Integer ๐‘— and ๐‘› tuple ๐‘—1 โ‹… โ‹… โ‹… ๐‘—๐‘› will be regarded as the same from here. The degree of monomial ๐‘ฅ๐‘— on ๐น2๐‘› is not ๐‘— itself but the number of support supp(๐‘—) = {๐‘˜ : ๐‘—๐‘˜ =ฬธ 0}, called the weight of ๐‘— and denoted as ๐‘ค๐‘ก(๐‘—). The degree of mapping ๐น : ๐น2๐‘› โ†’ ๐น2๐‘› is deg ๐น = max{๐‘ค๐‘ก(๐‘–) : ๐‘Ž๐‘– =ฬธ 0}, the highest degree of all nonzero monomials in its ANF. For example, linear mappings on ๐น2๐‘› are in the form ๐‘›โˆ’1 2๐‘– 2๐‘– ๐ฟ(๐‘ฅ) = โˆ‘๐‘›โˆ’1 ๐‘–=0 ๐‘™๐‘– ๐‘ฅ . Trace mapping tr(๐‘ฅ) = โˆ‘๐‘–=0 ๐‘ฅ is a linear mapping only of values 0 or 1.

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Discrete Dynamics in Nature and Society

Definition 4. Two mappings ๐น๓ธ€  and ๐น over ๐น2๐‘› are called extended affine- (EA-) equivalent if there are affine permutations ๐ด 1 , ๐ด 2 , and ๐ด over ๐น2๐‘› such that ๐น๓ธ€  = ๐ด 1 โˆ˜ ๐น โˆ˜ ๐ด 2 + ๐ด and affine-equivalent if ๐ด = 0.

[tr(๐‘Ž3 ) + 1] tr(๐‘Ž3 ๐ฟ(๐‘Žโˆ’1 )) = 0 for any ๐‘Ž. Obviously, ๐ฟ(๐‘ฅ) = ๐‘2 ๐‘ฅ + ๐‘๐‘ฅ2 and ๐ฟ(๐‘ฅ) = tr(๐‘๐‘ฅ) satisfy the identity. Theorem 8 will show that all kinds of ๐ฟ(๐‘ฅ) satisfying the identity are the adding of the two kinds above.

2. Methods and Tools

2 Theorem 8. One assumes that ๐‘› โ‰ฅ 9 and ๐ฟ(๐‘ฅ) = โˆ‘๐‘›โˆ’1 ๐‘–=0 ๐‘™๐‘– ๐‘ฅ 3 3 โˆ’1 over ๐น2๐‘› . If (tr(๐‘Ž ) + 1) tr(๐‘Ž ๐ฟ(๐‘Ž )) โ‰ก 0, then

๐‘–

Lemma 5. ๐น(๐‘ฅ) over ๐น2๐‘› is EA-equivalent to permutation if and only if there is linear mapping ๐ฟ(๐‘ฅ) such that ๐น(๐‘ฅ) + ๐ฟ(๐‘ฅ) is permuted on ๐น2๐‘› .

2

๐ฟ (๐‘ฅ) = (๐‘™1 + ๐‘) ๐‘ฅ + (๐‘™1 + ๐‘) ๐‘ฅ2 + tr (๐‘๐‘ฅ) . Proof. Initially,

Proof. If ๐ด 1 โˆ˜ ๐น โˆ˜ ๐ด 2 + ๐ด is permuted, then ๐ดโˆ’1 1 โˆ˜ (๐ด 1 โˆ˜ ๐น โˆ˜ โˆ’1 โˆ’1 ๐ด 2 + ๐ด) โˆ˜ ๐ดโˆ’1 2 = ๐น + ๐ด 1 โˆ˜ ๐ด โˆ˜ ๐ด 2 is also permuted over ๐น2๐‘› . ๐‘›โˆ’1 ๐‘›โˆ’1 โˆ’1 2๐‘– 2๐‘– ๐ดโˆ’1 1 โˆ˜ ๐ด โˆ˜ ๐ด 2 = โˆ‘๐‘–=0 ๐‘™๐‘– ๐‘ฅ + ๐‘ denotes that ๐น(๐‘ฅ) + โˆ‘๐‘–=0 ๐‘™๐‘– ๐‘ฅ , thereby permuted over ๐น2๐‘› .

(tr (๐‘Ž3 ) + 1) tr (๐‘Ž3 ๐ฟ (๐‘Žโˆ’1 )) = tr (๐‘Ž3 (tr (๐‘Ž3 ) + 1) ๐ฟ (๐‘Žโˆ’1 )) ๐‘›โˆ’1 ๐‘–=0

The exponents 3(2๐‘– +1)+(โˆ’2๐‘— ) and 3+(โˆ’2๐‘— ) (plus negative will be denoted as minus for convenience) can be divided into the following orbits:

Definition 6. Consider the circulation mapping ๐‘” : ๐‘2๐‘› โ†’ ๐‘2๐‘› defined as ๐‘”(๐‘—1 ๐‘—2 โ‹… โ‹… โ‹… ๐‘—๐‘› ) = ๐‘—2 โ‹… โ‹… โ‹… ๐‘—๐‘› ๐‘—1 , which means 2๐‘— when ๐‘— โ‰ค 2๐‘›โˆ’1 โˆ’ 1 or 2๐‘— โˆ’ 2๐‘› + 1 when ๐‘— โ‰ฅ 2๐‘›โˆ’1 . ๐‘(๐‘—) = {๐‘”๐‘– (๐‘—) : ๐‘– โˆˆ ๐‘} is called the circulation orbit of ๐‘— and its order |๐‘(๐‘—)| = min{๐‘– โˆˆ ๐‘โˆ— : ๐‘”๐‘– (๐‘—) = ๐‘—} minimal positive period of ๐‘—.

1: 3โˆ’20 , 3โˆ’21 , 3(20 +1)โˆ’21 , 3(20 +1)โˆ’22 , 3(21 +1)โˆ’20 , 3(21 + 1) โˆ’ 23 , 3(2๐‘›โˆ’1 + 1) โˆ’ 22 , 3(2๐‘›โˆ’1 + 1) โˆ’ 2๐‘›โˆ’1 . 101โ‹… โ‹… โ‹… 1: 3 โˆ’ 2๐‘— (2 โ‰ค ๐‘— โ‰ค ๐‘› โˆ’ 1), 3(20 + 1) โˆ’ 2๐‘— (๐‘— = 0 or 3 โ‰ค ๐‘— โ‰ค ๐‘› โˆ’ 1), 3(2๐‘– + 1) โˆ’ 2๐‘— (2 โ‰ค ๐‘– โ‰ค ๐‘› โˆ’ 2, ๐‘— = ๐‘– + 2) or ๐‘— = ๐‘– + 2 and 101 of 3(21 + 1) โˆ’ 22 and 3(2๐‘›โˆ’1 + 1) โˆ’ 21 and 1011 of 3(23 + 1) โˆ’ 24 and 3(2๐‘›โˆ’3 + 1) โˆ’ 21 .

The lemma below is obvious since tr(๐‘Ž๐‘– ๐‘ฅ๐‘– ) and tr(๐‘Ž๐‘— ๐‘ฅ๐‘— ) have no similar terms when ๐‘– โˆ‰ ๐‘(๐‘—). ๐‘›

2 โˆ’1 Lemma 7. tr(โˆ‘๐‘–=0 ๐‘Ž๐‘– ๐‘ฅ๐‘– ) = 0 if and only if tr(โˆ‘๐‘–โˆˆ๐‘(๐‘—) ๐‘Ž๐‘– ๐‘ฅ๐‘– ) = 0 ๐‘› for every 0 โ‰ค ๐‘— โ‰ค 2 โˆ’ 1.

10โ‹… โ‹… โ‹… 011: 3(2๐‘– + 1) โˆ’ 2๐‘— (2 โ‰ค ๐‘– โ‰ค ๐‘› โˆ’ 2, ๐‘— = 0, 1, ๐‘–, ๐‘– + 1), 100011 of 3(21 + 1) โˆ’ 2๐‘›โˆ’2 and 3(2๐‘›โˆ’1 + 1) โˆ’ 2๐‘›โˆ’3 , 111 of 3(21 + 1) โˆ’ 21 and 3(2๐‘›โˆ’1 + 1) โˆ’ 20 .

3. Main Result and Proof

101โ‹… โ‹… โ‹… 10โ‹… โ‹… โ‹… 011: 3(2๐‘– +1)โˆ’2๐‘— , 4 โ‰ค ๐‘– โ‰ค ๐‘›โˆ’2, 3 โ‰ค ๐‘— โ‰ค ๐‘–โˆ’1 or ๐‘– + 3 โ‰ค ๐‘— โ‰ค ๐‘› โˆ’ 1.

If ๐ท๐‘Ž ๐น(๐‘ฅ) = 0 has no solution for any ๐‘Ž โˆˆ ๐น2๐‘› \ {0}, then ๐น(๐‘ฅ) is permuted. If ๐‘Ž exists, such that ๐‘Ž๐‘‡ (๐ด๐‘ฅ + ๐‘) = 1, then ๐ด๐‘ฅ + ๐‘ = 0 has no solution based on linear algebra. If ๐ท๐‘Ž ๐น(๐‘ฅ) is 2 โˆ’ 1, then rank ๐ด = ๐‘› โˆ’ 1 and ๐‘Ž, which satisfies ๐‘Ž๐‘‡ ๐ด = 0, is unique. Furthermore, ๐‘Ž satisfies ๐‘Ž๐‘‡ ๐‘ = 1. If tr(๐‘Ž3 ) = 0, then tr(๐‘Ž3 ๐ท๐‘Žโˆ’1 (๐‘ฅ3 + tr(๐‘ฅ9 ) + ๐ฟ(๐‘ฅ))) = tr(๐‘Ž3 ๐ฟ(๐‘Žโˆ’1 ) + 1). If tr(๐‘Ž3 ๐ฟ(๐‘Žโˆ’1 )) = 0 when tr(๐‘Ž3 ) = 0, then

101 โ‹… โ‹… โ‹… 1: 101:

10001โ‹… โ‹… โ‹… 1: 3(21 + 1) โˆ’ 2๐‘— and 3(2๐‘›โˆ’1 + 1) โˆ’ 2๐‘— with ๐‘— unequal to ๐‘› โˆ’ 2 and ๐‘› โˆ’ 3. The condition ๐‘› โ‰ฅ 9 can distinguish this class from 10100011. The following equations were formulated based on Lemma 7:

1/4 ๐‘™01/2 + ๐‘™1 + ๐‘™11/4 + ๐‘™21/2 + ๐‘™01/8 + ๐‘™3 + ๐‘™๐‘›โˆ’1 + ๐‘™22 = 0 ๐‘—

1/2 ๐‘™๐‘— + ๐‘™๐‘—+1 + ๐‘™๐‘—+2 + ๐‘™22 = 0

(2 โ‰ค ๐‘— โ‰ค ๐‘› โˆ’ 3)

๐‘™๐‘›โˆ’1 + ๐‘™01/2 + ๐‘™21/2 + ๐‘™1 = 0

1011:

1/2 + ๐‘™0 + ๐‘™21/4 + ๐‘™41/4 + ๐‘™12 = 0 ๐‘™๐‘›โˆ’2 + ๐‘™๐‘›โˆ’1

100011:

2 + ๐‘™41/4 + ๐‘™04 + ๐‘™61/4 + ๐‘™18 = 0 ๐‘™๐‘›โˆ’2 + ๐‘™๐‘›โˆ’3

111: 10 โ‹… โ‹… โ‹… 011:

2 =0 ๐‘™1 + ๐‘™02 + ๐‘™3 + ๐‘™14 + ๐‘™01/2 + ๐‘™๐‘›โˆ’2

๐‘™3 + ๐‘™08 + ๐‘™5 + ๐‘™116 = 0 ๐‘–

๐‘–+1

๐‘™๐‘– + ๐‘™02 + ๐‘™๐‘–+2 + ๐‘™12 101 โ‹… โ‹… โ‹… 10 โ‹… โ‹… โ‹… 011:

(2)

๐‘–

= tr ((( โˆ‘ ๐‘Ž3(2 +1) ) + ๐‘Ž3 ) ๐ฟ (๐‘Žโˆ’1 )) .

Circulation and cycle are introduced to identify and 2๐‘› โˆ’1 combine similar terms in tr(โˆ‘๐‘–=0 ๐‘Ž๐‘– ๐‘ฅ๐‘– ).

1:

(1)

๐‘—

2 =0 ๐‘™๐‘– = ๐‘™๐‘›+๐‘–โˆ’๐‘—

=0

(5 โ‰ค ๐‘– โ‰ค ๐‘› โˆ’ 3)

(3 โ‰ค ๐‘– โ‰ค ๐‘› โˆ’ 3, 2 โ‰ค ๐‘— โ‰ค ๐‘› โˆ’ ๐‘– โˆ’ 1) .

(3)

Discrete Dynamics in Nature and Society ๐‘–โˆ’๐‘—

The last equation implies that ๐‘™๐‘– = ๐‘™๐‘—2 2 ; ๐‘™๐‘›โˆ’1

3 for all 3 โ‰ค ๐‘–, ๐‘— โ‰ค

2๐‘–

then ๐‘™๐‘– = ๐‘ for all 3 โ‰ค ๐‘– โ‰ค ๐‘› โˆ’ 1. ๐‘› โˆ’ 1. We let ๐‘ = 1/2 + We let ๐‘— = ๐‘› โˆ’ 3 in the second equation; thus, ๐‘™๐‘›โˆ’3 + ๐‘™๐‘›โˆ’2 2๐‘›โˆ’3 2๐‘›โˆ’3 4 ๐‘™๐‘›โˆ’1 + ๐‘™2 = ๐‘™๐‘›โˆ’1 + ๐‘™2 = 0. Therefore, ๐‘™2 = ๐‘ . Moreover, ๐‘™๐‘›โˆ’1 = ๐‘1/2 and ๐‘™2 = ๐‘4 are substituted into the third equation; thus, (๐‘™0 + ๐‘)1/2 = ๐‘™1 + ๐‘2 . Therefore, ๐ฟ(๐‘ฅ) = (๐‘™1 + ๐‘2 )2 ๐‘ฅ + (๐‘™1 + ๐‘2 )๐‘ฅ2 + tr(๐‘๐‘ฅ). ๐‘ฅ3 + tr(๐‘ฅ9 ) + ๐ฟ(๐‘ฅ) cannot be permuted unless ๐ฟ(๐‘ฅ) is in the form ๐‘Ž๐‘ฅ2 + ๐‘Ž2 ๐‘ฅ + tr(๐‘๐‘ฅ). Thus only ๐‘ฅ3 + tr(๐‘ฅ9 ) + ๐‘Ž๐‘ฅ2 + ๐‘Ž2 ๐‘ฅ + tr(๐‘๐‘ฅ) should be considered. If ๐‘ฆ = ๐‘ฅ + ๐‘Ž, then it is equal to ๐‘ฆ3 + tr(๐‘ฆ9 ) + tr(๐‘๐‘ฆ + ๐‘Ž8 ๐‘ฆ + ๐‘Ž๐‘ฆ8 ) + ๐‘Ž3 + tr(๐‘Ž9 + ๐‘Ž๐‘), only different in a constant with ๐‘ฆ3 +tr(๐‘ฆ9 )+tr(๐‘๓ธ€  ๐‘ฆ), in which ๐‘๓ธ€  = ๐‘ + ๐‘Ž8 + ๐‘Ž1/8 . Theorem 9. ๐‘ฅ3 + tr(๐‘ฅ9 ) + tr(๐‘๐‘ฅ) with ๐‘› โ‰ฅ 5 odd is not permuted. Proof. ๐‘ฅ3 is permuted on ๐น2๐‘› because ๐‘› is odd; its inverse is ๐‘ฅ๐‘ก in which supp(๐‘ก) = {1, 3, . . . , ๐‘›}. So the theorem is equal to ๐‘ฅ + tr(๐‘ฅ3 ) + tr(๐‘๐‘ฅ๐‘ก ) which is not permuted. There is ๐ท1 (๐‘ฅ + tr(๐‘ฅ3 ) + tr(๐‘๐‘ฅ๐‘ก )) = ๐ท1 tr(๐‘๐‘ฅ๐‘ก ) = tr(๐‘ โˆ‘๐‘—โˆˆฮ› ๐‘ฅ๐‘— ), in which ฮ› = {0 โ‰ค ๐‘— โ‰ค 2๐‘› โˆ’ 1 : supp(๐‘—) โŠ‚ supp(๐‘ก)}. Every ๐‘— with ๐‘—1 = ๐‘—๐‘› = 1 satisfies ๐‘– โˆ‰ ๐‘(๐‘—) if ๐‘– โˆˆ ฮ› \ {๐‘—}, which means tr(๐‘๐‘ฅ๐‘— ) has no similar terms with tr(๐‘๐‘ฅ๐‘– ); and |๐‘(๐‘—)| = ๐‘›, which means the terms in tr(๐‘๐‘ฅ๐‘— ) are not similar to each other. Since ๐‘—1 and ๐‘—๐‘› will be adjacent after circulation, when ๐‘—1 = ๐‘—๐‘› = 1 there is ฮ› โˆฉ ๐‘(๐‘—) = {๐‘—}, which means ๐‘๐‘ฅ๐‘— is not similar to other terms in tr(๐‘ โˆ‘๐‘—โˆˆฮ› ๐‘ฅ๐‘— ). Thus ANF of tr(๐‘ โˆ‘๐‘—โˆˆฮ› ๐‘ฅ๐‘— ) is not 1. According to Definitions 2 and 3, there exists ๐‘ฅ โˆˆ ๐น2๐‘› such that tr(๐‘๐‘ฅ๐‘ก ) = tr(๐‘(๐‘ฅ+1)๐‘ก ). However, ๐‘ฅ+tr(๐‘ฅ3 ) is equal to ๐‘ฅ+1+tr(๐‘ฅ+1)3 .

4. Conclusions The AB class in [11] when ๐‘› โ‰ฅ 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, 8โ€“10] and class with tr๐‘š/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, โ€œMaximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.),โ€ IEEE Transactions on Information Theory, vol. 14, no. 1, pp. 154โ€“156, 1968. [2] T. Kasami, โ€œThe weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes,โ€ Information and Control, vol. 18, no. 4, pp. 369โ€“394, 1971.

[3] A. Canteaut, P. Charpin, and H. Dobbertin, โ€œBinary m-sequences with three-valued crosscorrelation: a proof of Welchโ€™s conjecture,โ€ IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 4โ€“8, 2000. [4] Y. Niho, Multi-valued cross-correlation functions between two maximal linear recursive sequences [Ph.D. thesis], 1972. [5] H. Dobbertin, โ€œAlmost perfect nonlinear power functions on GF(2๐‘› ): a new case for n Divisible by 5,โ€ in Finite Fields and Applications, pp. 113โ€“121, Springer, Berlin, Germany, 2001. [6] C. Bracken, E. Byrne, N. Markin, and G. McGuire, โ€œNew families of quadratic almost perfect nonlinear trinomials and multinomials,โ€ Finite Fields and Their Applications, vol. 14, no. 3, pp. 703โ€“714, 2008. [7] L. Budaghyan, C. Carlet, and A. Pott, โ€œNew classes of almost bent and almost perfect nonlinear polynomials,โ€ IEEE Transactions on Information Theory, vol. 52, no. 3, pp. 1141โ€“1152, 2006. [8] L. Budaghyan, C. Carlet, P. Felke, and G. Leander, โ€œAn 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 โ€™06), pp. 2637โ€“2641, Seattle, Wash, USA, July 2006. [9] L. Budaghyan, C. Carlet, P. Felke, and G. Leander, โ€œAnother class of quadratic APN binomials over ๐น2๐‘› : the case divisible by 4,โ€ in Proceedings of the International Workshop on Coding and Cryptography (WCC โ€™07), pp. 49โ€“58, Versailles, France, 2007. [10] L. Budaghyan and C. Carlet, โ€œClasses of quadratic APN trinomials and hexanomials and related structures,โ€ IEEE Transactions on Information Theory, vol. 54, no. 5, pp. 2354โ€“2357, 2008. [11] L. Budaghyan, C. Carlet, and G. Leander, โ€œConstructing new APN functions from known ones,โ€ Finite Fields and Their Applications, vol. 15, no. 2, pp. 150โ€“159, 2009. [12] J. Dillon, โ€œAPN polynomials: an update,โ€ http://maths.ucd.ie/ โˆผgmg/Fq9Talks/Dillon.pdf. [13] T. P. Berger, A. Canteaut, P. Charpin, and Y. Laigle-Chapuy, โ€œOn almost perfect nonlinear functions over ๐น2๐‘› ,โ€ IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 4160โ€“4170, 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, โ€œThe nonexistence of permutations EAequivalent to certain AB functions,โ€ IEEE Transactions on Information Theory, vol. 59, no. 1, pp. 672โ€“679, 2013.

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