APN Permutations: Difference between revisions

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Let <math>\mathcal{H}</math> be a binary <math>(r\times n)</math> matrix. We say that a linear binary code <math>C</math> of length <math>n</math> is defined by the <i>parity check matrix</i> <math>\mathcal{H}</math> if <math>C=\{c\in\mathbb{F}_2^n\;|\;c\mathcal{H}^t=0\},</math> where <math>\mathcal{H}^t</math> is the transposed matrix of <math>\mathcal{H}</math>.
Let <math>\mathcal{H}</math> be a binary <math>(r\times n)</math> matrix. We say that a linear binary code <math>C</math> of length <math>n</math> is defined by the <i>parity check matrix</i> <math>\mathcal{H}</math> if <math>C=\{c\in\mathbb{F}_2^n\;|\;c\mathcal{H}^t=0\},</math> where <math>\mathcal{H}^t</math> is the transposed matrix of <math>\mathcal{H}</math>.


APN (and AB) properties were expressed in terms of codes in <ref name="CCZ">Carlet, Claude & Charpin, Pascale & Zinoviev, Victor. (1998). <i>Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems.</i> Designs, Codes and Cryptography, vol. 15, p. 125-156.</ref>. In particular, the following theorem is proved:
APN (and AB) properties were expressed in terms of codes in <ref name="CCZ">Carlet, Claude & Charpin, Pascale & Zinoviev, Victor. (1998). <i>Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems.</i> Designs, Codes and Cryptography, vol. 15, p. 125-156.</ref>. In particular, we mention the following result:




Let <math>F</math> be a function on <math>\mathbb{F}_{2^m}</math> such that <math>F(0) = 0</math> and let <math>C_F</math> be the
<strong>Theorem.</strong> Let <math>F</math> be a function on <math>\mathbb{F}_{2^m}</math> such that <math>F(0) = 0</math> and let <math>C_F</math> be the
<math>[2^m − 1,k,d]</math> code defined by the parity check matrix
<math>[2^m − 1,k,d]</math> code defined by the parity check matrix


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3. <math>F</math> is AB if and only if the weight of every codeword in <math>C_F^{\perp}</math> lies in <math>\{0, 2^{m-1}, 2^{m-1}\pm 2^{m-1/2}\}</math>.
3. <math>F</math> is AB if and only if the weight of every codeword in <math>C_F^{\perp}</math> lies in <math>\{0, 2^{m-1}, 2^{m-1}\pm 2^{m-1/2}\}</math>.


A binary linear <math>[2^k-1,k,2^{k-1}]-</math>code <math>C</math> in <math>\mathbb{F}_2^n</math> is called simplex.


Let <math>F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}</math> be APN, with <math>F(0)=0</math>.  
Let <math>F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}</math> be APN, with <math>F(0)=0</math>.  

Revision as of 15:04, 19 November 2024

Characterization of Permutations

Component Functions

An -function is a permutation if and only if all of its components Failed to parse (syntax error): {\displaystyle f_λ} for are balanced.

Autocorrelation Functions of the Directional Derivatives

For any boolean function , we denote by the following value related to the Fourier (or Walsh) transform of :

The characterization in terms of the component functions given above can be equivalently expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle λ\in\mathbb{F}^*_{2^n}} .

Equivalently [1], Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a permutation if and only if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in\mathbb{F}^*_{2^n}} .

Characterization of APN Permutations

Up to CCZ-equivalence, all of the APN permutations known so far belong to a few families, namely:[2]

1. APN monomial functions in odd dimension.

2. One infinite family of quadratic polynomials in dimension , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} odd and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gcd(n,3)=1} .[3]

3. Dillon's permutation in dimension 6.[4]

4. Two sporadic quadratic APN permutations in dimension 9.[5]

On the component functions

Clearly we have that no component function can be of degree 1. (This result is true for general APN maps)

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} even we have also that no component can be partially-bent[6]. This implies that, in even dimension, no component can be of degree 2.

Autocorrelation Functions of the Directional Derivatives

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n,n)} -function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is an APN permutation if and only if [1]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

and

for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in\mathbb{F}^*_{2^n}} .

On APN Power Functions

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} odd, all power APN functions and the known APN binomials are permutations. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is even, no APN function exists in a class of permutations including power permutations.

Specifically:

If a power function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=x^d} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^n}} is APN, then for every we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^d=1} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3=1} , that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{-1}(1)=\mathbb{F}_4\cap\mathbb{F}_{2^n}^*.}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gcd(d,2^n-1)=1} and, if is even, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gcd(d,2^n-1)=3} .

Consequently, APN power functions are permutations if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd, and are three-to-one over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^n}^*} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is even.[7]

APN Permutations and Codes

The (Hamming) weight of any vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathbb{F}_2^n} is denoted by , and the (Hamming) distance between any two vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_2^n} is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y)} . Any linear subspace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_2^n} of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is called a binary linear code of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [n,k,d]} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the minimum Hamming distance of .

Any linear code is associated with its dual code, denoted by and defined as

Let be a binary matrix. We say that a linear binary code Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is defined by the parity check matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=\{c\in\mathbb{F}_2^n\;|\;c\mathcal{H}^t=0\},} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}^t} is the transposed matrix of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} .

APN (and AB) properties were expressed in terms of codes in [8]. In particular, we mention the following result:


Theorem. Let be a function on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^m}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(0) = 0} and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_F} be the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2^m − 1,k,d]} code defined by the parity check matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}_F=\begin{pmatrix} 1 & \alpha & \alpha^2 & \cdots & \alpha^{2^m-2}\\ F(1) & F(\alpha) & F(\alpha^2) & \cdots & F(\alpha^{2^m-2}) \end{pmatrix}}

where each entry is viewed as a binary vector and is the primitive element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^m}.} Then:

1. The code Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_F} is such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \leq d \leq 5} .

2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is APN if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 5} .

3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is AB if and only if the weight of every codeword in lies in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0, 2^{m-1}, 2^{m-1}\pm 2^{m-1/2}\}} .


A binary linear Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2^k-1,k,2^{k-1}]-} code Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_2^n} is called simplex.

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}} be APN, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(0)=0} .

is CCZ-equivalent to an APN permutation if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_F^{\perp}} is a double simplex code (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_F^{\perp}=C_1\oplus C_2} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1, C_2} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2^n-1,n,2^{n-1}]-} codes).[9]

An APN Permutation in Dimension 6

In his paper [10], Hou conjectured that APN permutations did not exist in even dimension. He proved the following theorem that covers the case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4} :

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\in\mathbb{F}_{2^n}[x]} be a permutation polynomial with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2m} . Then:

1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is not APN.

2. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\in\mathbb{F}_{2^m}[x]} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is not APN.

The question of whether APN permutations exist in even dimension was a long-standing problem until, in 2009, Dillon presented an APN permutation (of algebraic degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-2} and nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{n-1}-2^{n/2}} ) in dimension 6[4].

This function is CCZ-equivalent to the Kim function (where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is a primitive element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^6}} ), whose associated code Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{\perp}_F} is therefore a double simplex code. It was used later in the cryptosystem Fides[11], which has been subsequently broken due to its weaknesses in the linear component.

Dillon's function is also EA-equivalent to an involution and it is studied further in the introduction of the butterfly construction[12]. Unfortunately, this construction does not allow obtaining APN permutations in more than six variables[13].

The Big Open APN Problem

The question of existence of APN permutations in even dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geq 8} remains open. There exist nonexistent results within the following classes:

1. Plateaued functions (when APN, they have bent components);

2. A class of functions including power functions;

3. Functions whose univariate representation coefficients lie in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^{n/2}}} , or in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_{2^4}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} divisible by 4; [10]

4. Functions whose univariate representation coefficients satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^{(2^n-1)/3}a_{3i}=0} ; [14]

5. Functions having at least one partially-bent component[6].

References

  1. ↑ 1.0 1.1 Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, On Almost Perfect Nonlinear Functions Over GF(2^n), IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70
  2. ↑ Bartoli, D., Timpanella, M. On a conjecture on APN permutations. Cryptogr. Commun. 14, 925–931 (2022)
  3. ↑ Budaghyan, L., Carlet, C., Leander, G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inf. Theory 54(9), 4218–4229 (2008)
  4. ↑ 4.0 4.1 Browning, K., Dillon, J.F., McQuistan, M., Wolfe, A.J.: An APN permutation in dimension six. In: Post-proceedings of the 9-th International conference on Finite Fields and their applications, American Mathematical Society, vol. 518, pp. 33–42 (2010)
  5. ↑ Beierle, C., Leander, G.: New instances of quadratic APN functions, arXiv:2009.07204 (2020)
  6. ↑ 6.0 6.1 Marco Calderini, Massimiliano Sala, Irene Villa, A note on APN permutations in even dimension, Finite Fields and Their Applications, vol. 46, 1-16, 2017
  7. ↑ H. Dobbertin. Private Communication, 1998.
  8. ↑ Carlet, Claude & Charpin, Pascale & Zinoviev, Victor. (1998). Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems. Designs, Codes and Cryptography, vol. 15, p. 125-156.
  9. ↑ Browning, K.A. & Dillon, J.F. & Kibler, R.E. & McQuistan, M.T.. (2009). APN polynomials and related codes. Journal of Combinatorics, Information & System Sciences. 34.
  10. ↑ 10.0 10.1 X.-D. Hou. Affinity of Permutations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_2^n} . Proceedings of Workshop on Coding and Cryptography WCC 2003, pp. 273-280, 2003. Completed version in Discrete Applied Mathematics 154 (2), pp. 313-325, 2006.
  11. ↑ B. Bilgin, A. Bogdanov, M. Knezevic, F. Mendel and Q. Wang. Fides: lightweight authenticated cipher with side-channel resistance for constrained hardware. Proceedings of International Workshop Cryptographic Hardware and Embedded Systems CHES 2013, Lecture Notes in Computer Science 8086, pp. 142-158, 2013.
  12. ↑ L. Perrin, A. Udovenko, A. Biryukov. Cryptanalysis of a theorem: decomposing the only known solution to the big APN problem. Proceedings of CRYPTO 2016, Lecture Notes in Computer Science 9815, part II, pp. 93-122, 2016.
  13. ↑ L. Perrin, A. Canteaut, S. Tian. If a generalized butterfly is APN then it operates on 6 bits. Special Issue on Boolean Functions and Their Applications 2018, Cryptography and Communications 11 (6), pp. 1147-1164, 2019.
  14. ↑ A. Canteaut. Differential cryptanalysis of Feistel ciphers and differentially uniform mappings. Proceedings of Selected Areas on Cryptography, SAC 1997, pp. 172-184, 1997.
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