Almost Perfect Nonlinear (APN) Functions: Difference between revisions

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Similar techniques can be used to characterize permutations and APN functions with plateaued components.
Similar techniques can be used to characterize permutations and APN functions with plateaued components.
= Characterization of Plateaued Functions =
== Characterization by the Derivatives <ref name="carletPlateaued">Carlet C. Boolean and vectorial plateaued functions and APN functions. IEEE Transactions on Information Theory. 2015 Nov;61(11):6272-89.</ref> ==
=== First characterization ===
Using the fact that two integer-valued functions over <math>\mathbb{F}_2^m</math> are equal precisely when their Fourier transforms are equal, one can obtain the following characterization.
Let <math>F</math> be an <math>(n,m)</math>-function. Then:
- <math>F</math> is plateaued if and only if, for every <math>v \in \mathbb{F}_2^m</math>, the size of the set
<div><math> \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \}</math></div>
does not depend on <math>x</math>;
- <math>F</math> is plateaued with single amplitude if and only if the size of the above set depends neither on <math>x</math> nor on <math>v</math> when <math>v \ne 0</math>.
Moreover:
- for any <math>(n,m)</math>-function <math>F</math>, the value distribution of <math>D_aD_bF(x)</math> equals the value distribution of <math>D_aF(b) + D_aF(x)</math> as <math>(a,b)</math> ranges over <math>(\mathbb{F}_2^n)^2</math>;
- if two plateuaed functions have the same distribution, then for every <math>u</math>, their component functions at <math>u</math> have the same amplitude.
=== Characterization in the Case of Unbalanced Components ===
Let <math>F</math> be an <math>(n,m)</math>-function. Then <math>F</math> is plateuaed with all components unbalanced if and only if, for every <math>v,x \in \mathbb{F}_2^n</math>, we have
<div><math> | \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \} | = | \{ (a,b) \in (\mathbb{F}_2^n)^2 : F(a) + F(b) = v \} |.</math></div>
Moreover, <math>F</math> is plateuaed with single amplitude if and only if, in addition, this value does not depend on <math>v</math> for <math>v \ne 0</math>.

Revision as of 13:21, 7 February 2019

Background and definition

Almost perfect nonlinear (APN) functions are the class of Vectorial Boolean Functions that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function is efficient when fixing some difference and computing the output of for all pairs of inputs whose difference is produces output pairs with a difference distribution that is far away from uniform.

The formal definition of an APN function is usually given through the values

which, 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 a \ne 0} , express the number of input pairs with difference 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} that map to a given 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 b} . The existence of a pair 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,b) \in \mathbb{F}_{2^n}^* \times \mathbb{F}_{2^n}} with a high value 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 \Delta_F(a,b)} makes the 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} vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity 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 \Delta_F = \max \{ \Delta_F(a,b) : a \in \mathbb{F}_{2^n}^*, b \in \mathbb{F}_{2^n} \}}

which clearly satisfies 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 \Delta_F \ge 2} for any 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} . The functions meeting this lower bound are called almost perfect nonlinear (APN).

Characterizations

Walsh transform[1]

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 (n,m)} -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} satisfies

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}, b \in \mathbb{F}_{2^m}^*} W_F^4(a,b) \ge 2^{2n}(3 \cdot 2^{n+m} - 2^{m+1} - 2^{2n})}

with equality characterizing APN functions.

In particular, 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,n)} -functions 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 \sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^n}^*} W_F^4(a,b) \ge 2^{3n+1}(2^n-1)}

with equality characterizing APN functions.

Sometimes, it is more convenient to sum through all 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 b \in \mathbb{F}_{2^m}} instead of just the nonzero ones. In this case, the inequality 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,m)} -functions becomes

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}, b \in \mathbb{F}_{2^m}} W_F^4(a,b) \ge 2^{2n + m}(3 \cdot 2^n - 2)}

and the particular case 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,n)} -functions becomes

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,b \in \mathbb{F}_{2^n}} W_F^4(a,b) \ge 2^{3n+1}(3 \cdot 2^{n-1} - 1)}

with equality characterizing APN functions in both cases.

Autocorrelation functions of the directional derivatives [2]

Given a Boolean 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 : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_2} , the autocorrelation function 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 f} is defined 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 \mathcal{F}(f) = \sum_{x \in \mathbb{F}_{2^n}} (-1)^{f(x)} = 2^n - 2wt(f).}

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 (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} satisfies

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^{2n+1}}

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}^*} . Equality occurs 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 F} is APN.

This allows APN functions to be characterized in terms of the sum-of-square-indicator 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 \nu(f)} defined 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 \nu(f) = \sum_{a \in \mathbb{F}_{2^n}} \mathcal{F}^2(D_aF) = 2^{-n} \sum_{a \in \mathbb{F}_{2^n}} \mathcal{F}^4(f + \varphi_a)}

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 \varphi_a(x) = {\rm Tr}(ax)} .

Then 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 (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} satisfies

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}^*} \nu(f_\lambda) \ge (2^n-1)2^{2n+1}}

and equality occurs 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 F} is APN.

Similar techniques can be used to characterize permutations and APN functions with plateaued components.

Characterization of Plateaued Functions

Characterization by the Derivatives [3]

First characterization

Using the fact that two integer-valued functions 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^m} are equal precisely when their Fourier transforms are equal, one can obtain the following characterization.

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} be 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,m)} -function. 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 plateaued if and only if, for every 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 v \in \mathbb{F}_2^m} , the size of the set

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,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \}}

does not depend 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 x} ;

- 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 plateaued with single amplitude if and only if the size of the above set depends neither 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 x} nor 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 v} 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 v \ne 0} .

Moreover:

- 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 (n,m)} -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} , the value distribution 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 D_aD_bF(x)} equals the value distribution 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 D_aF(b) + D_aF(x)} 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 (a,b)} ranges 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)^2} ;

- if two plateuaed functions have the same distribution, then for every 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} , their component functions at 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} have the same amplitude.

Characterization in the Case of Unbalanced Components

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} be 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,m)} -function. 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 plateuaed with all components unbalanced if and only if, for every 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 v,x \in \mathbb{F}_2^n} , 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 | \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \} | = | \{ (a,b) \in (\mathbb{F}_2^n)^2 : F(a) + F(b) = v \} |.}

Moreover, 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 plateuaed with single amplitude if and only if, in addition, this value does not depend 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 v} 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 v \ne 0} .

  1. ↑ Florent Chabaud, Serge Vaudenay, Links between differential and linear cryptanalysis, Workshop on the Theory and Application of Cryptographic Techniques, 1994 May 9, pp. 356-365, Springer, Berlin, Heidelberg
  2. ↑ 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
  3. ↑ Carlet C. Boolean and vectorial plateaued functions and APN functions. IEEE Transactions on Information Theory. 2015 Nov;61(11):6272-89.
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