Nonlinearity: Difference between revisions

From Boolean
(Created page with "= Background and Definition = Vectorial Boolean Functions play an essential role in the design of cryptographic algorithms, and as such should be resistant to various typ...")
 
No edit summary
Line 20: Line 20:
= Bounds on the Nonlinearity of Vectorial Boolean Functions =
= Bounds on the Nonlinearity of Vectorial Boolean Functions =


The covering radius bound for Boolean functions can naturally be extended to vectorial Boolean functions, stating
The ''covering radius bound'' for Boolean functions can naturally be extended to vectorial Boolean functions, stating


<div><math>nl(F) \le 2^{n-1} - 2^{n/2 - 1}</math></div>
<div><math>nl(F) \le 2^{n-1} - 2^{n/2 - 1}</math></div>


for any <math>(n,m)</math>-function <math>F</math>.
for any <math>(n,m)</math>-function <math>F</math>. [[Bent Functions]] are defined as those meeting this bound with equality.
 
The ''Sidelnikov-Chabaud-Vaudenay (SCV) bound'' <ref>Sidel'nikov VM. On mutual correlation of sequences. InDoklady Akademii Nauk 1971 (Vol. 196, No. 3, pp. 531-534). Russian Academy of Sciences.</ref> <ref>Chabaud F, Vaudenay S. Links between differential and linear cryptanalysis. Workshop on the Theory and Application of Cryptographic Techniques 1994 May 9 (pp. 356-365). Springer, Berlin, Heidelberg.</ref> bounds the nonlinearity of any <math>(n,m)</math>-function, with <math>m \ge n-1</math>, by
 
<div><math> nl(F) \le 2^{n-1} - \frac{1}{2} \sqrt{3 \cdot 2^n - 2 - 2 \frac{(2^n-1)(2^{n-1}-1)}{2^m-1}}.</math></div>
 
The SCV bound coincides with the covering radius bound for <math>m = n - 1</math>, and is strictly sharper than the covering radius bound for <math>m \ge n</math>. For <math>m > n</math>, the square root in the bound cannot be an integer, and thus the SCV bound can be, and is, tight only for <math>m=n</math>. This motivates the definition of [[Almost Bent Functions]] as those <math>(n,n)</math>-functions that meet the SCV bound with equality.

Revision as of 18:17, 10 February 2019

Background and Definition

Vectorial Boolean Functions play an essential role in the design of cryptographic algorithms, and as such should be resistant to various types of cryptanalytic attacks. The notion of nonlinearity is introduced by Nyberg [1] in order to measure the resistance of vectorial Boolean functions to Matsui's linear attack [2]. This attack attempts to approximate the function used in an encryption algorithm by a linear function (which, in turn, is easy to analyze), and is, therefore, applicable when the actual functions used in the encryption algorithm is "close" to linear in some sense. A natural measure of distance between two functions, and , is the Hamming distance, i.e. the metric

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{H}(F,G)=|\{x:F(x)\neq G(x)\}|.}

Formally, the nonlinearity of an -function is the minimum distance between any component function of and any affine Boolean function. In other words,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle nl(F)=\min\{d_{H}(u\cdot F,f_{a}):0\neq u\in \mathbb {F} _{2}^{m},f_{a}:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F} {\text{ affine }}\}.}

Properties

Nonlinearity remains invariant under CCZ-equivalence (and, therefore, under extended affine and affine equivalence as well). If is -permutation, then and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F^{-1}</mah>havethesamenonlinearity.Thenonlinearityofan<math>(n,m)} -function can be expressed in terms of its Walsh transform via the identity

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle nl(F)=2^{n-1}-{\frac {1}{2}}\max\{|W_{F}(u,v)|:u\in \mathbb {F} _{2}^{n},0\neq v\in \mathbb {F} _{2}^{m}\}.}

There is a relation [3] between the maximal possible nonlinearity of vectorial Boolean functions and the possible parameters of certain linear codes. If is a linear Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [2^{n},K,D]} containing the Reed-Muller code Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle RM(1,n)} as a subcode, let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (b_{1},b_{2},\ldots ,b_{K})} be a basis of completing a basis Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (b_{1},b_{2},\ldots ,B_{n+1})} of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle RM(1,n)} . Then the -variable Boolean functions corresponding to the vectors are the coordinate functions of an Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (n,K-n-1)} functions with nonlinearity . Conversely, given an function of nonlinearity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D>0} , the linear code obtained as the union of all cosets Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{v\cdot F+RM(1,n):v\in \mathbb {F} _{2}^{m}\}} has parameters 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,n+m+1,D]} .

Bounds on the Nonlinearity of Vectorial Boolean Functions

The covering radius bound for Boolean functions can naturally be extended to vectorial Boolean functions, stating

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 nl(F) \le 2^{n-1} - 2^{n/2 - 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 (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} . Bent Functions are defined as those meeting this bound with equality.

The Sidelnikov-Chabaud-Vaudenay (SCV) bound [4] [5] bounds the nonlinearity of 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, 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 m \ge n-1} , 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 nl(F) \le 2^{n-1} - \frac{1}{2} \sqrt{3 \cdot 2^n - 2 - 2 \frac{(2^n-1)(2^{n-1}-1)}{2^m-1}}.}

The SCV bound coincides with the covering radius bound 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 m = n - 1} , and is strictly sharper than the covering radius bound 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 m \ge n} . 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 m > n} , the square root in the bound cannot be an integer, and thus the SCV bound can be, and is, tight only 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 m=n} . This motivates the definition of Almost Bent Functions as those 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 that meet the SCV bound with equality.

  1. ↑ Nyberg K. On the construction of highly nonlinear permutations. Workshop on the Theory and Application of Cryptographic Techniques 1992 May 24 (pp. 92-98). Springer, Berlin, Heidelberg.
  2. ↑ Matsui M. Linear cryptanalysis method for DES cipher. Workshop on the Theory and Application of Cryptographic Techniques 1993 May 23 (pp. 386-397). Springer, Berlin, Heidelberg.
  3. ↑ Carlet C, Charpin P, Zinoviev V. Codes, bent functions and permutations suitable for DES-like cryptosystems. Designs, Codes and Cryptography. 1998 Nov 1;15(2):125-56.
  4. ↑ Sidel'nikov VM. On mutual correlation of sequences. InDoklady Akademii Nauk 1971 (Vol. 196, No. 3, pp. 531-534). Russian Academy of Sciences.
  5. ↑ Chabaud F, Vaudenay S. Links between differential and linear cryptanalysis. Workshop on the Theory and Application of Cryptographic Techniques 1994 May 9 (pp. 356-365). Springer, Berlin, Heidelberg.
Retrieved from "http://boolean.wiki.uib.no/index.php?title=Nonlinearity&oldid=178"