An Improvement of Correlation Analysis for Vectorial Boolean Functions

Abstract

This paper investigates the correlation of n-bit to m-bit vectorial Boolean functions denoted by F. At Crypto 2000, Zhang and Chan showed that the maximum of linear approximations for F with Boolean functions g have a higher bias than those based on the usual correlation attack. The correlation for this linear approximation has been named the maximum correlation and has been shown to be a useful tool for correlation attack resistance. In this work, we deal with two issues. Firstly, we show that combining F with any g does not always increase the bias as stated by several works. To justify such results, we demonstrate the exact correlation link between F, g and the combination of F by g. Secondly, we provide the exact condition in which the correlation coefficients for this approximation are maximum.

Appendices

A Appendix-A

See Tables 4 and 5

• ${\varGamma _1}$

Table 4. Hexadecimal Representation for $\varGamma _1$

• $\varGamma _2$

Table 5. Hexadecimal Representation for $\varGamma _2$

B Appendix-B

As Theorem 2 is linked to $\max \limits _b(c_F(a,b)\pm c_g(b))^2$, we fix $\max \limits _bc_F(a,b)$ and we vary $c_g(b)$. The y-axis indicates $c_{g\circ F}(a)$ and the x-axis indicates $\max \limits _bc_g(b)$. By computing the white area surface ($|\varepsilon _{F}|\le |\varepsilon _{g\circ F}|$), the probability $Pr_g$ is determined as the ratio of white area surface over the rectangle area surface (Fig. 2).

Fig. 2.

Computing Prg methods where $\varphi$ denotes the fixed $\max \limits _bc_F(a,b)$, i.e. $\max \limits _b (c_F(a,b)\pm c_g(b))^2=\max \limits _b(\varphi \pm c_g(b))^2$