Generalized Correlation Analysis of Vectorial Boolean Functions

We investigate the security of

n

-bit to

m

-bit vectorial Boolean functions in stream ciphers. Such stream ciphers have higher throughput than those using single-bit output Boolean functions. However, as shown by Zhang and Chan at Crypto 2000, linear approximations based on composing the vector output with any Boolean functions have higher bias than those based on the usual correlation attack. In this paper, we introduce a new approach for analyzing vector Boolean functions called generalized correlation analysis. It is based on approximate equations which are linear in the input

x

but of free degree in the output

z

 = 

F

(

x

). Based on experimental results, we observe that the new generalized correlation attack gives linear approximation with much higher bias than the Zhang-Chan and usual correlation attacks. Thus it can be more effective than previous methods.

First, the complexity for computing the generalized nonlinearity for this new attack is reduced from

$2^{2^m \times n+n}$

to 2

2

n

. Second, we prove a theoretical upper bound for generalized nonlinearity which is much lower than the unrestricted nonlinearity (for Zhang-Chan’s attack) or usual nonlinearity. This again proves that generalized correlation attack performs better than previous correlation attacks. Third, we introduce a generalized divide-and-conquer correlation attack and prove that the usual notion of resiliency is enough to protect against it. Finally, we deduce the generalized nonlinearity of some known secondary constructions for secure vector Boolean functions.