On the Resistance of Prime-variable Rotation Symmetric Boolean Functions against Fast Algebraic Attacks

Boolean functions used in stream ciphers should have many cryptographic properties in order to help resist different kinds of cryptanalytic attacks. The resistance of Boolean functions against fast algebraic attacks is an important cryptographic property. Deciding the resistance of an n-variable Boolean function against fast algebraic attacks needs to determine the rank of a square matrix of order ∑e i=0 (n i ) over binary field F2, where 1 6 e < dn2 e. In this paper, for rotation symmetric Boolean functions in prime n variables, exploiting the properties of partitioned matrices and circulant matrices, we show that the rank of such a matrix can be obtained by determining the rank of a reduced square matrix of order ( ∑e i=0 (n i ) )/n over F2, so that the computational complexity decreases by a factor of nω for large n, where ω ≈ 2.38 is known as the exponent of the problem of computing the rank of matrices.