In LFSR-based stream ciphers, the knowledge of the feedback equation of the LFSR plays a critical role in most attacks. In word-based stream ciphers such as those in the SNOW series, even if the feedback configuration is hidden, knowing the characteristic polynomial of the state transition matrix of the LFSR enables the attacker to create a feedback equation over $GF(2)$. This, in turn, can be used to launch fast correlation attacks. In this work, we propose a method for hiding both the feedback equation of a word-based LFSR and the characteristic polynomial of the state transition matrix. Here, we employ a $z$-primitive $\sigma$-LFSR whose characteristic polynomial is randomly sampled from the distribution of primitive polynomials over $GF(2)$ of the appropriate degree. We propose an algorithm for locating $z$-primitive $\sigma$-LFSR configurations of a given degree. Further, an invertible matrix is generated from the key. This is then employed to generate a public parameter which is used to retrieve the feedback configuration using the key. If the key size is $n$- bits, the process of retrieving the feedback equation from the public parameter has a average time complexity $\mathbb{O}(2^{n-1})$. The proposed method has been tested on SNOW 2.0 and SNOW 3G for resistance to fast correlation attacks. We have demonstrated that the security of SNOW 2.0 and SNOW 3G increases from 128 bits to 256 bits.

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