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Pseudo-random number generators (PRNGs) are a key component of stream ciphers used for encryption purposes. While linear feedback shift registers (LFSRs) combined with nonlinear feedback shift registers (NFSRs) have typically been utilized for PRNGs, the use of cellular automata (CA) is another viable option. This paper explores the combination of LFSRs and CA as the key components of an efficient stream cipher design for implementation on Field Programmable Gate Arrays (FPGAs). The proposed stream cipher design builds upon a recent published design known as A2U2. The suite of statistical tests found in the Diehard program is utilized to evaluate the quality of random numbers from the proposed CA-based stream cipher. Comparisons with the A2U2 design indicate that the use of CA have the potential to improve the quality of the random numbers generated and hence increase the security of the cipher.