In this paper, a novel method is being proposed to construct a substitution box or Boolean function for block ciphers using Gaussian distribution and linear fractional transform. The substitution box is constructed by employing a linear fractional transform based on Box-Muller transform, polarization decision, and central limit algorithm. The cryptographic strength of the proposed S-boxes is evaluated with standardized tests such as linear approximation probability, unified averaged changed intensity, bit independent criterion, histogram analysis, nonlinearity score, strict avalanche criterion, and differential approximation probability. The results show that the proposed substitution box achieves better cryptographic strength as compared with the state-of-the-art techniques.