In a recent paper [W. Zhang and E. Pasalic,Constructions of resilient S-Boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans Inf Theory 60, no. 3 (2014), pp. 1638–1651], by using disjoint linear codes, Zhang and Pasalic presented a method for constructingt-resilient S-boxes(even,with strictly almost optimal (currently best) nonlinearity exceeding the value. It was also shown that the algebraic degree and algebraic immunity of these resilient S-boxes are very good, but the resistance of these resilient S-boxes against fast algebraic attacks has not been treated in [W. Zhang and E. Pasalic,Constructions of resilient S-Boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans. Inf. Theory 60, no. 3 (2014), pp. 1638–1651]. In this work, we extend the method originally proposed in [E. Pasalic,Maiorana-McFarland class: Degree optimization and algebraic properties, IEEE Trans. Inf. Theory 52, no. 10 (2006), pp. 4581–4595] and used in deriving the upper bound on algebraic immunity of the Maiorana–McFarland class, for establishing the existence of low degree multiplier for the class of S-boxes that uses disjoint linear codes in the design. It is demonstrated that this class of functions has a substantial weakness against fast algebraic cryptanalysis. An alternative approach, based on the use of the associated dual codes is also developed. [ABSTRACT FROM AUTHOR]