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In several cryptographic primitives, Sboxes of small size are used to provide nonlinearity. After several iterations, all the output bits of the primitive are ideally supposed to depend in a nonlinear way on all of the input variables. However, in some cases, it is possible to find some output bits that depend in an affine way on a small number of input bits if the other input bits are fixed to a well-chosen value. Such situations are for example exploited in cube attacks or in attacks like the one presented by Fuhr against the hash function Hamsi. Here, we define a new property for nonlinear Sboxes, named \((v,w)\)-linearity, which means that \(2^w\) components of an Sbox are affine on all cosets of a \(v\)-dimensional subspace. This property is related to the generalization of the so-called Maiorana-McFarland construction for Boolean functions. We show that this concept quantifies the ability of an Sbox to propagate affine relations. As a proof of concept, we exploit this new notion for analyzing and slightly improving Fuhr’s attack against Hamsi and we show that its success strongly depends on the \((v,w)\)-linearity of the involved Sbox.

A New Criterion for Avoiding the Propagation of Linear Relations Through an Sbox