We investigate shift-invariant vectorial Boolean functions on $n$~bits that are lifted from Boolean functions on $k$~bits, for $k\leq n$. We consider vectorial functions that are not necessarily permutations, but are, in some sense, almost bijective. In this context, we define an almost lifting as a Boolean function for which there is an upper bound on the number of collisions of its lifted functions that does not depend on $n$. We show that if a Boolean function with diameter $k$ is an almost lifting, then the maximum number of collisions of its lifted functions is $2^{k-1}$ for any $n$. Moreover, we search for functions in the class of almost liftings that have good cryptographic properties and for which the non-bijectivity does not cause major security weaknesses. These functions generalize the well-known map $\chi$ used in the Keccak hash function.

翻译：我们研究了从$k$位布尔函数提升而来的$n$位移位不变向量布尔函数（其中$k\leq n$）。我们考察的向量函数不一定是置换函数，但在某种意义上是近似双射的。在此背景下，我们将近似提升定义为一种布尔函数，其提升函数的碰撞次数存在不依赖于$n$的上界。我们证明：若直径为$k$的布尔函数是近似提升函数，则对于任意$n$，其提升函数的最大碰撞次数为$2^{k-1}$。此外，我们在近似提升函数类中寻找具有良好密码学特性且非双射性不会导致重大安全弱点的函数。这类函数推广了Keccak哈希函数中使用的著名映射$\chi$。