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I'm also afraid you couldn't understand this as D.W., but let us start. I sometimes cannot understand your questions. Please restate them, if possible. The definition of the Ajtai hash functions Let $n$, $m$, and $q$ be positive integers. Let $R = \mathbb{Z}_q$ be the quotient ring of integers modulo $q$. Let us define a function, which maps a vector in ...