Let $\mathscr{H}$ be a family of hash functions, each with domain $U$ and range $\{0, 1, ..., m-1\}$, and let $h$ be any hash function that is picked uniformly at random from $\mathscr{H}$. The probabilities mentioned are probabilities over the picks of $h$.

• The family $\mathscr{H}$ is uniform (равномерное) if for any $key \in U$ and any slot $q \in \{0, 1, ..., m-1\}$, the probability that $h(k)=q$ is $\frac{1}{m}$.
• The family $\mathscr{H}$ is universal (универсальное) if for any distinct keys $key1, \space key2 \in U$ the probability that $h(key1)=h(key2)$ is at most $\frac{1}{m}$.
• The family $\mathscr{H}$ is $\bold\epsilon$-universal if for any distinct keys $key_1, \space key_2 \in U$ the probability that $h(key1)=h(key2)$ is at most $\epsilon$.
• The family $\mathscr{H}$ is d-independent if for any distinct keys $key_1, \space key_2, ..., key_d \in U$ and any slots $q_1, q_2, ..., q_d \in \{0, 1, ..., m-1\}$, not necessarily distinct, the probability that $h(key_i)=q_i$ for $i=1, 2, ..., d$ is $\frac{1}{m^d}$.