Meaning of $\operatorname{UOWH}(k; u_1, u_2)$ I do not understand the format used in Shoup's "Using Hash Functions as a Hedge against Chosen Ciphertext Attack (page 8 of the PDF) for universal one way hash functions (UOWH). What are $k$, $u_1$ and $u_2$ in $\operatorname{UOWH}(k; u_1, u_2)$?
If you look up the reference [BR97] in the paper, it clearly states that $k$ is the key which is used to select a hash function from a family of hash functions and the rest of the part of the input i.e. $u_1, u_2, u_3 ...$ is the input to the selected hash such that $u_i$ is the input to $UOWH(k;u_1,u_2,...,u_i-1)$