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In the BPS specification document it says

We denote by f the number of output bits of the internal function F

What is this if the internal function is AES? Would it be the block size (e.g. 128)?

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Yes, in BPS, if the internal function $F$ is a block cipher, then $f$ is its block width, thus 128 for AES (for all key width variants). That's clear enough in the paragraph quoted in the question:

We denote by $f$ the number of output bits of the internal function $F$. (...) If $F$ is a $f$-bit block cipher (...) We denote by $F_K(x)$ the application of the block cipher $E$ with the key $K$ on the plaintext $x$ ($F_K(x)=E_K(x)$), (...)

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