PRFs with long outputs and short keys

Weierstrass99

7/13/24, 8:18 AM

Assume I have a PRF $F$, with polynomial key length $s_{F}(n)\geq n$, and output length $l(n)$.

I need to construct a PRF $F'$, with key length $s_{F'}(n)=n$ and output length $l(n)$.

I thought about the following construction: $$F'(k,x)=F_{F(k',0)F(k',1)...F(k',m)}(x)$$ where $|k'| \leq |k|$ is an output of $s$ for some $n$, such that it will be possible key for $F'$, even if means discarding of some $k$'s random bits.

and where $m$ is such that $\sum_i|F(k',i)| \geq s_F(n)$.

I'm trying to use the given to me PRF, The intution is that I'm starting with "fixing" the key such that it'll be of a legitimate size, to be used with $F$. Then, I'm using a known construction of a PRG that generates a pseudorandom string of length $s_F(n)$, and using it as a key for $F$ .

Thanks!

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pseudo-random-generator

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EN: PRFs with long outputs and short keys

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