XOR of a secure PRF is modified weakly secure PRF

seanL

10/22/23, 6:17 PM

While reading A Graduate Course in Applied Cryptography by Dan Boneh and Victor Shoup. There was the next exercise (Ex. 4.2 (b)), let $F$ be a secure PRF over $(K,X,Y)$ where $Y := \{0,1\}^n$ and $|X|$ is super-poly. Define $F_1(k, (x,y)) := F(k,x) \oplus F(k,y)$, prove that $F_1$ is weakly secure even if we modify the weak PRF attack game and allow the adversary A to query $F_1$ at one chosen point in addition to Q random points.

I don't understand why the next adversary doesn't prove that $F_1$ is not weakly secure, $A$ gets a function $f$ and he need to find if $f = F_1$ or $f$ is a random function. As $A$ may choose one point he choose $(x,x)$ to get $f((x,x))$. If $f((x,x))= 0$ $A$ guess $f= F_1$ else he guess $f$ is a random function.

$mwRPFadv(A, F_1) = |1 - \frac{1}{|X|}|$, thus $F_1$ is not secure in the modified weak sense.

177

0 + 1

provable-security

pseudo-random-function

Maeher

10/23/23, 7:10 AM

I agree with you that, as stated, the exercise is wrong. It may be that they meant that the adversary gets to choose only $x$ and gets results for random $y$? But that's not made clear.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: XOR of a secure PRF is modified weakly secure PRF

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