# OWF by encrypting a constant?

This question is a generalization of this old, unanswered question.

Suppose we're given a strong PRP $E:\mathcal K\times\mathcal M\to\mathcal C,(K,M)\mapsto C=E(K,M)$. Suppose further we pick a constant parameter $S\in\mathcal M$.

Is $F_S:\mathcal K\to \mathcal C, X\mapsto E(X,S)$ a secure one-way-function?
If the above is not answerable: Is $F_S$ at least a secure OWF for practical purposes?

As example instances one can imagine $S=0^{128},E=\operatorname{AES}$ or $E$ being a fixed-key AES instance within an Even-Mansour construction.

• FWIW some X9 committee thought the variant of known and predictable but not constant data was safe when designing DUKPT decades ago; it uses K_(n+b) = DES(K_n, n+b) plus a little whitening as a claimed-irreversible 'transform' to derive a sequence of keys K_i that are intended to be forward-secure. Dec 21 '17 at 4:15

Since $E$ is a secure PRP, instead of giving $C=E(X, S)$ to the adversary, you can give a uniform random $C\gets\mathcal{C}$. To 'invert' this, adversary has to find a key $X'$ such that $E(X', S)=C$.
So, the construction being one-way function is equivalent to the property that, given a uniform random $C$, it should be hard to find $X$ such that $E(X, S)=C$. I don't think that this necessarily follows from $E$ being secure, strong PRP. For example, when the block cipher has polynomial-sized small domain (rather unusual case, I know), then this would not hold. On the other hand, most reasonable block cipher constructions would have this property, I guess. So, I think it could be considered as a secure OWF for practical purposes, in reasonable settings.
I will take ‘one-way function’ to mean preimage-resistant in the concrete setting, where for a preimage-finding algorithm $A$ we have $$\operatorname{Adv}_H^{\mathrm{pre}}(A) = \Pr[H_\mathit{salt}(A(\sigma)) = \sigma],$$ where $\mathit{salt}$ and $\sigma$ are uniform random bit strings, and presumably we hope $\operatorname{Adv}_H^{\mathrm{pre}}(A)$ to be bounded by $C(q)/2^n$ if $A$ makes $q$ queries with some plausibly small $C(q)$ when the output of $H$ has $n$ bits.
PRP security of $E$ does not imply preimage resistance of $H_\mathit{salt}\colon X \mapsto E_X(\mathit{salt})$, because there may be arbitrarily much structure known to the attacker about the input $X$, which violates the assumptions of PRP security. We have no reason to doubt the PRP security of AES-256, but it does not provide this preimage resistance.
An ideal cipher $E$ would imply preimage resistance of this construction, as shown by Black, Rogaway, and Shrimpton as case $j=17$ with $v = 0$, with the adversary's best probability in the interval $[0.15q^2/2^n, 9(q + 3)^2/2^n]$. But you need to find a better cipher than AES-256 for this to work out.