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Why we need DSPR or Eq. 14 for the proof of SPHNICS+?

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The paper says that

We show that preimage resistance (PRE) follows tightly from the conjunction of second-preimage resistance (SPR) and decisional second-preimage resistance (DSPR).

As I understand from here instead of assuming the hash function SPR and PRE we can assume DSPR and PRE to have secure scheme.

And the original proof of SPHNICS+ we need to assume PRE and Eq 14 to have SPR property.

Eq. 14: $(\forall k\in \{0,1\}^n)(\forall y\in \operatorname{IMG}(F_k)) ( \exists x,x'\in \{0,1\}^n): x\neq x' \land F_k(x)=F_k(x')$.

Why we need these, why we can't just assume SPR and PRE. Even, Eq.14 is very extreme assumption. Why the proof of SPHINCS+ assumed Eq.14 instead of SPR.

mangohost

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