# Where does the 8 come from? Generic Search Problem with Bounded Probabilities

I am working with lossy ID-schemes and their security in the QROM. Following the article of Kiltz et al. , I am at a loss of the number 8 appearing in most reductions throughout the article. I know it comes from the Generic Search Problem for Bounded Probabilites, however how?

The Lemma from the article as wee as the game for a quantum adversary is:

With the following proof in the appendix:

Any and all help would be greatly appreciated. I have been looking at line 02 for the $$GSPB_{\lambda}$$ game and wondering if it could have anything to do with the probability of $$\lambda(x)$$ being greater than $$\lambda$$ ?

I have also followed the proof to the original source, but nor here do I seem to find anything explaining the 8...

My main goal for looking at this was finding the bound for a classical adversary $$\mathcal{B}$$ against the GSPB. I would believe this to be simply $$Pr[GSPB^{\mathcal{B}}_{\lambda} \Rightarrow 1] = \lambda \cdot Q$$. Is this intuition right?

Theorem 3.2. Fix $$q$$, and let $$D_\lambda$$ be a family of distributions on $$H_{X,Y}$$ indexed by $$\lambda\in[0,1]$$. Suppose there are integers $$d$$ and $$\Delta$$ such that for every $$2q$$ pairs $$(x_i, r_i)\in X\times Y$$, the function $$p(\lambda) = \Pr_{H\gets D_\lambda}[H(x_i) = r_i\forall i\in\{1,\dots,2q\}]$$ satisfies:
• $$p$$ is a polynomial in $$\lambda$$ of degree at most $$d$$, and
• $$p^{(i)}(0)$$, the $$i$$th derivative of $$p$$ at 0, is 0 for each $$i\in\{1,\dots, \Delta-1\}$$.
Then any quantum algorithm $$A$$ making $$q$$ quantum queries can only distinguish $$D_\lambda$$ from $$D_0$$ with probability at most $$\frac{4^\Delta}{(2\Delta)!}\lambda^\Delta d^{2\Delta}.$$
It appears your paper later applies this with $$\Delta = 1$$ and $$d = 2(Q+1)$$, giving a bound of $$\frac{4}{2}\lambda (2(Q+1))^2 = 8\lambda (Q+1)^2$$.