Consider a probability distribution $D$ over $n$ bit strings. Denote $U$ to be the uniform distribution over $n$ bit strings and $U_{n}$ to be the uniform distribution over integers $\{1, 2, \ldots, n\}$.
Consider the following two equivalent statements (they are equivalent by Yao's theorem):
- There is no uniform polynomial time next bit predictor $A$ such that $$ \underset{X \sim D \\ M \sim U_{n} }{\text{Pr}}[A(X_1X_2.....X_{M-1})=X_M] \geq \frac{1}{2} + \frac{1}{\text{poly}(n)}. $$
- There is no uniform polynomial time distinguisher $B$ such that $$ \mid\underset{X \sim D}{\text{Pr}}[B(X)=1] - \underset{Y \sim U}{\text{Pr}}[B(Y)=1]\mid \geq \frac{1}{\text{poly}(n)}. $$
I was thinking about the centrality of the uniform distribution in the reductions. Will some form of these statements work when we replace the uniform distribution by a known and efficiently samplable distribution $D_2$? Let's say we can also efficiently sample from every marginal of $D_2$.
In other words, consider Statement 3 as follows.
Statement 3: There is no uniform polynomial time distinguisher $B$ such that $$ \mid\underset{X \sim D}{\text{Pr}}[B(X)=1] - \underset{Y \sim D_2}{\text{Pr}}[B(Y)=1]\mid \geq \frac{1}{\text{poly}(n)}. $$
Does this imply and/or is implied by a Statement 4 like the following (or some variant of this statement)?
Statement 4: There is no uniform polynomial time next bit predictor $A$ such that $$ \underset{X \sim D \\ M \sim U_{n} }{\text{Pr}}[A(X_1X_2.....X_{M-1})=X_M] \geq c, $$ where $c$ depends on the distribution $D_2$ and the distinguishing advantage of $B$.
If so, can we have an explicit form for $c$?