Suppose there exists a one-way function, show that there exists a one-way function with none of its input bit is a hardcore bit

Question

I just learnt the definition of hardcore bit, and I have no intuition about this. I want to know what are the possible approaches to this problem.

Answer 1

I get a good idea form my classmate.

Suppose that $f:\{0,1\}^n\to\{0,1\}^m$ is a one-way function.

We construct another one-way function $F:\{0,1\}^{n+1+\log(n+1)}\to\{0,1\}^{m+\log(n+1)+1}$ : For an input $x=x_0x_1\cdots x_{n-1}x_n||s$, where $s\in\{0,1,2,\cdots,n\}$ is a $\log(n+1)$-bit string, we have $$ F(x)=f(x_0x_1\cdots x_{s-1}x_{s+1}\cdots x_n)||s||x_s $$ Then we can prove that $F$ is a a one-way function with none of its input bit is a hardcore bit.