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

CaulCaul

7/24/23, 6:01 AM

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.

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one-way-function

hard-core-predicate

Maeher

7/24/23, 10:50 PM

Hint: For any fixed output, some input bits must be hard to predict. But each bit can be easy to predict on *average* over random inputs.

Score:1

Crypto

CaulCaul

7/25/23, 6:21 AM

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.

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Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: 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

TH: สมมติว่ามีฟังก์ชันทางเดียวอยู่ แสดงว่ามีฟังก์ชันทางเดียวอยู่โดยไม่มีบิตอินพุตที่เป็นฮาร์ดคอร์บิต

RO: Să presupunem că există o funcție unidirecțională, arătați că există o funcție unidirecțională cu niciunul dintre biții de intrare care nu este un bit hardcore

RU: Предположим, что существует односторонняя функция. Покажите, что существует односторонняя функция, ни один из входных битов которой не является хардкорным.

VI: Giả sử tồn tại hàm một chiều, hãy chỉ ra rằng tồn tại hàm một chiều mà không có bit đầu vào nào là bit khó

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