# Is the concatenation of two one-way functions a one-way function when each function takes different inputs?

Similar to this question, but having two seperate inputs for each length preserving one way function $$f$$ and $$g$$, i.e. $$h: \lbrace 0,1 \rbrace^{2\kappa} \to \lbrace 0,1 \rbrace^{2\kappa}, h(x) = f(x_1)||g(x_2)$$ where $$x_1$$ and $$x_2$$ are two $$\kappa$$ bit split halves of x.

I think $$h$$ will be one way, but am not sure on the appropriate reduction to demonstrate this.

I think I don't need to show that the probability of an attacker decrypting this being negligible in polynomial time but rather reduce to problems of one-wayness of $$f$$ and $$g$$ and knowing that they are one way should demonstrate that in fact $$h$$ is one way

• This seems like a good candidate for a proof by contradiction. Assume you were able to invert $h(x_1 || x_2)$ (with non-negligible probability). Are you then able to leverage this ability to invert either $f(x_1)$ or $g(x_2)$ (with non-negligible probability)? Commented Mar 21, 2022 at 8:16