Is the concatenation of two one-way functions a one-way function?

Question

Suppose we are given two one way functions $f$ and $g$. We define a new function h that is the concatenation of f and g. That is, $h(x)=f(x), g(x)$, where the comma indicates concatenation. We want to figure out whether this is a one way function or not. My teacher tells me this is not true generally. I know it is true if $f=g$, but otherwise I am clueless. I'd love some help.

Answer 1

But is h invertible more or less often? That is how I was asked the question.

When mathematicians say "This statement is not true generally" (or similar wordings) they mean "There exist cases for which the statement is false". So your teacher is talking about the statement "For all $f$ and $g$ which are one-way functions $h(x)=f(x),g(x)$ is also a one-way function" and says that this general statement is false, even if it might be true for some choices of $f$ and $g$.

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Looking at this specific statement:

• There are choices where it's true. You already found the trivial $f=g$.
• There are choices where it's false. When $g(x)=f(x)\oplus x$, $h$ is never a one-way function, since you can recover $x=f(x) \oplus g(x)$ from $h(x)$ (technically you still have to show that there exist $f$ and $g$ of with this relation that are both one-way functions).
• In practical terms, if you don't choose $f$ and $g$ deliberately to produce a weak $h$, $h$ will normally also be an OWF, but mathematicians avoid such vague and difficult to formalize claims.