A function $H(x)$ is given. If there is an algorithm $B(H(x))$ that get part of $x$, is $H(x)$ a one-way function? I came up with this question while I was reading this paper: Pilaram, Hossein, and Taraneh Eghlidos. &quot;An efficient lattice based multi-stage secret sharing scheme.&quot; IEEE Transactions on Dependable and Secure Computing 14.1 (2015): 2-8.
The theorem 1 in section 4, the author says this:
Therefore, on input $Ex$, the algorithm $\mathcal{B}$ outputs the first part of $x$, i.e. $x_1$, which contradicts with one-wayness of Ajtai's function.
I wonder if this is a correct way to prove a function satisfies one-wayness. I am a noob in cryptography, and English is not my first language. Deeply sorry for the trouble. Thanks for your time :P
If there is an algorithm $B(H(x))$ that get part of $x$, is $H(x)$ a one-way function?

That remains possible.
Simplifying the definition of a One-Way Function in Katz and Lindel's Introduction to Modern Cryptography, it's an efficiently computable function $f$ such that no algorithm $\mathcal A$ exists that, given as input $y=f(x)$ for a random $x$, outputs $x'$ with $f(x')=y$ with non-vanishing probability.
That definition is compatible with existence of an algorithm $\mathcal B$ that, given as input $y=f(x)$ for a random $x$, always outputs some segment of $x$ (e.g. it's first bit).
Illustration: assume a function $h$ that's indistinguishable from a random oracle (SHA-3 aspires to that). It's demonstrably a OWF. Construct a function $f$ with $f(x)$ the first bit of $x$ followed by $h(x)$. An algorithm $\mathcal B$ always producing the first bit of $x$ from $f(x)$ is trivial, yet an algorithm $\mathcal A$ breaking the one-wayness of $f$ can be turned into an algorithm  $\mathcal A'$ breaking the one-wayness of $h$, with at least half the success probability of $\mathcal A$.

The question's quote is in this online document. I'm ready to follow the OP that the proof made is locally wrong, but I do not exclude this can be repaired by some simple argument, on the tune of: if we can find a component $x_1$, by symmetry, we can find any $x_i$. Lattice based crypto is just too far from my comfort zone, and the time I spent on the paper too low, that I can have a useful opinion.
Note: the question mentions &quot;14.1 (2015): 2-8&quot;, but that's incompatible with this and this stating &quot;Volume: 14, Issue: 1, Jan.-Feb. 1 2017&quot;. Said online document is dated May 20, 2015, as well as another online version bearing &quot;Content may change prior to final publication&quot;, thus I can't exclude the document was later revised.