Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Existence of Weak One-Way Functions   This note is an attempt to unconditionally prove the existence of weak one
way functions (OWF). Starting from a provably intractable decision problem
$L_D$ (whose existence is nonconstructively assured from the well-known
discrete time-hierarchy theorem from complexity theory), we construct another
intractable decision problem $L\subseteq \{0,1\}^*$ that has its words
scattered across $\{0,1\}^\ell$ at a relative frequency $p(\ell)$, for which
upper and lower bounds can be worked out. The value $p(\ell)$ is computed from
the density of the language within $\{0,1\}^\ell$ divided by the total word
count $2^\ell$. It corresponds to the probability of retrieving a yes-instance
of a decision problem upon a uniformly random draw from $\{0,1\}^\ell$. The
trick to find a language with known bounds on $p(\ell)$ relies on switching
from $L_D$ to $L_0:=L_D\cap L'$, where $L'$ is an easy-to-decide language with
a known density across $\{0,1\}^*$. In defining $L'$ properly (and upon a
suitable G\"odel numbering), the hardness of deciding $L_D\cap L'$ is inherited
from $L_D$, while its density is controlled by that of $L'$. The lower and
upper approximation of $p(\ell)$ then let us construct an explicit threshold
function (as in random graph theory) that can be used to efficiently and
intentionally sample yes- or no-instances of the decision problem (language)
$L_0$ (however, without any auxiliary information that could ease the decision
like a polynomial witness). In turn, this allows to construct a weak OWF that
encodes a bit string $w\in\{0,1\}^*$ by efficiently (in polynomial time)
emitting a sequence of randomly constructed intractable decision problems,
whose answers correspond to the preimage $w$.

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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: physics