Let G be a finite cyclic group with generator \alpha and with
an encoding so that multiplication is computable in polynomial time. We
study the security of bits of the discrete log x when given \exp_{\alpha}(x),
assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way.
... more >>>

We study pseudorandom generator (PRG) constructions $G^f : {0,1}^l \to {0,1}^{l+s}$ from one-way functions $f : {0,1}^n \to {0,1}^m$. We consider PRG constructions of the form $G^f(x) = C(f(q_{1}) \ldots f(q_{poly(n)}))$
where $C$ is a polynomial-size constant depth circuit
and $C$ and the $q$'s are generated from $x$ arbitrarily.
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We investigate the average case complexity of a generalization of the compact knapsack problem to arbitrary rings: given $m$ (random) ring elements a_1,...,a_m in R and a (random) target value b in R, find coefficients x_1,...,x_m in S (where S is an appropriately chosen subset of R) such that a_1*x_1 ... more >>>

We prove a time hierarchy theorem for inverting functions
computable in polynomial time with one bit of advice.
In particular, we prove that if there is a strongly
one-way function, then for any k and for any polynomial p,
there is a function f computable in linear time
with one ... more >>>

Suppose you want to store a large file on a remote and unreliable server. You would like to verify that your file has not been corrupted, so you store a small private (randomized)``fingerprint'' of the file on your own computer. This is the setting for the well-studied authentication problem, and ... more >>>

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, ... more >>>

In this paper we study the possibility of proving the existence of
one-way functions based on average case hardness. It is well-known
that if there exists a polynomial-time sampler that outputs
instance-solution pairs such that the distribution on the instances
is hard on average, then one-way functions exist. We study ... more >>>

We establish new hardness amplification results for one-way functions in which each input bit influences only a small number of output bits (a.k.a. input-local functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain the input size of the original ... more >>>

Every pseudorandom generator is in particular a one-way function. If we only consider part of the output of the
pseudorandom generator is this still one-way? Here is a general setting formalizing this question. Suppose
$G:\{0,1\}^n\rightarrow \{0,1\}^{\ell(n)}$ is a pseudorandom generator with stretch $\ell(n)> n$. Let $M_R\in\{0,1\}^{m(n)\times \ell(n)}$ be a linear ... more >>>

A function $f$ mapping $n$-bit strings to $m$-bit strings can be constructed from a bipartite graph with $n$ vertices on the left and $m$ vertices on the right having right-degree $d$ together with a predicate $P:\{0,1\}^d\rightarrow\{0,1\}$. The vertices on the left correspond to the bits of the input to the ... more >>>

We revisit ``the randomized iterate'' technique that was originally used by Goldreich, Krawczyk, and Luby (SICOMP 1993) and refined by Haitner, Harnik and Reingold (CRYPTO 2006) in constructing pseudorandom generators (PRGs) from regular one-way functions (OWFs). We abstract out a technical lemma with connections to several recent work on cryptography ... more >>>

We prove that if the hardness of inverting a size-verifiable one-way function can
be based on NP-hardness via a general (adaptive) reduction, then coAM is contained in NP. This
claim was made by Akavia, Goldreich, Goldwasser, and Moshkovitz (STOC 2006), but
was later retracted (STOC 2010).

Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on ... more >>>

A one-way function is $d$-local if each of its outputs depends on at most $d$ input bits. In (Applebaum, Ishai, and Kushilevitz, FOCS 2004) it was shown that, under relatively mild assumptions, there exist $4$-local one-way functions (OWFs). This result is not far from optimal as it is not hard ... more >>>

We introduce the notion of pseudo-mixing time of a graph define as the number of steps in a random walk that suffices for generating a vertex that looks random to any polynomial-time observer, where, in addition to the tested vertex, the observer is also provided with oracle access to the ... more >>>

We prove the equivalence of two fundamental problems in the theory of computation:

- Existence of one-way functions: the existence of one-way functions (which in turn are equivalent to pseudorandom generators, pseudorandom functions, private-key encryption schemes, digital signatures, commitment schemes, and more).

- Mild average-case hardness of $K^{poly}$-complexity: ... more >>>

Let $\mktp[s]$ be the set of strings $x$ such that $K^t(x) \leq s(|x|)$, where $K^t(x)$ denotes the $t$-bounded Kolmogorov complexity of the truthtable described by $x$. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every $\varepsilon>0$, polynomial $t(n) \geq (1+\varepsilon)n$, and every ``nice'' class ... more >>>

Liu and Pass (FOCS'20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity---namely, Levin's notion of Kolmogorov Complexity---whose hardness is closely ... more >>>

We present the first natural $\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances implies the existence of one-way functions (OWF). In fact, we prove that the existence of OWFs is \emph{equivalent} to mild average-case hardness of this $\NP$-complete problem. The problem, which originated in the 1960s, is ... more >>>

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) ... more >>>

A one-way function is a function that is easy to compute but hard to invert *on average*. We establish the first characterization of a one-way function by *worst-case* hardness assumptions, by introducing a natural meta-computational problem whose NP-hardness (and the worst-case hardness of NP) characterizes the existence of a one-way ... more >>>

Pessiland is one of Impagliazzo's five possible worlds in which NP is hard on average, yet no one-way function exists. This world is considered the most pessimistic because it offers neither algorithmic nor cryptographic benefits.

In this paper, we develop a unified framework for constructing strong learning algorithms ... more >>>

Whether one-way functions (OWF) exist is arguably the most important
problem in Cryptography, and beyond. While lots of candidate
constructions of one-way functions are known, and recently also
problems whose average-case hardness characterize the existence of
OWFs have been demonstrated, the question of
whether there exists some \emph{worst-case hard problem} ... more >>>

A central result in the theory of Cryptography, by Hastad, Imagliazzo, Luby and Levin [SICOMP’99], demonstrates that the existence one-way functions (OWF) implies the existence of pseudo-random generators (PRGs). Despite the fundamental importance of this result, and several elegant improvements/simplifications, analyses of constructions of PRGs from OWFs remain complex (both ... more >>>

Consider the recently introduced notion of \emph{probabilistic
time-bounded Kolmogorov Complexity}, pK^t (Goldberg et al,
CCC'22), and let MpK^tP denote the language of pairs (x,k) such that pK^t(x) \leq k.
We show the equivalence of the following:
- MpK^{poly}P is (mildly) hard-on-average w.r.t. \emph{any} samplable
distribution $\D$;
- ... more >>>

A ZAP is a witness-indistinguishable two-message public-coin interactive proof with the following simple structure: the verifier sends a uniformly random string, the prover responds, and the verifier decides in polynomial time whether to accept or reject.

We show that one-way functions imply the existence of ... more >>>

The fundamental theorem of Goldreich, Micali, and Wigderson (J. ACM 1991) shows that the existence of a one-way function is sufficient for constructing computational zero knowledge ($\mathrm{CZK}$) proofs for all languages in $\mathrm{NP}$. We prove its converse, thereby establishing characterizations of one-way functions based on the worst-case complexities of ... more >>>

We introduce $\mathrm{pKt}$ complexity, a new notion of time-bounded Kolmogorov complexity that can be seen as a probabilistic analogue of Levin's $\mathrm{Kt}$ complexity. Using $\mathrm{pKt}$ complexity, we upgrade two recent frameworks that characterize one-way functions ($\mathrm{OWFs}$) via symmetry of information and meta-complexity, respectively. Among other contributions, we establish the following ... more >>>