We present the first constructions of *single*-prover proof systems that achieve *perfect* zero knowledge (PZK) for languages beyond NP, under no intractability assumptions:

1. The complexity class #P has PZK proofs in the model of Interactive PCPs (IPCPs) [KR08], where the verifier first receives from the prover a PCP and ... more >>>

Probabilistically Checkable Proofs (PCPs) [Babai et al. FOCS 90; Arora et al. JACM 98] can be used to construct asymptotically efficient cryptographic zero knowledge arguments of membership in any language in NEXP, with minimal communication complexity and computational effort on behalf of both prover and verifier [Babai et al. STOC ... more >>>

A Probabilistically Checkable Proof of Proximity (PCPP) for a linear code $C$, enables to determine very efficiently if a long input $x$, given as an oracle, belongs to $C$ or is far from $C$.
PCPPs are often a central component of constructions of Probabilistically Checkable Proofs (PCP)s [Babai et al. ... more >>>

We study *interactive oracle proofs* (IOPs) (Ben-Sasson, Chiesa, Spooner '16), which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and general techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs ... more >>>

The seminal result that every language having an interactive proof also has a zero-knowledge interactive proof assumes the existence of one-way functions. Ostrovsky and Wigderson (ISTCS 1993) proved that this assumption is necessary: if one-way functions do not exist, then only languages in BPP have zero-knowledge interactive proofs.

Ben-Or et ... more >>>

An $r$-simple $k$-path is a {path} in the graph of length $k$ that
passes through each vertex at most $r$ times. The \simpath{r}{k}
problem, given a graph $G$ as input, asks whether there exists an
$r$-simple $k$-path in $G$. We first show that this problem is
NP-Complete. We then show ... more >>>

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>

Let $F$ be the field of $q$ elements, where $q=p^{\ell}$ for prime $p$. Informally speaking, a polynomial source is a distribution over $F^n$ sampled by low degree multivariate polynomials. In this paper, we construct extractors for polynomial sources over fields of constant size $q$ assuming $p \ll q$.

More generally, ... more >>>

Let $\F$ be the field of $q$ elements. An \emph{\afsext{n}{k}} is a mapping $D:\F^n\ar\B$
such that for any $k$-dimensional affine subspace $X\subseteq \F^n$, $D(x)$ is an almost unbiased
bit when $x$ is chosen uniformly from $X$.
Loosely speaking, the problem of explicitly constructing affine extractors gets harder as $q$ gets ... more >>>

In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial ... more >>>

An $(n,k)$-bit-fixing source is a distribution $X$ over $\B^n$ such that
there is a subset of $k$ variables in $X_1,\ldots,X_n$ which are uniformly
distributed and independent of each other, and the remaining $n-k$ variables
are fixed. A deterministic bit-fixing source extractor is a function $E:\B^n
\ar \B^m$ which on ... more >>>

An $(n,k)$-affine source over a finite field $F$ is a random
variable $X=(X_1,...,X_n) \in F^n$, which is uniformly
distributed over an (unknown) $k$-dimensional affine subspace of $
F^n$. We show how to (deterministically) extract practically all
the randomness from affine sources, for any field of size larger
than $n^c$ (where ... more >>>