A {\em dot-product proof} (DPP) is a simple probabilistic proof system in which the input statement $\mathbf{x}$ and the proof $\boldsymbol{\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle \mathbf{q},(\mathbf{x} \| \boldsymbol{\pi}) \rangle$ jointly to $\mathbf{x}$ and $\boldsymbol{\pi}$. A DPP can be viewed as a 1-query fully linear PCP. We study the feasibility and efficiency of DPPs, obtaining the following results:
\begin{itemize}
\item
{\bf Small-field DPP.}
For any finite field $\mathbb{F}$ and Boolean circuit $C$ of size $S$, there is a $\DPP$ for proving that there exists
$\boldsymbol{v}$ such that $C(\boldsymbol{v}, \boldsymbol{v})=1$
with a proof $\piv$ of length
$S\cdot\poly(|\mathbb{F}|)$
and soundness error $\epsilon=O(1/\sqrt{|\mathbb{F}|})$. We show this error to be asymptotically optimal.

\smallskip In particular, and in contrast to the best known PCPs, there
exist strictly linear-length DPPs over constant-size fields.

\item
{\bf Large-field DPP.}
If $|\mathbb{F}|\ge\poly(S/\epsilon)$, there is a similar
DPP with soundness error $\epsilon$ and proof length $O(S)$ (in field elements).
\end{itemize}
The above results do not rely on the PCP theorem and their proofs are considerably simpler.
We apply our DPP constructions toward two kinds of applications.
\begin{itemize}
\item {\bf Hardness of approximation.} We obtain a simple proof for the NP-hardness of approximating MAXLIN (with dense instances) over any finite field $\mathbb{F}$ up to some constant factor $c>1$, independent of $\mathbb{F}$.
Unlike previous $\PCP$-based proofs, our proof yields {\em exponential-time} hardness under the exponential time hypothesis (ETH).
\item {\bf Succinct arguments.} We
improve the concrete efficiency of succinct
interactive arguments in the generic group model using input-independent preprocessing.
In particular, the communication is comparable to sending two group elements and the verifier's computation is dominated by a single group exponentiation. We also show how to use DPPs together with linear-only encryption
to construct succinct commit-and-prove arguments.
\end{itemize}

(1) Augmented the embedding transformation in Section 5.1 to make it more efficient and (2) Fixed the inaccuracy in our previous concrete instantiation of laconic arguments in the generic group model in Section 7.2.

A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement $x$ and the proof ${\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle {q},({x} \| {\pi})\rangle$ jointly to ${x}$ and ${\pi}$. A DPP can be viewed as a 1-query {\em fully linear} PCP. We study the feasibility and efficiency of DPPs, obtaining the following results:

* Small-field DPP: For any finite field $\mathbb{F}$ and Boolean circuit $C$ of size $S$, there is a DPP for proving that there exists ${w}$ such that $C({x}, {w})=1$ with a proof ${\pi}$ of length $S\cdot\text{poly}(|\mathbb{F}|)$ and soundness error $\epsilon=O(1/\sqrt{|\mathbb{F}|})$. We show this error to be asymptotically optimal. In particular, and in contrast to the best known PCPs, there exist strictly linear-length DPPs over constant-size fields.

* Large-field DPP: If $|\mathbb{F}|\ge\text{poly}(S/\epsilon)$, there is a similar DPP with soundness error $\epsilon$ and proof length $O(S)$ (in field elements).

The above results do not rely on the PCP theorem and their proofs are considerably simpler. We apply our DPP constructions toward two kinds of applications.

* Hardness of approximation: We obtain a simple proof for the NP-hardness of approximating MAXLIN (with dense instances) over any finite field $\mathbb{F}$ up to some constant factor $c>1$, independent of $\mathbb{F}$. Unlike previous PCP-based proofs, our proof yields exponential-time hardness under the exponential time hypothesis (ETH).

* Succinct arguments: We improve the concrete efficiency of succinct interactive arguments in the generic group model using input-independent preprocessing.In particular, the communication is comparable to sending two group elements and the verifier's computation is dominated by a single group exponentiation. We also show how to use DPPs together with linear-only encryption to construct succinct commit-and-prove arguments.

Minor revision, fixing inaccuracy in Theorem 3.5 and slightly simplifying the small-field DPP construction.

A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement $x$ and the proof ${\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle {q},({x} \| {\pi})\rangle$ jointly to ${x}$ and ${\pi}$. A DPP can be viewed as a 1-query {\em fully linear} PCP. We study the feasibility and efficiency of DPPs, obtaining the following results:

* Small-field DPP: For any finite field $\mathbb{F}$ and Boolean circuit $C$ of size $S$, there is a DPP for proving that there exists ${w}$ such that $C({x}, {w})=1$ with a proof ${\pi}$ of length $S\cdot\text{poly}(|\mathbb{F}|)$ and soundness error $\epsilon=O(1/\sqrt{|\mathbb{F}|})$. We show this error to be asymptotically optimal. In particular, and in contrast to the best known PCPs, there exist strictly linear-length DPPs over constant-size fields.

* Large-field DPP: If $|\mathbb{F}|\ge\text{poly}(S/\epsilon)$, there is a similar DPP with soundness error $\epsilon$ and proof length $O(S)$ (in field elements).

The above results do not rely on the PCP theorem and their proofs are considerably simpler. We apply our DPP constructions toward two kinds of applications.

* Hardness of approximation: We obtain a simple proof for the NP-hardness of approximating MAXLIN (with dense instances) over any finite field $\mathbb{F}$ up to some constant factor $c>1$, independent of $\mathbb{F}$. Unlike previous PCP-based proofs, our proof yields exponential-time hardness under the exponential time hypothesis (ETH).

* Succinct arguments: We improve the concrete efficiency of succinct interactive arguments in the generic group model using input-independent preprocessing.In particular, the communication is comparable to sending two group elements and the verifier's computation is dominated by a single group exponentiation. We also show how to use DPPs together with linear-only encryption to construct succinct commit-and-prove arguments.