134 lines
9.8 KiB
TeX
134 lines
9.8 KiB
TeX
% \section{Stern-like Proofs}
|
|
% \addcontentsline{tof}{section}{\protect\numberline{\thesection} Preuves à la Stern}
|
|
% \label{sse:stern}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
On the other hand, Stern's protocol has been originally introduced in the context of code-base cryptography~\cite{Ste96}.
|
|
\index{Syndrome Decoding Problem}
|
|
Initially, it was introduced for Syndrome Decoding Problem (\SDP): given a matrix $\mathbf P \in \FF_2^{n \times m}$ and a syndrome $\mathbf v \in \FF_2^n$, the goal is to find a binary vector $\mathbf x \in \FF_2^m$ with fixed hamming weight $w$ such that $\mathbf P \cdot \mathbf x = \mathbf v \bmod 2$.
|
|
|
|
This problem shows similarities with the $\ISIS$ problem defined in \cref{de:sis} where the constraints on the norm of $\mathbf x$ is a constraint on Hamming weight, and operations are in $\FF_2$ instead of $\Zq$.
|
|
|
|
After the first works of Kawachi, Tanaka and Xagawa~\cite{KTX08} that extended Stern's proofs to statements $\bmod q$, the work of Ling, Nguyen, Stehlé and Wang~\cite{LNSW13} enables the use of Stern's protocol to prove general $\SIS$ or $\LWE$ statements (meaning the knowledge of a solution to these problems).
|
|
These advances in the expressivity of Stern-like protocols has been used to further improve it and therefore enable privacy-based primitives for which no constructions existed in the post-quantum world, such as dynamic group signatures~\cite{LLM+16}, group encryption~\cite{LLM+16a}, electronic cash~\cite{LLNW17}, etc.
|
|
|
|
Unlike the Schnorr-like proof we saw in the previous section, Stern's proof is mainly combinatorial and relies on the fact that every permutation on a binary vector $\mathbf x \in \bit^m_{}$ leaves its Hamming weight $w$ invariant. As a consequence, for $\pi \in \permutations_m$, $\mathbf x$ satisfies these conditions if and only if $\pi(\mathbf x)$ also does.
|
|
Therefore, the randomness of $\pi$ is used to verify these two constraints (being binary and the hamming weight) in a zero-knowledge fashion.
|
|
We can notice that this can be extended to vectors $\mathbf x \in \nbit^m$ of fixed numbers of $-1$ and $1$ that allowed~\cite{LNSW13} to propose the generalization of this protocol to any $\ISIS_{n,m,q,\beta}$ statements.
|
|
In \cref{sse:stern-abstraction}, we describes these permutations while abstracting the set of ZK-provable statements as the set $\mathsf{VALID}$ that satisfies conditions~\eqref{eq:zk-equivalence}.
|
|
|
|
It is worth noticing that this argument on knowledge does not form a $\Sigma$-protocol, as the challenge space is ternary as described in \cref{sse:stern-abstraction}.
|
|
Thus standard theorems on $\Sigma$-protocols has to be adapted in this setting.
|
|
|
|
In this Section, we describe in a high-level view how Stern's protocol works, and then we detail it.
|
|
|
|
|
|
\subsection{Abstraction of Stern's Protocol} \label{sse:stern-abstraction}
|
|
\addcontentsline{tof}{subsection}{\protect\numberline{\thesubsection} Abstraction du protocole de Stern}
|
|
|
|
%%%% TODO
|
|
\begin{figure}[t]
|
|
|
|
\small
|
|
\begin{enumerate}
|
|
\item \textbf{Commitment:} Prover samples $\mathbf{r}_w \leftarrow U(\mathbb{Z}_q^D)$, $\phi \leftarrow U(\mathcal{S})$ and randomnesses $\rho_1, \rho_2, \rho_3$ for $\mathsf{COM}$.
|
|
Then he sends $\mathrm{CMT}= \big(C_1, C_2, C_3\big)$ to the verifier, where
|
|
\begin{gather*}
|
|
C_1 = \mathsf{COM}(\phi, \mathbf{M}\cdot \mathbf{r}_w \bmod q; \rho_1), \hspace*{5pt}
|
|
C_2 = \mathsf{COM}(\Gamma_{\phi}(\mathbf{r}_w); \rho_2), \\
|
|
C_3 = \mathsf{COM}(\Gamma_{\phi}(\mathbf{w} + \mathbf{r}_w \bmod q); \rho_3).
|
|
\end{gather*}
|
|
|
|
\item \textbf{Challenge:} The verifier sends a challenge $Ch \leftarrow U(\{1,2,3\})$ to the prover.
|
|
\item \textbf{Response:} Depending on $Ch$, the prover sends $\mathrm{RSP}$ computed as follows:
|
|
\smallskip
|
|
\begin{itemize}
|
|
\item $Ch = 1$: Let $\mathbf{t}_{w} = \Gamma_{\phi}(\mathbf{w})$, $\mathbf{t}_{r} = \Gamma_{\phi}(\mathbf{r}_w)$, and $\mathrm{RSP} = (\mathbf{t}_w, \mathbf{t}_r, \rho_2, \rho_3)$. \smallskip
|
|
|
|
\item $Ch = 2$: Let $\phi_2 = \phi$, $\mathbf{w}_2 = \mathbf{w} + \mathbf{r}_w \bmod q$, and
|
|
$\mathrm{RSP} = (\phi_2, \mathbf{w}_2, \rho_1, \rho_3)$. \smallskip
|
|
\item $Ch = 3$: Let $\phi_3 = \phi$, $\mathbf{w}_3 = \mathbf{r}_w$, and
|
|
$\mathrm{RSP} = (\phi_3, \mathbf{w}_3, \rho_1, \rho_2)$.
|
|
\end{itemize}
|
|
\end{enumerate}
|
|
\textbf{Verification:} Receiving $\mathrm{RSP}$, the verifier proceeds as follows:
|
|
\smallskip
|
|
\begin{itemize}
|
|
\item $Ch = 1$: Check that
|
|
\begin{gather*}
|
|
\mathbf{t}_w \in \mathsf{VALID},\\
|
|
C_2 = \mathsf{COM}(\mathbf{t}_r; \rho_2), \qquad
|
|
{C}_3 = \mathsf{COM}(\mathbf{t}_w + \mathbf{t}_r \bmod q; \rho_3).
|
|
\end{gather*}
|
|
|
|
\item $Ch = 2$: Check that
|
|
\[
|
|
C_1 = \mathsf{COM}(\phi_2, \mathbf{M}\cdot \mathbf{w}_2 - \mathbf{v} \bmod q; \rho_1),\qquad
|
|
{C}_3 = \mathsf{COM}(\Gamma_{\phi_2}(\mathbf{w}_2); \rho_3).
|
|
\]
|
|
|
|
\item $Ch = 3$: Check that
|
|
\[
|
|
C_1 = \mathsf{COM}(\phi_3, \mathbf{M}\cdot \mathbf{w}_3; \rho_1), \qquad
|
|
C_2 = \mathsf{COM}(\Gamma_{\phi_3}(\mathbf{w}_3); \rho_2).
|
|
\]
|
|
|
|
\end{itemize}
|
|
In each case, the verifier outputs $1$ if and only if all the conditions hold.
|
|
\caption{Stern-like \textsf{ZKAoK} for the relation $\mathrm{R_{abstract}}$.}
|
|
\label{Figure:Interactive-Protocol}
|
|
\end{figure}
|
|
|
|
|
|
Let $K$, $D$, $q$ be positive integers with $D \geq K$ and $q \geq 2$, and let $\mathsf{VALID}$ be a subset of $\mathbb{Z}^D$. Suppose that $\mathcal{S}$ is a finite set such that every $\phi \in \mathcal{S}$ can be associated with a permutation $\Gamma_\phi \in \permutations_D$ satisfying the following conditions:
|
|
\begin{eqnarray}\label{eq:zk-equivalence}
|
|
\begin{cases}
|
|
\mathbf{w} \in \mathsf{VALID} ~ \iff ~ \Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}, \\
|
|
\text{If } \mathbf{w} \in \mathsf{VALID} \text{ and } \phi \text{ is uniform in } \mathcal{S}, \text{ then } \Gamma_\phi(\mathbf{w}) \text{ is uniform in } \mathsf{VALID}. \quad
|
|
\end{cases}
|
|
\end{eqnarray}
|
|
We aim to construct a statistical Zero-Knowledge Argument of Knowledge (\textsf{ZKAoK}) for the following abstract relation:
|
|
\begin{eqnarray*}
|
|
\mathrm{R_{abstract}} = \big\{ \big((\mathbf{M}, \mathbf{v}), \mathbf{w} \big) \in \mathbb{Z}_q^{K \times D} \times \mathbb{Z}_q^D \times \mathsf{VALID}: \mathbf{M}\cdot \mathbf{w} = \mathbf{v} \bmod q.\big\}
|
|
\end{eqnarray*}
|
|
|
|
Note that, Stern's original protocol corresponds to the special case when the set
|
|
$\mathsf{VALID} = \{
|
|
\mathbf{w} \in \{0,1\}^D: \mathsf{wt}(\mathbf{w}) = k\}$, where $\mathsf{wt}(\cdot)$ denotes the Hamming weight and $k < D$ is a given integer, $\mathcal{S} = \mathcal{S}_D$ is the set of all permutations of~$D$ elements and $\Gamma_{\phi}(\mathbf{w}) = \phi(\mathbf{w})$.
|
|
|
|
The conditions in \eqref{eq:zk-equivalence} play a crucial role in proving in \textsf{ZK} that $\mathbf{w} \in \mathsf{VALID}$. To this end, the prover samples a random $\phi \hookleftarrow U(\mathcal{S})$ and lets the verifier check that $\Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}$ without learning any additional information about $\mathbf{w}$ due to the randomness of $\phi$. Furthermore, to prove in a zero-knowledge manner that the linear equation is satisfied, the prover samples a masking vector $\mathbf{r}_w \hookleftarrow U(\mathbb{Z}_q^D)$, and convinces the verifier instead that $\mathbf{M}\cdot (\mathbf{w} + \mathbf{r}_w) = \mathbf{M}\cdot \mathbf{r}_w + \mathbf{v} \bmod q.$
|
|
|
|
|
|
The interaction between prover $\mathcal{P}$ and verifier $\mathcal{V}$ is described in Figure~\ref{Figure:Interactive-Protocol}. The protocol uses a statistically hiding and computationally binding string commitment scheme \textsf{COM} (e.g., the \textsf{SIS}-based scheme from~\cite{KTX08}).
|
|
|
|
\begin{theorem}\label{Theorem:zk-protocol}
|
|
The protocol in Figure~\ref{Figure:Interactive-Protocol} is a statistical \emph{\textsf{ZKAoK}} with perfect completeness, soundness error~$2/3$, and communication cost~$\mathcal{O}(D\log q)$. Namely:
|
|
\begin{itemize}
|
|
\item There exists a polynomial-time simulator that, on input $(\mathbf{M}, \mathbf{v})$, outputs an accepted transcript statistically close to that produced by the real prover.
|
|
\item There exists a polynomial-time knowledge extractor that, on input a commitment $\mathrm{CMT}$ and $3$ valid responses $(\mathrm{RSP}_1,\mathrm{RSP}_2,\mathrm{RSP}_3)$ to all $3$ possible values of the challenge $Ch$, outputs $\mathbf{w}' \in \mathsf{VALID}$ such that $\mathbf{M}\cdot \mathbf{w}' = \mathbf{v} \bmod q.$
|
|
\end{itemize}
|
|
\end{theorem}
|
|
The proof of the theorem relies on standard simulation and extraction techniques for Stern-like protocols~\cite{KTX08,LNSW13,LLM+16}.
|
|
\vspace{-0.1 cm}
|
|
|
|
|
|
%%%% END TODO
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%
|
|
%%%% Recap Table %%%%
|
|
%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{figure}
|
|
\begin{itemize}
|
|
\item $\mathsf B^{2}_{\mathfrak m}$: the set of vectors in $\bit^{2\mathfrak m}$ with Hamming weight $\mathfrak m$.
|
|
\item $\mathsf B^{3}_{\mathfrak m}$: the set of vectors in $\nbit^{3\mathfrak m}$ which has exactly $\mathfrak m$ coordinates equal to $j$ for each $j \in \nbit$.
|
|
\end{itemize}
|
|
\caption{Notations for Stern-like protocols.}
|
|
\label{fig:stern-notations}
|
|
\end{figure}
|
|
|
|
\subsection{The Decomposition-Extension Framework}
|
|
\addcontentsline{tof}{subsection}{\protect\numberline{\thesubsection} Méthode de décomposition-extension}
|
|
|
|
A method used in~\cite{LNSW13} to prove knowledge of an \ISIS preimage consists in first \textit{decomposing} the secret $\mathbf{x} = (x_1, \ldots, x_m) \in [-B,B]^m$ into a vector $\tilde{\mathbf x}$ of $\nbit^{m \delta_B}$ such that $\tilde{\mathbf x} = [ \tilde{\mathbf u}_1^{T} \mid \cdots \mid \tilde{\mathbf u}_{\delta_B}^T]^T$ and for all $j \in \{1, \ldots, m\}$, $(1, 2, \ldots, 2^{\delta_B - 1})^T \cdot \tilde{\mathbf u}_j^{} = x_j$.
|
|
Once that is done, we fix the hamming weight of the resulting vector by \textit{extending} its components $\tilde{\mathbf u}_j^{}$ into $\mathbf u_j \in \mathsf B^3_{m}$.
|