Rotate #

Recap of types #

$ \mathcal{R}_{\mathtt{rotate}} := \left\{ \begin{array}{l} (K_\mathsf{Arr}, K_\mathsf{Arr'}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}],\\ \mathsf{Arr'}[i] = \mathsf{Arr}[i +\alpha], \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT.Interp}(\omega,\mathsf{Arr}),\\ \mathsf{Poly}_\mathsf{Arr'}=\mathsf{FFT.Interp}(\omega,\mathsf{Arr'}), \\ K_\mathsf{Arr}=\mathsf{KZG.Commit}(\mathsf{Poly}_\mathsf{Arr}),\\ K_\mathsf{Arr'}=\mathsf{KZG.Commit}(\mathsf{Poly}_\mathsf{Arr'}),\\ \end{array} \right\} $

Intuition #

Assume $\mathsf{Arr}$ is an array of data of size $n$. It is encoded as the y-coordinates into univariant polynomials where the x-coordinates (called the domain $\mathcal{H}_\kappa$) are chosen as the multiplicative group of order $\kappa$ with generator $\omega\in\mathbb{G}_\kappa$ (see Background for more). In short, $\omega^0$ is the first element and $\omega^{\kappa-1}$ is the last element of $\mathcal{H}_\kappa$. We call our polynomial $\mathsf{Poly}_\mathsf{Arr}(X)$. The goal is to construct an output polynomial $\mathsf{Poly_\mathsf{Arr'}}$ where all the elements in $\mathsf{Arr}$ have been rotated by $\alpha$ positions; i.e. $\mathsf{Arr'}[i] \leftarrow \mathsf{Arr}[i + \alpha]$. Here, the index is defined mod $n$.

Written in terms the polynomials, the goal is that $\mathsf{Poly_\mathsf{Arr'}}(\omega^i) = \mathsf{Poly_\mathsf{Arr}}(\omega^i\cdot\omega^\alpha)$ for all $i \in [0, n-1]$. To this end, we define $\mathsf{Poly_\mathsf{Arr}}(X) = \mathsf{Poly_\mathsf{Arr}}(X)\cdot\omega^\alpha$ and demonstrate below why this satisfies the requirements of the output polynomials.

Consider $\mathsf{Poly_\mathsf{Arr}}(X) = c_{n-1}X^{n-1} + \dots + c_2X^2 + c_1X + c_0$. Then:

$\mathsf{Poly_\mathsf{Arr'}}(X)$

$= \mathsf{Poly_Arr}(X) \cdot \omega^\alpha$

$= (c_{n-1}X^{n-1} + \dots + c_2X^2 + c_1X + c_0)(\omega^\alpha) $

$= c_{n-1}X^{n-1}\cdot\omega^\alpha + \dots + c_2X^2\cdot\omega^\alpha + c_1X\cdot\omega^\alpha + c_0\cdot\omega^\alpha$

$= \mathsf{Poly_\mathsf{Arr}}(X\cdot\omega^\alpha)$

In particular, $\mathsf{Poly_{Arr'}}(\omega^i) = \mathsf{Poly_\mathsf{Arr}}(\omega^i\cdot\omega^\alpha)$ for $i \in [0, n-1]$. We also note that $\omega^n = \omega^0$, since $\omega$ is of order $\kappa$. Thus the exponent of $\omega$ is defined mod $n$, like the indexing of $\mathsf{Arr}$. This means that $\omega^i\cdot\omega^\alpha$ wraps around to have the rotation defined as it should be for $i + \alpha \gt n-1$.

To prove the relation between $\mathsf{Arr}$ and $\mathsf{Arr'}$, the prover must commit to the two polynomials, $\mathsf{Poly_{Arr}}$ and $\mathsf{Poly_{Arr'}}$ as $K_{\mathsf{Arr}}$ and $K_\mathsf{Arr'}$. The verifier ($\mathcal{V}$) cannot check either array directly (they may contain secret information, and even if they do not, it is too long to check) so the verifier only sees $K_\mathsf{Arr_1}$ and $K_\mathsf{Arr_2}$. The relation will be proven by showing $K_\mathsf{Arr_1}$ and $K_\mathsf{Arr_2}$ and consistent; this mean verifying that $\mathsf{Arr'}[i] \leftarrow \mathsf{Arr}[i + \alpha]$.

Protocol Details #

Array Level #

* $\mathcal{P}$ holds an array $\mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}]$ of $n$ integers ($a_{i} \in \mathbb{Z}_q$)

* $\mathcal{P}$ holds or computes $\mathsf{Arr}'$ such that:

* $\mathsf{Arr}'[i]= \mathsf{Arr}[i+\alpha]$ for $i \in [0, n-1]$

Polynomial Level #

We assume that $\mathsf{Arr}$ and $\mathsf{Arr'}$ are encoded as the y-coordinates into a univariant polynomial where the x-coordinates (called the domain $\mathcal{H}_\kappa$) are chosen as the multiplicative group of order $\kappa$ with generator $\omega\in\mathbb{G}_\kappa$ (see Background for more). In short, $\omega^0$ is the first element and $\omega^{\kappa-1}$ is the last element of $\mathcal{H}_\kappa$. If $\kappa$ is larger than the length of the arrays, then $\mathsf{Arr}$ can be padded with any value (say, all 1s) as long as this is done before the rotation to create $\mathsf{Arr'}$ is computed.

Recall the constraint we want to prove:

1. $\mathsf{Arr}'[i]= \mathsf{Arr}[i+\alpha]$ for $i \in [0, n-1]$

We write the constraint in polynomial form:

1. For all $X$ from $\omega^0$ to $\omega^{\kappa-1}$: $\mathsf{Poly}_\mathsf{Arr'}(X) = \mathsf{Poly}_\mathsf{Arr}(X)\cdot\omega^\alpha$

We adjust the constraint to show an equality with 0 and label it:

1. $\mathsf{Poly}_\mathsf{Vanish}(X)= \mathsf{Poly}_\mathsf{Arr'}(X) - \mathsf{Poly}_\mathsf{Arr}(X)\cdot\omega^\alpha = 0$

This equation is true for every value of $X \in \mathcal{H}_\kappa$ (but not necessarily true outside of these values). To show this, we divide the polynomial by $X^\kappa - 1$, which is a minimal vanishing polynomial for $\mathcal{H}_\kappa$ that does not require interpolation to create. If the quotient is polynomial (and not a rational function), then $\mathsf{Poly}_\mathsf{Vanish}(X)$ must be vanishing on $\mathcal{H}_\kappa$ too. Specifically, the prover computes:

1. $Q(X) = \frac{\mathsf{Poly}_\mathsf{Vanish}(X)}{X^\kappa - 1}$

By rearranging, we can get $\mathsf{Poly}_\mathsf{Zero}(X)$ as a true zero polynomial (zero at every value both in $\mathcal{H}_\kappa$ and outside of it):

$\mathsf{Poly}_\mathsf{Zero}(X)=\mathsf{Poly}_\mathsf{Vanish}(X) - Q(X)\cdot (X^\kappa - 1)=0$

Ultimately the rotate argument will satisfy the following constraints at the Commitment Level:

1. Show $Q(X)$ exists (as a polynomial that evenly divides the divisor)
2. Show $\mathsf{Poly}_\mathsf{Zero}(X)$ is correctly constructed from $\mathsf{Poly}_\mathsf{Arr}(X)$ and $\mathsf{Poly}_\mathsf{Arr'}(X)$
3. Show $\mathsf{Poly}_\mathsf{Zero}(X)$ is the zero polynomial

Commitment Level #

The verifier will never see the arrays or polynomials themselves. They are undisclosed because they either (i) contain private data or (ii) they are too large to examine and maintain a succinct proof system. Instead the prover will use commitments.

The prover will write the following commitments to the transcript:

• $K_\mathsf{Arr}=\mathsf{KZG.Commit}(\mathsf{Poly}_\mathsf{Arr}(X))$

• $K_\mathsf{Arr'}=\mathsf{KZG.Commit}(\mathsf{Poly}_\mathsf{Arr'}(X))$

• $K_Q=\mathsf{KZG.Commit}(Q(X))$

The prover will generate a random challenge evaluation point (using strong Fiat-Shamir) on the polynomials that is outside of $\mathcal{H}_\kappa$. Call this point $\zeta$. The prover will write the point and opening proofs to the transcript:

• $\zeta$

• $\mathsf{Poly}_\mathsf{Arr}(\zeta)=\mathsf{KZG.Open}(K_\mathsf{Arr},\zeta)$

• $\mathsf{Poly}_\mathsf{Arr'}(\zeta)=\mathsf{KZG.Open}(K_\mathsf{Arr'},\zeta)$

• $Q(\zeta)=\mathsf{KZG.Open}(K_Q,\zeta)$

To check the proof, the verifier uses the transcript to construct the value $Y_\mathsf{Zero}$ as follows:

• $Y_\mathsf{Vanish}= \mathsf{Poly}_\mathsf{Arr'}(\zeta) - \mathsf{Poly}_\mathsf{Arr}(\zeta)\cdot\omega^\alpha$

• $Y_\mathsf{Zero}=Y_\mathsf{Vanish} - Q(\zeta)\cdot (\zeta^\kappa - 1)$

Finally, if the constraint system is true, the following constraint will be true (and will be false otherwise with overwhelming probability, due to the Schwartz-Zippel lemma on $\zeta$) :

• $Y_\mathsf{Zero}\overset{?}{=}0$

Security Proof #

Completeness #

If $Y_\mathsf{Zero}$ is zero, then $\mathcal{V}$ will accept. Therefore, to show completeness, we show that any prover who holds $\mathsf{Arr'}$ and $\mathsf{Arr}$ such that $\mathsf{Arr'}[i] = \mathsf{Arr}[i +\alpha] \space \forall i \in [0, n-1]$ can follow the steps outlined in the above protocol and the resulting $Y_\mathsf{Zero}$ will be equal to zero. To see this, observed that $Y_\mathsf{Zero}$

$ = Y_\mathsf{Vanish} - Q(\zeta)(\zeta^\kappa - 1)$

$ = \mathsf{Poly_{Arr'}}(\zeta) - \mathsf{Poly_{Arr}}(\zeta)\cdot \omega^\alpha - Q(\zeta)(\zeta^\kappa - 1)$

$= \mathsf{Poly_{Arr'}}(\zeta) - \mathsf{Poly_{Arr}}(\zeta)\cdot \omega^\alpha - \frac{\mathsf{Poly_{Vanish}}(\zeta)}{\zeta^\kappa - 1}\cdot(\zeta^\kappa - 1)$

$= \mathsf{Poly_{Arr'}}(\zeta) - \mathsf{Poly_{Arr}}(\zeta)\cdot \omega^\alpha - \mathsf{Poly_{Arr'}}(\zeta) + \mathsf{Poly_{Arr}}(\zeta)\cdot \omega^\alpha$

$= 0$

Where the third equality relies on the fact that $\mathsf{Poly_{Vanish}}(X)$ is divisible by $X^\kappa - 1$. This is true if $\mathsf{Poly_{Vanish}}(X)$ is vanishing on $\mathcal{H_\kappa}$, i.e. if $\mathsf{Poly}_\mathsf{Arr'}(X) - \mathsf{Poly}_\mathsf{Arr}(X)\cdot\omega^\alpha = 0 \space \forall X \in \mathcal{H_\kappa}$. This is true if $\mathsf{Poly}_\mathsf{Arr'}(X) - \mathsf{Poly}_\mathsf{Arr}(X\cdot\omega^\alpha) = 0 \space \forall X \in \mathcal{H_\kappa}$, which is in turn true if $\mathsf{Arr'}[i] - \mathsf{Arr}[i + \alpha] = 0 \space \forall i \in [0, \kappa -1]$, since $\mathsf{Poly_Arr}(\omega^i)=\mathsf{Arr}[i] \space \forall i \in [0, \kappa -1]$. But $\mathsf{Arr'}[i] - \mathsf{Arr}[i + \alpha] = 0 \space \forall i \in [0, n-1]$ is precisely the relation between $\mathsf{Arr'}$ and $\mathsf{Arr}$ that we assumed held for our prover (if $\kappa \gt n$ then the arrays get padded such that this relation still holds), thus the $Y_\mathsf{Zero}(X)$ it creates by following the protocol is the zero polynomial, and its transcript will be accepted.

Soundness #

We prove knowledge soundness in the Algebraic Group Model (AGM). To do so, we must prove that there exists an efficient extractor $\mathcal{E}$ such that for any algebraic adversary $\mathcal{A}$, the probability of $\mathcal{A}$ winning the following game is $\mathsf{negl}(\lambda)$.

1. Given $[g, g^\tau, g^{\tau^2}, \dots,g^{\tau^{n-1}}]$ $\mathcal{A}$ outputs commitments to $\mathsf{Poly}_\mathsf{Arr}(X)$, $\mathsf{Poly}_\mathsf{Arr'}(X)$, $Q(X)$

2. $\mathcal{E}$, given access to $\mathcal{A}$’s outputs from the previous step, outputs $\mathsf{Poly}_\mathsf{Arr}(X)$, $\mathsf{Poly}_\mathsf{Arr'}(X)$, $Q(X)$

3. $\mathcal{A}$ plays the part of the prover in showing that $Y_{\mathsf{Zero}}$ is zero at a random challenge $\zeta$

4. $\mathcal{A}$ wins if:

   i) $\mathcal{V}$ accepts at the end of the protocol

   ii) $\mathsf{Arr'}[i] \neq \mathsf{Arr}[i + \alpha]$ for some $i \in [0, n-1]$

Our proof is as follows:

For the second win condition to be fulfilled, there must be some $i \in [0, n-1]$ such that $\mathsf{Arr'}[i] \neq \mathsf{Arr}[i + \alpha]$. But then $\mathsf{Poly}_\mathsf{Vanish}(X)$ is not vanishing on $\mathcal{H}_\kappa$, so $Q(X)$ is not a polynomial (it is a rational function). This means that $\mathcal{A}$ cannot calculated the correct commitment value $g^{Q(\tau)}$ without solving the t-SDH. Thus, $\mathcal{A}$ chooses an arbitrary value for $Q(\tau)$ and writes $K_Q = g^{Q(\tau)}$ to the transcript. Before this, it also writes commitments to $\mathsf{Poly}_\mathsf{Arr}(X)$ and $\mathsf{Poly}_\mathsf{Arr'}(X)$. All commitments $\mathcal{A}$ has written are linear combinations of the elements in $[g, g^\tau, g^{\tau^2}, \dots,g^{\tau^{n-1}}]$. $\mathcal{E}$ is given these coefficients (since $\mathcal{A}$ is an algebraic adversary) so $\mathcal{E}$ can output the original polynomials.

$\mathcal{A}$ then obtains the random challenge $\zeta$ (using strong Fiat-Shamir). By the binding property of KZG commitments, $\mathsf{Poly}_\mathsf{Arr}(\zeta)$ and $\mathsf{Poly}_\mathsf{Arr'}(\zeta)$ can each only feasibly be opened to one value. For $\mathcal{A}$ to have the verifier accept, it must produce a proof that $Q(\zeta)$ opens to $Q(\zeta) = \frac{Y_\mathsf{Vanish}} {(\zeta^\kappa - 1)}$. This means being able to produce $g^{q(\tau)}$ where $q(\tau) = \frac{Q(\tau) - Q(\zeta)}{\tau - \zeta}$ and $Q(\zeta) = \frac{Y_\mathsf{Vanish}}{(\zeta^\kappa - 1)}$. Since $Q(\tau)$ and $Q(\zeta)$ are known, this implies knowing $g^{\frac{1}{\tau - \zeta}}$. Thus $\mathcal{A}$ would have found $\langle\zeta,g^{\frac{1}{\tau - \zeta}}\rangle$, which is the t-SDH problem. We have shown that creating an accepting proof reduces to the t-SDH, so $\mathcal{A}$’s probability of success is negligible.

Zero-Knowledge #

We prove that the above protocol is zero-knowledge when $\mathsf{PolyCommit}_\mathsf{Ped}$ (from the KZG paper) is used for the polynomial commitments. We do so by constructing a probabilistic polynomial time simulator $\mathcal{S}$ that knows the trapdoor $\tau$, which, for every (possibly malicious) verifier $\mathcal{V}$, can output a view of the execution of the protocol that is indistinguishable from the view produced by the real execution of the program.

The simulator $\mathcal{S}$ chooses arbitrary values for ${\mathsf{Poly}_\mathsf{Arr}(\tau)}$ and ${\mathsf{Poly}_\mathsf{Arr'}(\tau)}$, then computes $g^{\mathsf{Poly}_\mathsf{Arr}(\tau)}$ and $g^{\mathsf{Poly}_\mathsf{Arr'}(\tau)}$ to write as the commitments $ K_\mathsf{Arr}$ and $K_\mathsf{Arr'}$. $\mathcal{S}$ computes $Q(\tau)$ using the values it chose for ${\mathsf{Poly}_\mathsf{Arr}(\tau)}$ and ${\mathsf{Poly}_\mathsf{Arr'}(\tau)}$. $\mathcal{S}$ writes the commitment $K_Q = g^{Q(\tau)}$.

Now, $\mathcal{S}$ generates the random challenge point $\zeta$ (which we assume is not in $\mathcal{H}_\kappa$; if it is in $\mathcal{H}_\kappa$, $\mathcal{S}$ simply restarts and runs from the beginning). This is by strong Fiat-Shamir. $\mathcal{S}$ then create fake opening proofs for ${\mathsf{Poly}_\mathsf{Arr}(\zeta)}$ and ${\mathsf{Poly}_\mathsf{Arr'}(\zeta)}$, to arbitrary values. This is done using the knowledge of $\tau$, calculating the respective witness $q(\tau) = \frac{{f(\tau) - f(\zeta)}}{\tau - \zeta}$ for each of the polynomials.

Finally, $\mathcal{S}$ creates a fake opening proof for $Q(\zeta) = \frac{Y_\mathsf{Vanish}}{(\zeta^\kappa - 1)}$. This is done using knowledge of $\tau$ to calculate an accepting witness $q(\tau)$, as above. This means that $Y_\mathsf{Zero}$ will be zero, and the transcript will be accepted by the verifier. It is indistinguishable from a transcript generates from a real execution, since $\mathsf{PolyCommit}_\mathsf{Ped}$ has the property of Indistinguishability of Commitments due to the randomization by $h^{\hat{\phi}(x)}$.