---
title: Cyclic-Addition Degree Tables (CAT)
url: https://www.emergentmind.com/topics/cyclic-addition-degree-tables-cat
type: topic
---

# Cyclic-Addition Degree Tables (CAT)

Cyclic-Addition Degree Tables (CAT) are a family of degree table constructions used in coding schemes for private distributed matrix multiplication (PDMM), particularly designed to reduce communication overhead and worker count while preserving information-theoretic privacy. The CAT framework exploits cyclic (modular) addition of degree exponents together with root-of-unity evaluation points in finite fields, enabling more efficient polynomial code construction than classical integer-addition-based schemes, especially in the low-privacy regime where the number of colluding adversaries is small relative to the number of matrix partitions [2511.23406], [2501.12371].

## 1. Formulation and Definition

The CAT approach encodes two private input matrices, $A \in \mathbb{F}_q^{m \times \kappa}$ and $B \in \mathbb{F}_q^{\kappa \times \ell}$, for distributed matrix multiplication across $N$ servers, in such a way that any $T$ colluding servers do not learn anything about $A$ or $B$. The matrices are partitioned into $K$ row blocks and $L$ column blocks respectively, yielding $K \times L$ block products $A_i B_j$ that must be recovered.

CAT codes assign integer degree vectors $r = (r_1, \ldots, r_K) \in \mathbb{Z}_M^K$ and $s = (s_1, \ldots, s_L) \in \mathbb{Z}_M^L$ for the polynomial encoding of $A$ and $B$, forming a degree table $D$:

$$
d(i,j) := (r_i + s_j) \bmod M,\quad i = 1,\ldots,K,\ j = 1,\ldots,L
$$

Each entry $d(i,j)$ uniquely designates the polynomial degree corresponding to the block product $A_i B_j$. The cyclic addition modulo $M$ is the critical innovation, enabling the mapping of all possible sums into a smaller index set and facilitating efficient Vandermonde interpolation.

## 2. Encoding, Evaluation, and Decoding Procedures

The user defines two polynomials for encoding:

- $f(x) = \sum_{i=1}^K A_i x^{r_i} + \sum_{t=1}^T R_t x^{u_t}$
- $g(x) = \sum_{j=1}^L B_j x^{s_j} + \sum_{t=1}^T S_t x^{v_t}$

Random matrices $R_t$, $S_t$ with degree exponents $u_t$, $v_t$ (chosen from the interference set $\mathrm{IS}$ disjoint from the block-product degrees) serve as privacy masks. The user chooses $N = M$ field elements $\{\alpha_n\}_{n=1}^N \subset \mathbb{F}_q^*$ as evaluation points, commonly chosen as distinct roots of unity of order $M$.

Each server $n$ receives the tuple $(f(\alpha_n), g(\alpha_n))$ and returns $h(\alpha_n) = f(\alpha_n)g(\alpha_n)$. The user collects all $N$ responses and interpolates the univariate polynomial $h(x) = f(x)g(x)$, extracting the coefficients at the degrees $d(i,j)$ to recover the block products $A_iB_j$.

## 3. Privacy and Feasibility Conditions

CAT codes guarantee $T$-privacy: any $T$ colluding servers, even with access to all their $(f(\alpha_n), g(\alpha_n))$ values, obtain no information about $A$ or $B$. This security is derived from the random mask terms spanning a $T$-dimensional subspace that completely obscures any possible combination revealed to the adversaries.

For correct matrix recovery (decodability), the $K \cdot L$ message degrees $\{d(i,j)\}$ must be pairwise distinct. Privacy demands that the interference set size satisfies $|\mathrm{IS}| \geq 2T$, and the noise exponents are distinct from the message degrees as well as from each other. For the quantum setting, a further feasibility requirement is imposed: the longest consecutive chain $|\mathrm{LCC}(\mathrm{IS})| \geq \lceil N/2 \rceil$ to support super-dense coding for quantum rates [2511.23406].

## 4. Parameter Selection and Structural Distinctions

To satisfy all constraints, the modulus $M$ is set as

$$
M = K^* L^* + (T-1)^2
$$

with $K^* = K + 1 + \kappa$ and $L^* = L + 1 + \lambda$, such that $K^*, L^*$ are coprime to $T-1$. The auxiliary variables $\kappa, \lambda \geq 0$ are chosen to meet these coprimality requirements.

Exponents $r$ and $s$ are constructed via selectable parameters $x$ and $y$ (with $x$ coprime to $M$ and $y$ such that $x + yK^* \equiv 0 \pmod{M}$) to ensure both non-collision in $D$ and invertibility of the evaluation/interpolation matrix. This explicit construction enables tight worker counts in contrast to GASP or DOG schemes [2501.12371].

## 5. Comparative Performance and Asymptotics

CAT coding schemes are advantageous in the low-privacy regime ($T \ll K, L$), where the number of colluding servers is much less than the block partitioning. The worker count and rate for the CAT scheme are:

$$
N_{\mathrm{CAT}} = K^* L^* + (T-1)^2,\quad R_{\mathrm{CAT}} = \frac{K L}{K^* L^* + (T-1)^2}
$$

For comparison, the GASP and DOG coding schemes yield:

| Scheme | Worker Count $N$ | Rate $R$ |
|--------|--------------------------|----------|
| GASP   | $KL + 2K + 3T - 2$       | $KL/(KL+2K+3T-2)$ |
| DOG    | $KL + K + T$             | $KL/(KL+K+T)$ |
| CAT    | $K^*L^* + (T-1)^2$       | $KL/(K^*L^* + (T-1)^2)$ |

Numerical results demonstrate that for small $T$, CAT can save up to $5$–$15\%$ in the number of workers relative to GASP, and may also outperform DOG for moderate $K, L$ [2501.12371]. In the high-privacy regime ($T \geq L$), CAT becomes suboptimal due to diminished interference dimensions, in which case GASP or multivariate schemes are preferable [2511.23406].

## 6. Quantum Adaptation and Super-Dense Coding

Cyclic-addition degree tables admit a natural extension to quantum protocols. When the feasibility condition on the interference set’s longest consecutive chain is satisfied, CAT codes enable an entanglement-assisted protocol where servers, sharing an $N$-partite entangled state, perform Pauli operations parameterized by the encoded messages and masks. The user then recovers two classical symbols per quantum transmission via super-dense coding, doubling the download rate:

$$
R_{\mathrm{QCAT}} = 2\,R_{\mathrm{CAT}}
$$

This quantum extension leverages the SSO-generator matrix for the dual GRS code on CAT exponents, with the key property that the interference set structure supports the required stabilizer measurements [2511.23406].

## 7. Extensions, Optimizations, and Example

Potential optimizations to the baseline CAT design include:

- **Enlarged Modulus**: Replacing $M$ by a larger divisor of $\mathbb{F}_q^*$ can reduce collisions in $D$ and enhance flexibility.
- **Discretely Optimized Parameters (DOG)**: Allowing “gaps” in degree assignments interpolates between CAT and GASP, optimizing performance for intermediate $T$.
- **Explicit CATx Construction**: The paper "CAT and DOG: Improved Codes for Private Distributed Matrix Multiplication" introduces CATx, realizing these improvements in practical settings [2501.12371].

**Illustrative Example** (for $K = L = T = 2$):

- $K^* = L^* = 3$, $T-1 = 1$, $M = 10$
- Degree vectors: $r = [0,7]$, $s = [0,3]$ mod $10$
- Degree table: $D = \{0, 3, 7, 0\}$; interference set $IS = \{1,2,4,5,6,8,9\}$
- Classical rate: $4/10 = 0.4$; Quantum rate (via super-dense coding): $0.8$

A plausible implication is that such parameter choices yield practical schemes with reduced worker requirements and strong privacy guarantees for distributed matrix multiplication in both classical and quantum-augmented regimes [2511.23406], [2501.12371].

## References

- "Quantum Private Distributed Matrix Multiplication With Degree Tables" [2511.23406]
- "CAT and DOG: Improved Codes for Private Distributed Matrix Multiplication" [2501.12371]

Source: https://www.emergentmind.com/topics/cyclic-addition-degree-tables-cat