The analyses conducted thus far have examined four different approaches for data encoding and sampling methods within the Nomos DA v2 framework:
During these analyses, we evaluated these methods using different matrix sizes: 256x256, 512x512, and 1024x1024. Increasing the matrix size has the advantage of reducing the total amount of data sent to subnetworks. However, for fixed total data sizes, this increase in matrix size leads to smaller chunk sizes, causing potential issues with the subspace distance check.
While this issue is negligible for large datasets (greater than 1 MB), it poses a significant problem for smaller datasets (less than 1 MB). To address this, we employed field extension techniques specifically for smaller data sizes. This is because when the chunk size is small, the finite field ($2^k$) used in encoding also becomes small, increasing the probability of failing the subspace distance check. Such failures could potentially allow adversaries to introduce undetectable errors.
By applying field extensions (utilizing a larger finite field), we substantially decrease the risk of incorrect encodings passing unnoticed through the verification process. This ensures robust data integrity and security, even for datasets smaller than 1 MB.
From our comprehensive analysis, the most optimal solution in terms of both performance and security—and which also aligns effectively with Nomos DA v1's existing subnetwork structure—was identified as the "1024x1024 rows-only with field extension" method. The only disadvantage observed in this method for small datasets is that reconstruction can only be performed in one direction (either row-wise or column-wise). For larger datasets, both row-wise and column-wise reconstruction are fully supported.
Upon examining the data prepared in the tables, for datasets larger than 1 MB, the ratio of total data sent to each subnetwork compared to the original dataset size is approximately 1/32. However, as the dataset sizes decrease, this ratio increases up to about 1/6 for datasets around 128 KB. As anticipated, this method is more efficient for large datasets. Yet, considering the encoding time gains for small datasets, the method still delivers acceptable performance.
In comparison, Nomos DA v1 maintained a consistent ratio of approximately 1/256 between the original dataset size and total data sent per subnetwork, regardless of dataset size.
Extended Matrix: 2048 x 2048
Sampled Rows and Columns: 17-1
Sparsity Check Error: $2^{-7}$
$\text{Total Data per Subnetwork} = 17 \times \text{Row Data} + \text{Column Data} + \text{Merkle Root for Column} + \text{Merkle Root for Row} + 17 \times \text{Merkle Proof for Rows} + \text{Merkle Proof for Column}$
| Data Size | Chunk Size | Row Size | Column Size | Subspace Distance Check | Field Size $2^k$ | Merkle Root Size | Merkle Proof Size for Rows/Colms (KB) | Total Data per Subnetwork | Light Clients 1 row+ 1 column check (KB) | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | 128 KB | 1 bits | 0.25 KB | 8.00 KB | $2^{-20.0}$ | $2^1$ | 0.03 KB | 0.34 KB | 18.50 KB | 9.28 KB | | 256 KB | 2 bits | 0.50 KB | 8.00 KB | $2^{-20.0}$ | $2^2$ | 0.03 KB | 0.34 KB | 22.75 KB | 9.53 KB | | 512 KB | 4 bits | 1.00 KB | 8.00 KB | $2^{-20.0}$ | $2^4$ | 0.03 KB | 0.34 KB | 31.25 KB | 10.03 KB | | 1 MB | 8 bits | 2.00 KB | 8.00 KB | $2^{-20.0}$ | $2^8$ | 0.03 KB | 0.34 KB | 48.25 KB | 11.03 KB | | 8 MB | 64 bits | 16.00 KB | 16.00 KB | $2^{-52.0}$ | $2^{64}$ | 0.03 KB | 0.34 KB | 292.25 KB | 33.03 KB | | 16 MB | 128 bits | 32.00 KB | 32.00 KB | $2^{-116.0}$ | $2^{128}$ | 0.03 KB | 0.34 KB | 564.25 KB | 65.03 KB | | 32 MB | 256 bits | 64.00 KB | 64.00 KB | $2^{-244.0}$ | $2^{256}$ | 0.03 KB | 0.34 KB | 1108.25 KB | 129.03 KB |