Rank analysis for leakage exploiting addition

Analyze the rank condition governing the image of the extension map for the product code associated with the additive relation between three masked secrets, and determine when the condition has full rank so that the proposed leakage attack is guaranteed to exist.

Background

For the additive computation w=u+v, the paper derives a sufficient dimension bound for obtaining punctured dual codewords whose share-coordinate columns lie in one-dimensional subspaces. This bound alone does not guarantee that the associated extension map has a two-dimensional image over the extension field, which is the condition needed to recover both independent input secrets.

The authors therefore use randomized searches and simulations rather than a theoretical characterization of the rank condition. Establishing such an analysis would clarify the exact parameter range in which additive computational structure yields leakage beyond the base-code attack.