cronokirby

Better Notation for Matrices

Let [l]:={1,…,l}[l] := \{1, \ldots, l\}.

Given an r×cr \times c matrix MM, let MiM^i denote the ith column, with i∈[c]i \in [c], and MjM_j denote the jth row, with j∈[r]j \in [r] and MjiM^i_j denotes a single entry. This makes it a lot easier to swap between transposes when doing calculations. For example, if Δi\Delta_i is a vector of bits, with ⋅\cdot denoting multiplication, and ∗* denoting convolution, you can do:

Mi=Ti+Δi⋅XiMji=Tji+Δi⋅XjiMj=Tj+Δ∗Xj\begin{aligned} &M^i = T^i + \Delta_i \cdot X^i \cr &M^i_j = T^i_j + \Delta_i \cdot X^i_j \cr &M_j = T_j + \Delta * X_j \end{aligned}

This is a lot more convenient.

The only issue is that this conflicts with tensor notation. Thankfully, there's not much overlap between general relativity and Cryptography.