Element-wise Multiply

Definition

Given $a, b \in R^N$ over a ring $R$, compute $c \in R^N$ with

Each output element depends on exactly one element of each input. There is no summation and no interaction between positions: this is neither an inner product ($\sum_i a_i b_i$) nor a convolution (position $k$ of a convolution sums over $i + j = k$). Both are separate kernels.

Complexity

$N$ ring multiplications, $2N$ reads, $N$ writes. No algorithm improves on this bound. Performance differences between implementations at a fixed point reflect memory traffic and vectorisation, not arithmetic.

Role in the catalogue

The correct answer is inexpensive to compute independently, so the kernel is also used to validate the measurement pipeline itself — generator, oracle, verifier and runner.