# Gram Moments and Pairwise Mixing of AES > Source: > Published: 2026-10-01 22:45:14+00:00 Almost pairwise independence measures how closely a block cipher reproduces the joint output distribution of a uniform random permutation on two distinct plaintexts. Recent analyses of AES with independent round keys combine smoothing from a fixed input difference with contraction over later rounds. The central difficulty is to control the dependencies created by diffusion. We develop a bound for local difference channels on additive maximum distance separable codes that retains the squared overlaps between channel responses, expressed as the second moment of a Gram matrix. This refines the activity-space analysis and strengthens both the initial estimate and the repeated contraction. We prove that, for every integer $k\ge0$, $4k+4$ rounds of AES with independent uniform round keys, sampled once and retained across evaluations, have pairwise total variation distance below $2^{-14.3-73.1k}$. The proof separates activity-pattern error from residual variation involving one, two, three, or four words. A fixed five-component comparison propagates these errors through each additional four-round block. Consequently, 12, 16, and 20 rounds suffice for statistical accuracy $2^{-128}$, $2^{-192}$, and $2^{-256}$, respectively. These bounds control the complete view of any classical algorithm with a finite private seed making at most two adaptive forward queries.