NAIBI: Binding Reconciliation KEMs and Ephemeral Key Agreement over Non-Split Commutative Algebras

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We propose NAIBI-Full, a lattice-based key encapsulation mechanism (KEM) together with its forward-secure ephemeral key-agreement protocols, built on the regular representation 𝜌 of the non-split commutative algebra \cA𝛼 =\Rq⁢[𝑦]/(𝑦𝑘 −𝛼) over \Rq =\Z𝑞⁢[𝑥]/(𝑥𝑛 +1), 𝑘 ∈{2,3}, 𝛼 a non-𝑘 -th power. Each party publishes the full matrix \bft =𝐴⁢𝜌⁡(\bfs) +\bfe ∈\Rq𝑘×𝑘 ; because 𝜌⁡(\cA𝛼) is commutative, the cross-product collapses to small noise and a Peikerthint closes the gap to ex