Engel p-adic Isogeny-based Cryptography over Laurent Series: Foundations, Security, and an ESP32 Implementation

Securing the Internet of Things (IoT) against quantum attacks requires public-key cryptography that (i) remains compact and (ii) runs efficiently on microcontrollers, capabilities many post-quantum (PQ) schemes lack due to large keys and heavy arithmetic. We address both constraints simultaneously with, to our knowledge, the first-ever isogeny framework that encodes super-singular elliptic-curve isogeny data via novel Engel expansions over the p-adic Laurent series. Engel coefficients compress torsion information, thereby addressing the compactness constraint, yielding public keys of ~1.1 - 16.9 kbits preserving the hallmark small sizes of isogeny systems. Engel arithmetic is local and admits fixed-precision p-adic operations, enabling micro-controller efficiency with low-memory, branch-regular kernels suitable for embedded targets.