Twin-Prime Double Frobenius Vanishing for Non-CM Elliptic Curves and Frobenius Sync Cryptography (FSC) Proposal

This work investigate the simultaneous vanishing of Frobenius traces 𝑎p(𝐸) = 0, 𝑎𝑝+2(𝐸) = 0, forelliptic curves 𝐸/ℚ and twin primes 𝑝, 𝑝 + 2. Extensive computation suggests that such coincidences are veryrare for non-CM curves and impossible for CM curves due to modular-congruence obstructions. It wasformalizated this phenomenon by introducing the notion of a Frobenius-double-annihilating twin prime (FDpair)for a curve 𝐸. The experiments up to 𝑝 < 5000 identify exactly one such pair for a simple non-CM family,namely (𝑝, 𝑝 + 2) = (17,19) for curves 𝐸: 𝑦2 = 𝑥3 + A(p)𝑥 + B(p). The work then propose several conjecturesabout the structural uniqueness, finiteness, and Hecke-theoretic characterization of FD-pairs, and suggest a newproposal to use this possible invariant in a criptography application.