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The authors present a multivariate version of Hidden Field Equations (HFE) over a finite field of odd characteristic, with an extra "Embedding" modifier. Combining these known ideas makes the new MPKC (Multivariate Public Key Cryptosystem) more efficient and scalable than any other extant multivariate encryption scheme. Switching to odd characteristics in HFE-like schemes affects how an attacker can make use of field equations. Extensive empirical tests (using magma-2.14, the best commercially available F4 implementation) suggests that the new construction is indeed secure against algebraic attacks using Gr?bner Basis algorithms. The "Embedding" serves both to narrow down choices of pre-images and to guard against a possible Kipnis-Shamir type (rank) attack. The authors thus reasonably argue that for practical sizes, prior attacks take exponential time.
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