Odd-Char Multivariate Hidden Field Equations

We present a multivariate version of Hidden Field Equations (HFE) over a nite eld of odd characteristic, with an extra embedding modi er. Combining these known ideas makes our new MPKC (multivariate public key cryptosystem) more e cient and scalable than any other extant multivariate encryption scheme. Switching to odd characteristics in HFE-like schemes a ects how an attacker can make use of eld equations. Extensive empirical tests (using magma-2.14, the best commercially available F4 implementation) suggests that our new construction is indeed secure against algebraic attacks using Gröbner Basis algorithms. The embedding serves both to narrow down choices of pre-images and to guard against a possible Kipnis-Shamir type (rank) attack. We may hence reasonably argue that for practical sizes, prior attacks take exponential time. We demonstrate that our construction is in fact e cient by implementing practical-sized examples of our odd-char HFE with 3 variables ( THFE ) over F31. To be precise, our preliminary THFE implementation is 15× 20× the speed of RSA-1024.