In this paper, we study how the algebraic attack on the HFE multivariate public key cryptosystem works if we build an HFE cryptosystem on a finite field whose characteristic is not two. Using some very basic algebraic geometry we argue that when the characteristic is not two the algebraic attack should not be polynomial in the range of the parameters which are used in practical applications. We further support our claims with extensive experiments using the Magma implementation of F_4, which is currently the best publicly available implementation of the Gröbner basis algorithm. We present a new variant of the HFE cryptosystems, where we project the public key of HFE to a space of one dimension lower. This protects the system from the Kipnis-Shamir attack and makes the decryption process avoid multiple candidates for the plaintext. We propose an example for a practical application on GF(11) and suggest a test challenge on GF(7).

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TTM (tame transformation method) is a type of multivariate public key cryptosystem. In 2007, the inventor of TTM proposed two new instances of TTM to resist the existing attacks, in particular, the Nie et al attack. The two instances are claimed to achieve a security of 2109 against Nie et al attack. Through computation, we found that the instance II satisfied First Order Linearization Equations. After finding all linearization equations, we can perform a ciphertext-only attack to break it.

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