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.

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).

We explore ideas for speeding up HFE-based signature schemes. In particular, we propose an HFEv^− system with odd characteristic and a secret map of degree 2. Changing the characteristic of the system has a profound effect, which we attempt to explain and also demonstrate through experiment. We discuss known attacks which could possibly topple such systems, especially algebraic attacks. After testing the resilience of these schemes against F4, we suggest parameters that yield acceptable security levels.

In 1989, Tsujii, Fujioka, and Hirayama proposed a family of multivariate public key cryptosystems, where the public key is given as a set of multivariate rational functions of degree 4. These cryptosystems are constructed via composition of two quadratic rational maps. In this paper, we present the cryptanalysis of this family of cryptosystems. The key point of our attack is to transform a problem of decomposition of two rational maps into a problem of decomposition of two polynomial maps. We develop a new improved 2R decomposition method and other new techniques, which allows us to find an equivalent decomposition of the rational maps to break the system completely. For the example suggested for practical applications, it is very fast to derive an equivalent private key, and it requires only a few seconds on a standard PC.

MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008. This paper proposes two substantial improvements to this algorithm over GF(2) that result in significantly reduced memory usage. We present experimental results comparing MXL2 to the XL algorithm, the MutantXL algorithm and Magma’s implementation of F_4. For this comparison we have chosen small, randomly generated instances of the MQ problem and quadratic systems derived from HFE instances. In both cases, the largest matrices produced by MXL2 are substantially smaller than the ones produced by MutantXL and XL. Moreover, for a significant number of cases we even see a reduction of the size of the largest matrix when we compare MXL2 against Magma’s F_4 implementation.