In this paper, we present a new algorithm, such that, for the small integer solution (SIS) problem, if the solution is bounded ( by an integer \beta in l_\infty norm, which we call a bounded SIS (BSIS) problem, {\it and if the difference between the row dimension n and the column dimension m of the corresponding matrix is relatively small with respect the row dimension m}, we can solve it easily with a complexity of polynomial in m.

MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008 and improved in PQC 2008. This article gives an overview over the MutantXL algorithm. It also presents experimental results comparing the behavior of the MutantXL algorithm to the $F_4$ algorithm on HFE and randomly generated multivariate systems. In both cases MutantXL is faster and uses less memory than the Magma's implementation of $F_4$.

In this paper, we present an efficient attack to the multivariate Quadratic Quasigroups (MQQ) cryptosystem. Our cryptanalysis breaks MQQ cryptosystems by solving systems of multivariate quadratic polynomial equations using a modified version of the MutantXL algorithm. We present experimental results comparing the behavior of our implementation of MutantXL to Magma's implementation of F_4 on MQQ systems (\ge 135 bit). Based on our results we show that the MutantXL implementation solves with much less memory than Magma's implementation of F_4 algorithm.

We show that Multivariate Public Key Cryptosystems (MPKCs) over fields of small odd prime characteristic, say 31, can be highly efficient. Indeed, at the same design security of 2^{80} under the best known attacks, odd-char MPKC is generally faster than prior MPKCs over \GF{2^k}, which are in turn faster than "traditional'' alternatives. This seemingly counter-intuitive feat is accomplished by exploiting the comparative over-abundance of small integer arithmetic resources in commodity hardware, here embodied by SSE2 or more advanced special multimedia instructions on modern x86-compatible CPUs. We explain our implementation techniques and design choices in implementing our chosen MPKC instances modulo small a odd prime. The same techniques are also applicable in modern FPGAs which often contains a large number of multipliers.

We 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 our 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 \mathbold{F_4} 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-based) attack. We may hence reasonably argue that for practical sizes, prior attacks take exponential time. We demonstrate that our construction is in fact efficient by implementing practical-sized examples of our "odd-char HFE'' with 3 variables (`"THFE'') over GF(31). To be precise, our preliminary THFE implementation is 15x--20x the speed of RSA-1024.