In this paper, we present a new algorithm to solve algebraically the following lattice-related problems: 1) the small integer solution (SIS) problem under the condition: if the solution is bounded by an integer β in l_∞ norm, which we call a bounded SIS (BSIS) problem, and if the difference between the row dimension n and the column dimension m of the corresponding basis matrix is relatively small with respect the row dimension m; 2) the learning with errors (LWE) problems under the condition: if the errors are bounded - the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D < q), which we call a learning with bounded errors (LWBE) problem. We will show that we can solve these problems with polynomial complexity.

We propose a new efficient hardware implementation of Rainbow signature scheme. We enhance the implementation in three directions. First, we develop a new parallel hardware design for the Gauss-Jordan elimination, and solve a 12 ×12 system of linear equations with only 12 clock cycles. Second, a novel multiplier is designed to speed up multiplication of three elements over a finite field. Third, we design a novel partial multiplicative inverter to speed up the multiplicative inversion of finite field elements. Through further other minor optimizations of the parallelization process and by integrating the major optimizations above, we build a new hardware implementation, which takes only 198 clock cycles to generate a Rainbow signature, a new record in generating digital signatures and four times faster than the 804-clock-cycle Balasubramanian-Bogdanov-Carter-Ding-Rupp design with similar parameters.

In this paper, we present an experimental analysis of HFE Challenge 2 (144 bit) type systems. We generate scaled versions of the full challenge fixing and guessing some unknowns. We use the MXL_3 algorithm, an efficient algorithm for computing Gröbner basis, to solve these scaled versions. We review the MXL_3 strategy and introduce our experimental results.

Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.

Recently, the Little Dragon Two and Poly-Dragon multivariate based public-key cryptosystems were proposed as efficient and secure schemes. In particular, the inventors of the two schemes claim that Little Dragon Two and Poly-Dragon resist algebraic cryptanalysis. In this paper, we show that MXL2, an algebraic attack method based on the XL algorithm and Ding’s concept of Mutants, is able to break Little Dragon Two with keys of length up to 229 bits and Poly-Dragon with keys of length up to 299. This contradicts the security claim for the proposed schemes and demonstrates the strength of MXL2 and the Mutant concept. This strength is further supported by experiments that show that in attacks on both schemes the MXL2 algorithm outperforms the Magma’s implementation of F4.

We give exact formulas for the growth of the ideal Aλ for λ a quadratic element of the algebra of Boolean functions over the Galois field GF(2). That is, we calculate dim A_k λ where A_k is the subspace of elements of degree less than or equal to k. These results clarify some of the assertions made in the article of Yang, Chen and Courtois [22,23] concerning the efficiency of the XL algorithm.

In this paper we present a new variant of the Zhuang-Zi algorithm, which solves multivariate polynomial equations over a finite field by converting it into a single variable problem over a large extension field. The improvement is based on the newly developed concept of mutant in solving multivariate equations.

MFE, a multivariate public key encryption scheme proposed by Wang et al in CT-RSA 2006, was conquered by second order linearization equation (SOLE) attack by Ding et al in PKC 2007. To resist this attack, many improved schemes were proposed. Wang et al in [WFW09 and Wang in [Wan07] both modified MFE and raised the public key from quadratic to quartic equations. We call the two quartic schemes Quartic-1 and Quartic-2 respectively for convenience. They are indeed immune to the SOLE attack. However, we find that there exist many quadratization equations (QEs), which are quadratic in plaintext variables and linear in ciphertext variables and can be derived from the public keys of Quartic-1 and Quartic-2. In this paper, we utilize QEs to recover the corresponding plaintext for a given ciphertext. For Quartic-1, we firstly find there are at least 2r SOLEs, which was regarded as impossible by the original authors, furthermore, we can find at least 35r QEs with a complexity O((90r^2(15r+1)+180r^2+15r(15r+1)/2+27r+1)^w), where r is a small number denoting the degree of extension of finite fields and w ≈ 2.732. The computational complexity of deriving these equations is about 2^{37}. But to find the original plaintext, there still needs 240 times Gröbner basis computations, which needs practically 1.328 seconds each time. For Quartic-2, we make a theoretical analysis and find 18r QEs with a computational complexity O((15r+1)6r(12r+1)+180r^2+27r+1)^w. The complexity is 2^{36} for the parameter proposed in [Wan07], and we can break the scheme practically in 3110.734 seconds. Finally, we show that another improved version of MFE in [WZY07] is insecure against the linearization equation attack although its authors claimed it is secure against high order linearization equation attack. Our attack on the two quartic schemes illustrates that non-linearization equations like quadratization equations which are not degree one in plaintext variables can also be used efficiently to analyze multivariate cryptosystems.

This paper discusses two encryption schemes to fix the Square scheme. Square+ uses the Plus modification of appending randomly chosen polynomials. Double-Layer Square uses a construction similar to some signature schemes, splitting the variables into two layers, one of which depends on the other.

In this paper, we present an efficient attack on the multivariate Quadratic Quasigroups (MQQ) public key cryptosystem. Our cryptanalysis breaks the MQQ cryptosystem by solving a system of multivariate quadratic polynomial equations using both the MutantXL algorithm and the F4 algorithm. We present the experimental results that show that MQQ systems is broken up to size n equal to 300. Based on these results we show also that MutantXL solves MQQ systems with much less memory than the F4 algorithm implemented in Magma.