MathSciDoc: An Archive for Mathematician ∫

This paper summarizes most of the main developments in the cryptanalysis of multivariate cryptosystems and discuss some problems that remain open. A strong emphasis is put on the symbolic computation tools that have been used to achieve these advances.

This paper introduces a new efficient algorithm, called MXL_3, for computing Gröbner bases of zero-dimensional ideals. The MXL_3 is based on XL algorithm, mutant strategy, and a new sufficient condition for a set of polynomials to be a Gröbner basis. We present experimental results comparing the behavior of MXL_3 to F_4 on HFE and random generated instances of the MQ problem. In both cases the first implementation of the MXL_3 algorithm succeeds faster and uses less memory than Magma’s implementation of F_4.

The SMS4 block cipher is part of the Chinese WAPI wireless standard. This paper describes the specification and offers a specification for a toy version called simplified SMS4 (S-SMS4). We explore algebraic attacks on SMS4 and S-SMS4 using Gröbner basis attacks on equation systems over GF(2) and GF(2^8), as well as attacks using a SAT solver derived from the GF(2) model. A comparison of SAT and Gröbner basis attacks is provided.

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F(X) = X^2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.

Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers. It also has been known for efficiency compared to “traditional” alternatives. However, this advantage seems to erode with the increase of arithmetic resources in modern CPUs and improved algorithms, especially with respect to Elliptic Curve Cryptography (ECC). In this paper, we show that hardware advances do not just favor ECC. Modern commodity CPUs also have many small integer arithmetic/logic resources, embodied by SSE2 or other vector instruction sets, that are useful for MPKCs. In particular, Intel’s SSSE3 instructions can speed up both public and private maps over prior software implementations of Rainbow-type systems up to 4×. Furthermore, MPKCs over fields of relatively small odd prime characteristics can exploit SSE2 instructions, supported by most modern 64-bit Intel and AMD CPUs. For example, Rainbow over F_{31} can be up to 2× faster than prior implementations of similarly-sized systems over F_{16}. Here a key advance is in using Wiedemann (as opposed to Gauss) solvers to invert the small linear systems in the central maps. We explain the techniques and design choices in implementing our chosen MPKC instances over fields such as F_{31}, F_{16} and F_{256}. We believe that our results can easily carry over to modern FPGAs, which often contain a large number of small multipliers, usable by odd-field MPKCs.

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.

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.

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.

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.

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.