A new fundamental and secure group key management approach with a group controller GC using the theory of polynomial functions over a vector space over finite field is developed, where each member in the group corresponds to a vector in the vector space and the GC computes a central vector, whose inner product with every member's ID vector are identical. The central vector is published and each member can compute a common group key via inner product. The security relies on the fact that any illegitimate user cannot calculate this value without the legitimate vector, therefore cannot derive the group key. This approach is secure and its backward and forward secrecy can be guaranteed. The performance of our approach is analyzed to demonstrate its advantages in comparison with others, which include: 1) it requires both small memory and little computations for each group member; 2) it can handle massive membership change efficiently with only two re-keying messages, i.e., the central vector and a random number; 3) it is very efficient and very scalable for large size groups. Our experiments confirm these advantages and the implementation of our prototype presents very satisfactory performance for large size groups.

In this paper, we prove that the degree of regularity of the family of Square systems, an HFE type of systems, over a prime finite field of odd characteristics q is exactly q, and therefore prove that \begin{itemize} \item inverting Square systems algebraically is exponential, when q=O(n), where n is the number of variables of the system. \end{itemize}

In this paper, we prove a closed formula for the degree of regularity of the family of HFE- (HFE Minus) multivariate public key cryptosystems over a finite field of size q. The degree of regularity of the polynomial system derived from an HFE- system is less than or equal to \begin{eqnarray*} \frac{(q-1)(\lfloor \log_q(D-1)\rfloor +a)}2 +2 & & \text{if is even and is odd,} \\ \frac{(q-1)(\lfloor \log_q(D-1)\rfloor+a+1)}2 +2 & & \text{otherwise.} \end{eqnarray*} Here q is the base field size, D the degree of the HFE polynomial, r=\lfloor \log_q(D-1)\rfloor +1 and a is the number of removed equations (Minus number). This allows us to present an estimate of the complexity of breaking the HFE Challenge 2: \begin{itemize} \item the complexity to break the HFE Challenge 2 directly using algebraic solvers is about 2^{96}. \end{itemize}

In this paper, we present a new algorithm, such that, for the learning with errors (LWE) problems, 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 the learning with bounded errors (LWBE) problems, we can solve it with complexity O(n^D).

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.

Micciancio defined a generalization of cyclic lattices, called ideal lattices. These lattices can be used in cryptosystems to decrease the number of parameters necessary to describe a lattice by a square root, making them more efficient. He proves that the computational intractability of classic lattice problems for these lattices gives rise to provably secure one-way and collision-resistant hash functions. This provable security relies on the assumption that reducing bases of ideal lattices is similar to reducing bases of random lattices. We give an indication that lattice problems in ideal lattices do not represent the general case by providing a distinguisher, which decides in time O(n^4) whether a given basis of rank spans an ideal lattice or not. Using this algorithm we perform a statistical analysis for several dimensions and show that randomly generated lattices are practically never ideal.

Sflash had recently been broken by Dubois, Stern, Shamir, etc., using a differential attack on the public key. The C^{*-} signature schemes are hence no longer practical. In this paper, we will study the new attack from the point view of symmetry, then (1) present a simple concept (projection) to modify several multivariate schemes to resist the new attacks; (2) demonstrate with practical examples that this simple method could work well; and (3) show that the same discussion of attack-and-defence applies to other big-field multivariates. The speed of encryption schemes is not affected, and we can still have a big-field multivariate signatures resisting the new differential attacks with speeds comparable to Sflash.

A Tamed Transformation Method (TTM) cryptosystem was proposed by T.T.Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 2^{33} computations on the finite field with 2^8 elements.

We show all the existing TTM implementation schemes have a defect that there exist linearization equations \sum_{i=1,j=1}^{n,m} a_{ij} x_i y_j(x_1,...,x_n) + \sum_{i=1}^n b_i x_i + \sum_{j=1}^m c_j y_j(x_1,...,x_n) + d = 0, which are satisfied by the components y_i(x_1,...,x_n) of the ciphers of the TTM schemes. We further demonstrate that, for the case of the most recent two implementation schemes in two versions of the paper \cite{CM}, where the inventor of TTM used them to refute a claim in \cite{CG}, if we do a linear substitution with the linear equations derived from the linearization equations for a given ciphertext, we can find the plaintext easily by an iteration of the procedure of first search for linear equations by linear combinations and then linear substitution. The computation complexity of the attack on these two schemes is less than 2^{35} over a finite field of size 2^8.

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.

In a (t,n) threshold scheme any t or more shares can reconstruct the secret s, but less than t shares reveal no information about s. However, an unauthenticated adversary can pretend to be the shareholder at the reconstruction stage. If there were more than t honest shareholders, the unauthenticated adversary without valid share can obtain the secret. To deal with this type of attacks, a model of (t,m,n) group oriented secret sharing (GOSS) scheme was proposed by Miao et al. in 2015. Here the group oriented property means that if m>t parties try to reconstruct the secret, they should all have the authentic shares in advance. It was claimed by Miao et al. that the group oriented property in their GOSS schemes holds in the information-theoretic sense. In this paper, we revisit two instantiations of (t,m,n) group oriented secret sharing schemes and show that these constructions cannot provide the so-called “group oriented property”. Specifically, we develop concrete attacks which allow an unauthenticated adversary with no valid share to participate in the reconstruction phase and obtain the secret provided that there are at least t honest shares presented at the reconstruction phase.

Multivariate public key cryptography is a candidate for post-quantum cryptography, and it allows generating particularly short signatures and fast verification. The Rainbow signature scheme proposed by Ding and Schmidt is such a multivariate cryptosystem, and it is considered secure against all known attacks. The Rainbow-Band-Separation attack recovers a secret key of Rainbow by solving certain systems of quadratic equations, and its complexity is estimated by the well-known theoretical value called the degree of regularity. However, the degree of regularity is generally larger than the solving degree in experiments, and an accurate estimation cannot be obtained. In this article, we propose a new theoretical value for the complexity of the Rainbow-Band-Separation attack using a Gröbner basis algorithm, which provides a more precise estimation compared to that using the degree of regularity. This theoretical value is deduced by the two-variable power series \frac{\prod_{i=1}^m (1 - t_1^{d_{i1}} t_2^{d_{i2}})}{(1-t_1)^{n_1} (1-t_2)^{n_2}}. Since the two-variable power series coincides with the one-variable power series at t_1=t_2 deriving the degree of regularity, the theoretical value is less than or equal to the degree of regularity under a certain condition. Moreover, we show a relation between the Rainbow-Band-Separation attack using the hybrid approach and the HighRank attack. By considering this relation and our theoretical value, we obtain a new complexity estimation for the Rainbow-Band-Separation attack. Furthermore, applying our theoretical value to the complexity formula used in the NIST PQC 2nd round, we show that a slight modification of the proposed Rainbow parameter sets is required. Consequently, we provide a new theoretical value for generally estimating the solving degree of a bi-graded polynomial system, which can influence the parameter selection of Rainbow in the NIST PQC standardization project.

We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al [10], which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.

Research on key mismatch attacks against lattice-based KEMs is an important part of the cryptographic assessment of the ongoing NIST standardization of post-quantum cryptography. There have been a number of these attacks to date. However, a unified method to evaluate these KEMs’ resilience under key mismatch attacks is still missing. Since the key index of efficiency is the number of queries needed to successfully mount such an attack, in this paper, we propose and develop a systematic approach to find lower bounds on the minimum average number of queries needed for such attacks. Our basic idea is to transform the problem of finding the lower bound of queries into finding an optimal binary recovery tree (BRT), where the computations of the lower bounds become essentially the computations of a certain Shannon entropy. The optimal BRT approach also enables us to understand why, for some lattice-based NIST candidate KEMs, there is a big gap between the theoretical bounds and bounds observed in practical attacks, in terms of the number of queries needed. This further leads us to propose a generic improvement method for these existing attacks, which are confirmed by our experiments. Moreover, our proposed method could be directly used to improve the side-channel attacks against CCA-secure NIST candidate KEMs.

The HFE cryptosystem is one of the most popular multi- variate schemes. Especially in the area of digital signatures, the HFEv- variant offers short signatures and high performance. Recently, an instance of the HFEv- signature scheme called GeMSS was selected as one of the alternative candidates for signature schemes in the third round of the NIST Post-Quantum Crypto (PQC) Standardization Project. In this paper, we propose a new key recovery attack on the HFEv- signature scheme. Our attack shows that both the Minus and the Vinegar modification do not enhance the security of the basic HFE scheme significantly. This shows that it is very difficult to build a secure and efficient signature scheme on the basis of HFE. In particular, we use our attack to show that the proposed parameters of the GeMSS scheme are not as secure as claimed.

In 2017, Ward Beullens et al. submitted Lifted Unbalanced Oil and Vinegar [3], which is a modification to the Unbalanced Oil and Vinegar Scheme by Patarin. Previously, Ding et al. proposed the Subfield Differential Attack [22] which prompted a change of parameters by the authors of LUOV for the second round of the NIST post quantum standardization competition [4]. In this paper we propose a modification to the Subfield Differential Attack called the Nested Subset Differential Attack which fully breaks half of the parameter sets put forward. We also show by experimentation that this attack is practically possible to do in under 210 min for the level I security parameters and not just a theoretical attack. The Nested Subset Differential attack is a large improvement of the Subfield differential attack which can be used in real world circumstances. Moreover, we will only use what is called the “lifted” structure of LUOV, and our attack can be thought as a development of solving “lifted” quadratic systems.

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In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

In 2017, Ward Beullens et al. submitted Lifted Unbalanced Oil and Vinegar (LUOV) [4], a signature scheme based on the famous multivariate public key cryptosystem (MPKC) called Unbalanced Oil and Vinegar (UOV), to NIST for the competition for post-quantum public key scheme standardization. The defining feature of LUOV is that, though the public key P works in the extension field of degree r of F_2, the coefficients of P come from F_2. This is done to significantly reduce the size of P. The LUOV scheme is now in the second round of the NIST PQC standardization process. In this paper we introduce a new attack on LUOV. It exploits the “lifted” structure of LUOV to reduce direct attacks on it to those over a subfield. We show that this reduces the complexity below the targeted security for the NIST post-quantum standardization competition.

In 2017 Kyung-Ah Shim et al. proposed a multivariate signature scheme called Himq-3 which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems. The Himq-3 signature scheme can be classified into the oil vinegar signature scheme family. Similar to the rainbow signature scheme, the Himq-3 signature scheme uses a multilayer structure to shorten the signature size. Moreover the signing process is very fast due to a special system called L-inveritble cycle system that is used to invert the central map. In this paper, we provide a complete cryptanalysis to the Himq-3 signature scheme. We describe a new attack method called the singularity attack. This attack is based on the observation that the variables in the L-invertible cycle system are not allowed to be zero in a valid signature. For the completeness, we show step by step how variables and layers can be separated so that signature forgery can be performed. We claim that the complexity of our attack is much lower than the proposed security level.

In this paper, we proposed an idea to construct a general multivariate public key cryptographic (MPKC) scheme based on a user’s identity. In our construction, each user is distributed a unique identity by the key distribution center (KDC) and we use this key to generate user’s private keys. Thereafter, we use these private keys to produce the corresponding public key. This method can make key generating process easier so that the public key will reduce from dozens of Kilobyte to several bits. We then use our general scheme to construct practical identity-based signature schemes named ID-UOV and ID-Rainbow based on two well-known and promising MPKC signature schemes, respectively. Finally, we present the security analysis and give experiments for all of our proposed schemes and the baseline schemes. Comparison shows that our schemes are both efficient and practical.