In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.

Secure group communication systems have become increasingly important for many emerging network applications. An efficient and robust group key management approach is indispensable to a secure group communication system. Motivated by the theory of hyper-sphere, this paper presents a new group key management approach with a group controller (GC). In our new design, a hyper-sphere is constructed for a group and each member in the group corresponds to a point on the hyper-sphere, which is called the member's private point. The GC computes the central point of the hyper-sphere, intuitively, whose “distance” from each member's private point is identical. The central point is published such that each member can compute a common group key, using a function by taking each member's private point and the central point of the hyper-sphere as the input. This approach is provably secure under the pseudo-random function (PRF) assumption. Compared with other similar schemes, by both theoretical analysis and experiments, our scheme (1) has significantly reduced memory and computation load for each group member; (2) can efficiently deal with massive membership change with only two re-keying messages, i.e., the central point of the hyper-sphere and a random number; and (3) is efficient and very scalable for large-size groups.

In this paper, we propose an improved version of the Simple Matrix encryption scheme of PQCrypto2013. The main goal of our construction is to build a system with even stronger security claims. By using square matrices with random quadratic polynomials, we can claim that breaking the system using algebraic attacks is at least as hard as solving a set of random quadratic equations. Furthermore, due to the use of random polynomials in the matrix A, Rank attacks against our scheme are not feasible.

In this paper we propose a new multivariate public key encryption scheme named ZHFE. The public key is constructed using as core map two high rank HFE polynomials. The inversion of the public key is performed using a low degree polynomial of Hamming weight three. This low degree polynomial is obtained from the two high rank HFE polynomials, by means of a special reduction method that uses Hamming weight three polynomials produced from the two high rank HFE polynomials. We show that ZHFE is relatively efficient and that it is secure against the main attacks that have threatened the security of HFE. We also propose parameters for a practical implementation of ZHFE.

In this paper, we try to clarify some of the questions related to a key concept in multivariate polynomial solving algorithm over a finite field: the degree of regularity. By the degree of regularity, here we refer to a concept first presented by Dubois and Gama, namely the lowest degree at which certain nontrivial degree drop of a polynomial system occurs. Currently, it is somehow commonly accepted that we can use this degree to estimate the complexity of solving a polynomial system, even though we do not have systematic empirical data or a theory to support such a claim. In this paper, we would like to clarify the situation with the help of experiments. We first define a concept of solving degree for a polynomial system. The key question we then need to clarify is the connection of solving degree and the degree of regularity with focus on quadratic systems. To exclude the cases that do not represent the general situation, we need to define when a system is degenerate and when it is irreducible. With extensive computer experiments, we show that the two concepts, the degree of regularity and the solving degree, are related for irreducible systems in the sense that the difference between the two degrees is indeed small, less than 3. But due to the limitation of our experiments, we speculate that this may not be the case for high degree cases.