MathSciDoc: An Archive for Mathematician ∫

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 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 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}

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.

We use the learning with errors (LWE) problem to build a new simple and provably secure key exchange scheme. The basic idea of the construction can be viewed as certain extension of Diffie-Hellman problem with errors. The mathematical structure behind comes from the commutativity of computing a bilinear form in two different ways due to the associativity of the matrix multiplications: (x^t \times A)\times y = x^t \times (A\times y), where x,y are column vectors and A is a square matrix. We show that our new schemes are more efficient in terms of communication and computation complexity compared with key exchange schemes or key transport schemes via encryption schemes based on the LWE problem. Furthermore, we extend our scheme to the ring learning with errors (RLWE) problem, resulting in small key size and better efficiency.

We present a generalized learning with errors (LWE) problem, which is essentially a simple and direct extension of the original LWE problem to the case of matrices. Then we use this new version of LWE problem, which we call matrix LWE problem to build new cryptographic schemes, which include a new scalable key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.

We present a new method for building pairs of HFE polynomials of high degree, such that the map constructed with such a pair is easy to invert. The inversion is accomplished using a low degree polynomial of Hamming weight three, which is derived from a special reduction via Hamming weight three polynomials produced by these two HFE polynomials. This allows us to build new candidates for multivariate trapdoor functions in which we use the pair of HFE polynomials to fabricate the core map. We performed the security analysis for the case where the base field is GF(2) and showed that these new trapdoor functions have high degrees of regularity, and therefore they are secure against the direct algebraic attack. We also give theoretical arguments to show that these new trapdoor functions over GF(2) are secure against the MinRank attack as well.

In this paper, we propose a new algorithm for solving the general approximate common divisors (GACD) problems, which is based on lattice reduction algorithms on certain special lattices and linear equation solving algorithms over integers. Through both theoretical arguments and experimental data, we show that our new algorithm works in polynomial time but under roughly the following condition: \begin{itemize} \item There is a positive integer t such that \frac{\gamma+\eta}{t} + \frac{t}{H} + \rho < \eta; \item We have more than t GACD samples. \end{itemize} or equivalently \begin{itemize} \item H(\eta-\rho)^2 - 4(\gamma+\eta) > 0; \item We have more than t = \lceil\frac{H(\eta-\rho) - \sqrt{H^2(\eta-\rho)^2 - 4H(\gamma+\eta)}{2}\rceil GACD samples. \end{itemize} Here \gamma, \eta and \rho are parameters describing a GACD problem, H = 1/\log_2 F and F is the Hermite Factor. In our experiments, H is shown to be roughly 40 when using the LLL reduction algorithm and it should be around 80 when using Deep or BKZ algorithms. % We use our algorithm to solve concrete problems that no other algorithm could solve before. We show how our algorithm can be applied to attack the fully homomorphic encryption schemes which are based on the general approximate common divisors problem and its limitations.

In this paper we analyze the Kahrobaei-Lam-Shpilrain (KLS) key exchange protocols that use extensions by endomorpisms of matrices over a Galois field proposed in \cite{Kahrobaei-Lam-Shpilrain:2014}. We show that both protocols are vulnerable to a simple linear algebra attack.

Authenticated Key Exchange (AKE) is a cryptographic scheme with the aim to establish a high-entropy and secret session key over a insecure communications network. \emph{Password}-Authenticated Key Exchange (PAKE) assumes that the parties in play share a simple password, which is cheap and human-memorable and is used to achieve the authentication. PAKEs are practically relevant as these features are extremely appealing in an age where most people access sensitive personal data remotely from more-and-more pervasive hand-held devices. Theoretically, PAKEs allow the secure computation and authentication of a high-entropy piece of data using a low-entropy string as a starting point. In this paper, we apply the recently proposed technique introduced in~\cite{DXX2012} to construct two lattice-based PAKE protocols enjoying a very simple and elegant design that is an parallel extension of the class of Random Oracle Model (ROM)-based protocols \msf{PAK} and \msf{PPK}~\cite{BMP2000,M2002}, but in the lattice-based setting. The new protocol resembling \msf{PAK} is three-pass, and provides \emph{mutual explicit authentication}, while the protocol following the structure of \msf{PPK} is two-pass, and provides \emph{implicit authentication}. Our protocols rely on the Ring-Learning-with-Errors (RLWE) assumption, and exploit the additive structure of the underlying ring. They have a comparable level of efficiency to \msf{PAK} and \msf{PPK}, which makes them highly attractive. We present a preliminary implementation of our protocols to demonstrate that they are both efficient and practical. We believe they are suitable quantum safe replacements for \msf{PAK} and \msf{PPK}.