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}.

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.

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.

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.

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.