We survey some recent developments in the study of the complexity of certain lattice problems. We focus on the recent progress on complexity results of intractability. We will discuss Ajtai’s worstcase/average-case connections for the shortest vector problem, similar results for the closest vector problem and short basis problem, NP-hardness and non-NP-hardness, transference theorems between pr...

Recent results of Ajtai on the hardness of lattice problems have inspired several cryptographic protocols. At Crypto ’97, Goldreich, Goldwasser and Halevi proposed a public-key cryptosystem based on the closest vector problem in a lattice, which is known to be NP-hard. We show that there is a major flaw in the design of the scheme which has two implications: any ciphertext leaks information on ...

in the present paper, we introduce and study a fuzzy vector equilibrium problem and prove some existence results with and without convexity assumptions by using some particular forms of results of textit{kim} and textit{lee} [w.k. kim and k.h. lee, generalized fuzzy games and fuzzy equilibria, fuzzy sets and systems, 122 (2001), 293-301] and textit{tarafdar} [e. tarafdar, fixed point theorems i...

We propose two trapdoors for the Closest-Vector-Problem in lattices (CVP) related to the lattice tensor product. Using these trapdoors we set up a lattice-based cryptosystem which resembles to the McEliece scheme.

This paper shows the problem of nding the closest vector in an n-dimensional lattice to be NP-hard to approximate to within factor n c= log log n for some constant c > 0.

GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...

In this paper, we propose new classes of trapdoor functions to solve the closest vector problem in lattices. Specifically, we construct lattices based on properties of polynomials for which the closest vector problem is hard to solve unless some trapdoor information is revealed. We thoroughly analyze the security of our proposed functions using state-of-theart attacks and results on lattice red...

In the present paper, we introduce and study a fuzzy vector equilibrium problem and prove some existence results with and without convexity assumptions by using some particular forms of results of textit{Kim} and textit{Lee} [W.K. Kim and K.H. Lee, Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets and Systems, 122 (2001), 293-301] and textit{Tarafdar} [E. Tarafdar, Fixed point theorems i...

We prove that for an arbitrarily small constant ε > 0, assuming NP 6⊆DTIME(2log n), the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor better than 2log 1−ε n. This improves upon the previous hardness factor of (log n)δ for some δ > 0 due to [AKKV05].