In this paper we introduce a new parallel variant of the LLL lattice basis reduction algorithm. Lattice theory and in particular lattice basis reduction continues to play an integral role in cryptography. Not only does it provide effective cryptanalysis tools but it is also believed to bring about new cryptographic primitives that exhibit strong security even in the presence of quantum computer...

By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we shall see, integer relation algorithms have a variety of interesting applications, including the recognition of a numeric constant in terms of the mathematical formula that it satisfies. The prob...

We consider the complexity of a Lenstra Lenstra Lovász lattice reduction algorithm ([LLL]) in which the vectors are allowed to be linearly dependent and in which one also asks for the matrix of the transformation from the given generators to the reduced basis. The main problem will be to show that the entries of the transformation matrix remain bounded through the algorithm, with a reasonable b...

25 years ago, Lenstra, Lenstra and Lovász presented their celebrated LLL lattice reduction algorithm. Among the various applications of the LLL algorithm is a method due to Coppersmith for finding small roots of polynomial equations. We give a survey of the applications of this root finding method to the problem of inverting the RSA function and the factorization problem. As we will see, most o...

25 years ago, Lenstra, Lenstra and Lovasz presented their celebrated LLL lattice reduction algorithm. Among the various applications of the LLL algorithm is a method due to Coppersmith for finding small roots of polynomial equations. We give a survey of the applications of this root finding method to the problem of inverting the RSA function and the factorization problem. As we will see, most o...

The Lovász Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good “local” properties. In many cases, one wants more information about these structures, other than that they exist. In such case, using the “LLL-distribution”, one can show that the resulting combinatorial structures have good...

Lattice reduction has a wide range of applications. In this paper, we first present a polynomial time Jacobi method for lattice basis reduction by modifying the condition for the Lagrange reduction and integrating the size reduction into the algorithm. We show that the complexity of the modified Jacobi algorithm is O(n5 logB), where n is the dimension of the lattice and B is the maximum length ...

Reduction of lattice bases of rank 2 in R was given by Lagrange and Gauss. The algorithm is closely related to Euclid’s algorithm and we briefly present it in Section 17.1. The main goal of this section is to present the lattice basis reduction algorithm of Lenstra, Lenstra and Lovász, known as the LLL or L algorithm. This is a very important algorithm for practical applications. Some basic ref...

While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algor...