Adaptive Precision Floating Point LLL
نویسندگان
چکیده
The LLL algorithm is one of the most studied lattice basis reduction algorithms in the literature. Among all of its variants, the floating point version, also known as L, is the most popular one, due to its efficiency and its practicality. In its classic setting, the floating point precision is a fixed value, determined by the dimension of the input basis at the initiation of the algorithm. We observe that a fixed precision overkills the problem, since one does not require a huge precision to handle the process at the beginning of the reduction. In this paper, we propose an adaptive way to handle the precision, where the precision is adaptive during the procedure. Although this optimization does not change the worst-case complexity, it reduces the average-case complexity by a constant factor. In practice, we observe an average 20% acceleration in our implementation.
منابع مشابه
Adaptive precision LLL and Potential-LLL reductions with Interval arithmetic
Lattice reduction is fundamental in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (called LLL or L) has been improved in many ways through the past decades and remains one of the central tool for reducing lattice basis. In particular, its floating-point variants — where the long-integer arithmetic requi...
متن کاملFloating-Point LLL: Theoretical and Practical Aspects
The text-book LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the Gram-Schmidt orthogonalisation by floating-point approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floating-point approximations seems to be natural for LLL even from the theoretical point of view: it is t...
متن کاملPerturbation Analysis of the QR factor R in the context of LLL lattice basis reduction
In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and László Lovász introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorization of the basis matrix. In order to circumvent the use of rational/exact arithmetic with large bit-sizes, it is tempting to consi...
متن کاملAdaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates
Exact computer arithmetic has a variety of uses including, but not limited to, the robust implementation of geometric algorithms. This report has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive-precision arithmetic that can often speed these ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013