3 research outputs found
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
The elliptic curve primality proving (ECPP) algorithm is one of the current
fastest practical algorithms for proving the primality of large numbers. Its
running time cannot be proven rigorously, but heuristic arguments show that it
should run in time O ((log N)^5) to prove the primality of N. An asymptotically
fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4).
The aim of this article is to describe this version in more details, leading to
actual implementations able to handle numbers with several thousands of decimal
digits
A Simple and Fast Algorithm for Computing the -th Term of a Linearly Recurrent Sequence
We present a simple and fast algorithm for computing the -th term of a
given linearly recurrent sequence. Our new algorithm uses arithmetic operations, where is the order of the recurrence, and
denotes the number of arithmetic operations for computing the
product of two polynomials of degree . The state-of-the-art algorithm, due
to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant
factor. Our algorithm is simpler, faster and obtained by a totally different
method. We also discuss several algorithmic applications, notably to polynomial
modular exponentiation, powering of matrices and high-order lifting.Comment: 34 page
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