4 research outputs found
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
A fast algorithm for computing large Fibonacci numbers
Abstract We present a fast algorithm for computing large Fibonacci numbers. It is known that the product of Lucas numbers algorithm uses the fewest bit operations to compute the Fibonacci number F n . We show that the number of bit operations in the conventional product of Lucas numbers algorithm can be reduced by replacing multiplication with the square operation